Several Complex Variables II: Function Theory in Classical Domains Complex Potential Theory

Encyclopaedia of Mathematical Sciences

Volume 8

Editor-in-Chief: R. V. Gamkrelidze

G.M. Khenkin A.G. Vitushkin (Eds.)

Several Complex Variables II

Function Theory in Classical Domains Complex Potential Theory

With 19 Figures

Springer-Verlag Berlin Heidelberg GmbH

Consulting Editors of the Series: A.A. Agrachev, AA Gonchar, E.F. Mishchenko, N. M. Ostianu, V. P. Sakharova, A B. Zhishchenko

Title of the Russian edition: Itogi nauki i tekhniki, Sovremennye problemy matematiki,

Fundamental'nye napravleniya, VoI. 8, Kompleksnyj analiz - mnogie peremennye 2 Publisher VINITI, Moscow 1985

Mathematics Subject Classification (1991): 32-02, 32A07, 32A27, 32A35, 32A40, 32F05

ISBN 978-3-642-63391-1

Library of Congress Cataloging-in-Publication Data Kompleksnyi analiz-mnogie peremennye 2. English

Several complex variables II: function theory in c1assieal domains: complex potentialtheory / G. M. Khenkin, A. G. Vitushkin (eds.)

p. cm. - (Encyclopaedia of mathematical sciences; v. 8) Includes bibliographical references and indexes.

ISBN 978-3-642-63391-1 ISBN 978-3-642-57882-3 (eBook) DOI 10.1007/978-3-642-57882-3

1. Functions of several complex variables. 1. Khenkin, G. M. II. Vitushkin, A. G. (Anatolii Georgievich) III. Title. IV. Title: Several complex variables 2. V. Series.

QA33I.K7382513 1994 515'.94-dc20 92-45735

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List of Editors, Authors and Translators

Editor-in-Chief

RV. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow, Institute for Scientific Information (VINITI), ul. Usievicha 20a, 125219 Moscow, Russia

Consulting Editors

G. M. Khenkin, Central Economic and Mathematical Institute of the Russian Academy of Sciences, ul. Krasikova 32,117418 Moscow, Russia

A. G. Vitushkin, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow, Russia

Authors

L. A. Aizenberg, Akademgorodok, Institute of Physics, 660036 Krasnoyarsk 36, Russia

A. B. Aleksandrov, Petrodvorets, S1. Petersburg State University, 198904 S1. Petersburg, Russia

A. Sadullaev, Vuzgorodok, Tashkent State University, 700095 Tashkent, Usbekistan

A. G. Sergeev, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow, Russia

A. K. Tsikh, Akademgorodok, Institute of Physics, 660036 Krasnoyarsk 36, Russia V. S. Vladimirov, Steklov Mathematical Institute, ul. Vavilova 42,

117966 Moscow, Russia A. P. Yuzhakov, Akademgorodok, Institute of Physics, 660036 Krasnoyarsk 36,

Russia

Translators

P. M. Gauthier, Departement de Mathematiques et de Statistique, Universite de Montreal, CP 6128-A, Montreal QC H3C 3J7, Canada

J. R. King, Department of Mathematics, GN-50, Seattle, WA 98195, USA

Contents

I. Multidimensional Residues and Applications L. A. Aizenberg, A. K. Tsikh, A. P. Yuzhakov

1

ll. Plurisubharmonic Functions A. Sadullaev

59

Ill. Function Theory in the Ball A. B. Aleksandrov

107

IV. Complex Analysis in the Future Tube A. G. Sergeev, V. S. Vladimirov

179

Author Index 255

Subject Index 258

I. Multidimensional Residues and Applications

L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov

Translated from the Russian by 1.R. King

Contents

Chapter 1. Methods for Computing Multidimensional Residues (A.P. Yuzhakov) ............................................ 3

Introduction .................................................. 3 § 1. Leray Theory. Froissart Decomposition Theorem ............... 4

1.1. Leray Coboundary ..................................... 4 1.2. Form-Residue, Class-Residue, Leray Residue Formula ....... 5 1.3. Tests for Leray Coboundaries. Froissart Decomposition

Theorem .............................................. 6 1.4. Cohomological Lowering of Pole Order .................... 7 1.5. Generalization of the Leray Theory to the Case of Submanifolds

of Codimension q > 1 ................................... 9 § 2. Application of Alexander-Pontryagin Duality and De Rham

Duality ................................................... 10 2.1. Application of Alexander-Pontryagin Duality ............... 10 2.2. Residues of Rational Functions of Two Variables ............ 11 2.3. Application of De Rham Duality .......................... 13

§ 3. Homological Methods for Studying Integrals that Depend upon Parameters. Application to Combinatorial Analysis .............. 15 3.1. Analytic Continuation of Integrals Depending on Parameters.

Isotopy Theorem ....................................... 16 3.2. Foliation near a Landau Singularity. Picard-Lefschetz Formula 18 3.3. Some Examples of Integrals Depending on Parameters ....... 20 3.4. Application of Residues to Combinatorial Analysis .......... 22

Chapter 2. Multidimensional Logarithmic Residues and Their Applications (L.A. Aizenberg) ................................ 24

§ 1. Multidimensional Logarithmic Residues ....................... 24 § 2. Series Expansion ofImplicit Functions ......................... 31

2 L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov

§ 3. Application of the Multidimensional Logarithmic Residue to Systems of Nonlinear Equations ..................................... 33

§ 4. Computation of the Zero-Multiplicity of a Holomorphic Mapping. 37 § 5. Application of the Multidimensional Logarithmic Residue to the

Theory of Numbers ......................................... 38

Chapter 3. The Grothendieck Residue and its Applications to Algebraic Geometry (A.K. Tsikh) ...................................... 39

Introduction .................................................. 39 § 1. Integral Definition and Fundamental Properties of the Local

Residue ................................................... 40 1.1. Definitions ............................................ 40 1.2. Representation of the Local Residue by an Integral over the

Boundary of a Domain .................................. 41 1.3. Transformation Formula for the Local Residue ............. 41 1.4. Local Duality Theorem .................................. 42

§ 2. Using the Trace to Express the Local Residue ................... 43 2.1. Definition of the Trace and its Fundamental Properties ....... 43 2.2. Algebraic Interpretation ................................. 44

§ 3. The Total Sum of Local Residues ............................. 45 3.1. The Total Sum of Residues on a Compact Manifold. The Euler-

Jacobi Formula ........................................ 45 3.2. Applications to Plane Projective Geometry ................. 47 3.3. The Converse of the Theorem on Total Sum of Residues ...... 47 3.4. Abel's Theorem and its Converse .......................... 48 3.5. Residue Theorem for Vector Bundles ...................... 50 3.6. The Total Sum of Residues Relative to a Polynomial Mapping

in cn .............•••.•.•..••••••••••...••••.•......•. 51 § 4. Application of the Grothendieck Residue to the Algebra of

Polynomials and to the Local Ring (!)a ....•.....•.•..•••.••.••• 52 4.1. Macauley's Theorem .................................... 52 4.2. Noether-Lasker Theorem in ClPn .......................... 52 4.3. Verification of the Local Noether Condition ................ 53 4.4. A Consequence of Global Duality ......................... 54

Bibliography .................................................. 55

I. Multidimensional Residues and Applications 3

Chapter 1 Methods for Computing Multidimensional Residues

A.P. Yuzhakov

Introduction

One of the problems in the theory of multidimensional residues is the problem of studying and computing integrals of the form

(1)

where w is a closed differential form of degree p on a complex analytic manifold X with a singularity on an analytic set SeX, and where Y is a compact pdimensional cycle in X\S. A special case of this problem is computing the integral (1) when w is a holomorphic (meromorphic) form of degree p = n =

dime X; in local coordinates the form can be written as w = I(z) dz =

I(z l' ... , zn) dz 1 /\ ... /\ dzn, where 1 is a holomorphic (meromorphic) function. According to the Stokes formula, the integral (1) depends only on the homology class l [y] E Hp(X\S) and the De Rham cohomology class [w] E HP(X\S). Thus in integral (1) the cycle Y can be replaced by a cycle Yl homologous to it (Yl '" y) in X\S and the form w can be replaced by a cohomologous form W 1(W 1 '" w) which may perhaps be simpler; for example, it could have poles of first order on S (see § 1, Subsection 4). If {Yj} is a basis for the p-dimensional homology of the manifold X\S, then by Stokes formula for any compact cycle Y E Zp(X\S) the integral (1) is equal to

(2)

where the kj are the coefficients of the cycle Y as a combination of the basis elements {yJ, Y '" Lj kjYj. Formula (2) shows that the problem of computing integral (1) can be reduced to

1) studying the homology group Hp(X\S) (finding its dimension and a basis); 2) determining the coefficients of the cycle Y with respect to a basis; 3) computing the integrals over the cycles in the basis.

Solving problems 1) and 2) is a difficult topological problem in the multidimensional case and requires the machinery of algebraic topology. In some

1 In this chapter we will denote by Hp the group of compact singular homology; this group was denoted by H~ in the contribution of Dolbeault (Dolbeault, 1985) in Volume 7 of the Encyclopaedia of Mathematical Sciences.

cases, to solve this it helps to apply the dualities of Alexander-Pontryagin and De Rham (§ 2). Simple and multiple Leray coboundaries (Subsection 1.1) give a construction of standard cycles in X\S. The general structure of the homology group Hp(X\S) is described in "good cases" by the decomposition theorem of Froissard (Subsection 1.3). Integrals on coboundary cycles can be reduced to integrals of lower degree by the simple and multiple Leray residue formulas (Subsection 1.2). The computation of an important class of residues, the Grothendieck residues, and a special case of them, the logarithmic residue, is considered in Section 2 and Section 3 of this article; § 3 is devoted to the application of residues to the study of integrals depending on parameters and to combinatorial analysis.

§ 1. Leray Theory. Froissart Decomposition Theorem

Here we will pause to study in more detail the computational side of the Leray theory of residues expounded in (Dolbeault (1985)). To start with, we consider the case of co dimension 1.

1.1. Leray Coboundary. We give a constructive description of the coboundary homomorphism b which was introduced in (Dolbeault (1985), Sect. 0.3). In the one-dimensional case the simplest cycle (contour) of integration is a circle of sufficiently small radius around an isolated singular point. Leray (1959) constructed the analog of this for complex analytic manifolds, the co boundary homomorphism b. The construction of b-1 was first considered by Poincare (1887).

Let X be a complex analytic manifold of complex dimension n. Let S be a complex-analytic submanifold of X of co dimension 1. We consider a tubular neighborhood V of the submanifold S, which is a locally-trivial fiber bundle with base S and fiber Ya, a E S, homeomorphic to the disk. In order to construct such a fiber bundle we choose a Riemannian metric on X and take as Ya the union of geodesic segments of length p(a), beginning at a and orthogonal to S, where p(a) is sufficiently small. We assume that the function p(a) is smooth; this implies the smoothness of OV To each (p - I)-dimensional element of a chain (a simplex, a rectangle) <Tp- 1 in S we associate a p-dimensional chain in X\S. The chain is b<TP _ 1 = UaE!up_tI ba where ba = oYa; it is homeomorphic to oYa x <Tp - 1

with the natural orientation. Thus a homomorphism of homology groups is defined,

b : Hp- 1 (S) ~ Hp(X\S),

since Ob = -bO. Then the Leray homology exact sequence is defined: ro a i

... ~ Hp+1(X) ~ Hp_1(S) ~ Hp(X\S) ~ Hp(X) ~... (3)

where i is the homomorphism induced by the inclusion X\S c X and OJ in induced by the intersection of chains in X, transversal to S, with the submanifold S.


If a family of S l' ... , Sm of submanifolds of codimension 1 is in general position, the multiple Leray co boundary is defined:

which is anticommutative with respect to the order of S l' ... , Sm (for the cohomological multiple coboundary, see (Dolbeault (1985), Sect. 03).

1.2. Form-Residue, Class-Residue, Leray Residue Formula. As was pointed out in (Dolbeault (1985), Sect. 03), if rfJ is a closed regular differential form in X\S with a pole of first order on S, then in some neighborhood of any point a E S the form rfJ can be represented as

ds rfJ = - 1\ t/J + e,

s (4)

where s = sa(z) is the defining function of the manifold Sin Ua and t/J, e are forms which are regular on Ua. Here the form 1/1 Is is globally defined, is closed, and is uniquely determined by the form rfJ. This restriction 1/1 Is is called the form-residue of the form rfJ and is denoted by res[rfJJ. We remark that if rfJ is holomorphic on X\S, then the form residue res[rfJJ is holomorphic on S.

Example 1. Let X = cn, with S = {z E Cn : s(z) = O} and rfJ = f(z) dZ 1 1\

... 1\ dzn/s(z). Since rfJ = ( _1)j-1 ds 1\ dZ!J1/s· S~j' where dZ!J1 = dz 1 1\ •.• 1\

dZj - 1 1\ dZj +1 1\ ... 1\ dzn, then res[rfJJ = (-1y-1f(z) dZ!J1/s~)s at the points where s~. =1= O.

J

Remark. The map

f(z) dz/s(z) --+ (-1y-1f(z) dZ!J1/s;)s

is called the Poincare residue map and is denoted by P.R. If we denote by ,gx, ,gx(S), ,gs-I, the sheaves of germs, respectively, of holomorphic n-forms on X, meromorphic n-forms having only simple poles on S, and holomorphic (n - 1)forms on S, then there is an exact sequence of sheaves

o --+ ,gx --+ ,gx (S) --+ ,gs -1 --+ 0

which defines an exact sequence on cohomology

HO(X, ,gx(S)) P~. HO(S, ,gS-1) ~ H1(X, ,gx).

Therefore, the Poincare residue map is surjective on global sections if H1(X, ,gx) = O. In particular, this is true for projective space X = cpn, n> 1. Thus for n > 1 every holomorphic form of degree n - 1 on the submanifold S is the Poincare residue of a merom orphic n-form on cpn.

By the theorem of (Dolbeault (1985), Sect. 0.3), for every closed regular differentiable form rfJ on X\S, there is a form ~ cohomologous to it which has a pole of first order on S. In this case the cohomology class of the form res[~J depends only on the cohomology class of the form rfJ.


The cohomology class of the form res[~] is called the class residue ofthe form r/J and is denoted by Res[r/J]' Since the operator res is linear, Res: HP(X\S)--+ HP-1 (S) is a homomorphism.

We observe that the form residue and the class residue also exist in the case when S is an analytic subset. However in this case the form residue has singularities on the set S* of singular points of S. If the singular set S* of the set S is resolved to a divisor with normal crossings, then the form residue only has simple poles on the resolution of S* (Gordan, 1974).

The abstract residue formula (0.3.2) of (Dolbeault (1985)) is written thus:

Theorem 1.1 (Leray residue formula). For an arbitrary closed form r/J of degree p on X \ S and a cycle (J E Zp -1 (S) there is a formula

f r/J = 2ni f Res[r/J]' lJa a

(5)

If the form r/J E ZP(X\(Sl U .. · U Sm)) has a pole of first order on Sl' ... , Sm' then by applying formula (4) one can define iterated form residues resm[r/J] E

zp-m(Sl n'" n Sm) and a homomorphism

as the composition of homomorphisms

HP(X\(Sl U··· U Sm))~ HP-1(Sl \(S2 U··· U Sm)) --+ .••

--+ Hp-m+1(Sl n'" n Sm-1 \Sm)~ Hp-m(Sl n'" n Sm).

Iterating Theorem 1.1 we obtain the compound Leray residue.

Theorem 1.2 (Leray (1959)). For an arbitrary form r/JEZP(X\(Sl U .. · U Sm)) and any cycle (J E Zp-m(Sl n'" n Sm), a compound Leray formula holds:

f r/J = (2nir f Resm[r/J]' ~mO' a

(6)

The Leray formulas (5) and (6) allow one to lower the degree of the multiple integral (1) when the cycle of integration belongs to a co boundary class: y E

bHp_1(S), YEbmHp_m(Sln"'nSm)' Since, for a form r/J having a first-order pole, the form-residue res[r/J] (resm[r/J]) is found constructively, the problem arises of how to lower the order of poles of semimeromorphic forms r/J E ZP(X\S) (r/J E ZP(X\(Sl n'" n Sm))); in other words, how can one find a form r/J1 cohomologous to r/J which has a first-order pole.

1.3. Tests for Leray Coboundaries. Froissart Decomposition Theorem. From the Leray exact sequence (3) it follows that a cycle Y E Zp(X\S) is a coboundary (y '" b(J for some (J E Zp-1 (S)) if and only if y '" ° in X. If Hp(X) = 0, then Hp(X\S) = bHp_1(S), i.e., every cycle in X\S is a co boundary. If Hp+1(X) = 0, then b is a monomorphism.

Proposition 1.3 (Griffiths (1969)). If S is an algebraic manifold in complex projective space ClPn, then c5: Hn- 1 (S) --+ Hn(ClPn\S) is always surjective and is injective for even n.

Proposition 1.4. For a cycle y E Zp(X\(SI U··· U Sm)) to be a compound Leray co boundary ([y] E c5 mHp_m(SI n'" n Sm)), it is necessary (and sufficient in the case when X is a Stein manifold) that

y '" 0 in X\(SI U ... uSj - 1 U Sj+l U ... U Sm),j = 1, ... , m.

Under some assumptions about X and S l' ... , Sm' the structure of the homology group Hp(X\(SI U··· U Sm)) is described by the following theorem.

Theorem 1.5 (Froissart decomposition (Fotiadi et al. (1965)). Let So, SI' ... , Sm and L 1, ... , Lk be two families of submanifolds of codimension 1 in the complex projective space ClPn such that the families are in general position. So = CIP!-1 = ClPn\ e is the hyperplane at infinity. Let Y = Lin'" n Lk. X = cn n Y = Y\So. Then the cohomology group

Hp(X\O Sj) ~ EB c5lhIHp_lhl (n (Sj n X)) )-1 hc:(I, .... m} )Eh

m

= Hp(X) $ L c5Hp_1 (Sj n X) $ L c5Hp_2(X n Sj n Sq) j=1 l,;j,;q';m

$'"

where Ihl is the number of elements in the set h.

Example 2. Let X = e, and let Sj = {z E cn: Lj(z) = Lk=1 ajkzk + bj = O}, j = 1, ... , m be analytic hyperplanes in general position (if Sj, n'" n Sjk -# 0, then Ljt , Ljk are linearly independent). Since Hn(cn) = 0, Hn-r(Sj, n'" n Sj) = 0 for r < n, and Ho(Sj, n'" n SjJ ~ 7L for Sj, n'" n Sjn -# 0, then there is a basis of the group Hn(cn\(SI U··· U Sm)) consisting of cycles oftheform c5n(Sj, n'" n SjJ = {z: ILv(z)1 = e, v = jl' '" ,jn}' The residue of the form ,p = h dz/L~' ... L~m, where hE A(cm), relative to a basis cycle c5 n (Sj, n ... n SjJ is the Grothendieck residue (see Chapter 3) at the point Sjt n ... n Sjn' After a linear change of variables, it is computed as the derivative of a multiple Cauchy integral.

1.4. Cohomological Lowering of Pole Order. In Subsection 1.2 only the existence of the class residue of a form ,p E ZP(X\S) was discussed, bu\no algorithm was demonstrated for computing it, that is, for finding a form ,pI '" ,p having a pole of first order. The same is true for the compound class residue of a form ,p E ZP(X\(S 1 U ... U Sm)). In some cases the problem of the cohomological reduction of a semimeromorphic form (see (Dolbeault (1985)) Subsection 3.5) to a form having a pole of order 1 can be solved constructively. For example,

Proposition 1.6 (Pham (1967)). Let S = {z: s(z) = O}, where s is a holomorphic function in a neighborhood V(S) of the manifold S, with grad sis -# 0, then any


form ¢> E ZP(X\S) having a pole of order k on S can be represented in the form

ds () ( '" ) ()1 ()1 ¢> = Sk A '" + Sk-l = d (k _ 1)Sk 1 + Sk-l - Sk-l' (7)

where () and", are regular forms in V(S) and ()1 = d",/(k - 1) + (). Theorem 1.7 (Leray (1959)). Let the submanifolds Sj in a neighborhood Vof

the set SIn'" n Sm be defined by the equations Sj(z) = 0, where the Sj are functions holomorphic in v,j = 1, ... , m. If the form ¢> E ZP(X\(SI U'" U Sm)) is represented in V as

then

1 ar1 + ... +rmw I Resm [¢>] '3 ,. ,r r '

r1 ····rm· as11 ... asmm S1 n ''' nSm

where the ar1 + ... +rm/as~1 ... as~m are found recursively from the equation

def", '" dw = dS 1 A WI + ... + dSm A Wm' Wj = UW/USj, (8)

which is a consequence of the condition dw A dS 1 A ..• A dSm = 0 (d¢> = 0).

We observe that in order to find the forms", and () in (7) and Wj in (8) it is necessary to employ a partition of unity. The apparatus of partial derivatives for exterior differntial forms was developed by Norguet (1959).

Theorem 1.8 (Leinart as, Yuzhakov (cf. Aizenberg-Yuzhakov (1979)). Let Ql' ... , Qm be irreducible polynomials in Cn and let the Sj = {z E cn: Qj(z) = O},j = 1, ... , m be manifolds in general position. Then the form

is co homologous in cn\ (S 1 U ... U Sm) to a form of type

w* = L PJ dZ 1 A ... A dzn/Qj1 ... Qjk' J

where J = {jl' ... ,jk}' k ~ n, and the PJ are polynomials.

Remark. The form w* is found constructively using elimination theory (the Hilbert Nullstellensatz).

Theorem 1.9 (Griffiths (1969)). Let Q be an irreducible homogeneous polynomial and let S = g E ClPn : Q(O = O} be a manifold. Then any closed rational n-form W in ClPn with poles on S,

(9)

where Q(O = L'J=1 (-1Y'j d,o A ..• [j] ... A d'n, can be replaced by another form of type (9) which is co homologous to it but which has m ~ n - E(n/q), where q = deg Q (the symbol [j] signifies that the term d'j is omitted).


We remark that in ICIPn it is not always possible to lower the order of the pole to one. This follows from the fact that in Hn(ICIPn\s) there can exist classes which contain no rational forms with first order poles. For example, let

Since S is a curve of genus 1, then

Since any rational form w with a pole of first order on S can be written as canst·Q/((6 + n + n), the rational forms with first order poles cannot generate the entire group H 2 (ICIP 2 \S).

In the general case, if S is an algebraic submanifold of ICIPn, with q(z) =

q(z 1, ... , Zn) = 0 being its equation in affine coordinates, then any differential form of degree n in ICIPn\S with a pole of order one on S has the form w = p(z) dz/q(z) in the coordinates z = (z l' ... , zn), where deg p S deg q - n - 1. The latter condition means that w does not have a pole on the hyperplane at infinity. The class residues of such forms are represented by Poincare residues: P.R.[wJ = [(p dzUl/(oq/OZj) Is]. According to the remark in 1.2, this coincides with the cohomology of holomorphic forms of degree n - 1 on S. In general, on a compact manifold S this part of the cohomology comprises a proper subspace of Hn-1(s).

Let A!:(S) be the set of rational p-forms with poles of order k on a submanifold S c ICIPn, Ylk = A~(S)/dA~=HS). Then the image of the mapping P.R. : £1 ~ Hn-1(S) lies in what is called the primitive subgroup H~ri~(S) (see Griffiths (1969)). Moreover there exists a natural map Rk : Ylk ~ F(n-1,k-1)(S), mapping Ylk to the Hodge filtration

F(n-1,O)(S) c F(n-1,1)(S) c ... c F(n-1,n-1)(S) = Hn-1(S)

of the group W- 1 (S), where R1 = P.R. and the image Rk(Ylk) is the primitive subgroup F(n.-k,k -l)(S)

prIm .

1.5. Generalization of the Leray Theory to the Case of Submanifolds of Codimension q > 1 (Norguet (1971)). Let X be a complex analytic manifold, dime X = n, and let S be a complex submanifold of codimension q (in some neighborhood Va of any point a E S the set S (') Va = {z E Va: Sl (z) = ... = Sq(z) = O}, where the Sj are holomorphic functions in Va and the vectors grad Sj' j = 1, ... , q are linearly independent). There is an exact homology sequence analogous to (3), where the homomorphism (y is induced from a fiber bundle with base S and fiber (ya homeomorphic to the (2q - I)-dimensional sphere. A differential form rjJ, regular in X\S, is called a simple form (cf. [Dolbeault (1985), 4.1J) if there exists a form t/I, regular on X, such that for any point a E S, in some neighborhood Va of the point, the form rjJ can be represented as


where ()a is a form regular in Ua and

K = "s·s· " (_I)V-1s ds A ···[V] .. · A ds A ds A'" A ds ( q )-q q

a L., J J L. v 1 q 1 q' j=1 v=1

Theorem 1.10 (Norguet). Every form ¢l E ZP(X\S) is cohomologous to a simple form ¢l1 E ZP(X\S). Thus for any cycle a E Zp-2q+1 (S), the residue formula holds:

f (2ni)q f ¢l = ( _ 1)' ¢lIs·

ba q . a

§ 2. Application of Alexander-Pontryagin Duality and De Rham Duality

2.1. Application of Alexander-Pontryagin Duality.2 To find the dimension and a basis for the group Hp(X\S), it is sometimes useful to apply the topological Alexander-Pontryagin duality theory which establishes an isomorphism of homology groups:

where sn is a manifold homeomorphic to the n-dimensional sphere and T is a compact subset of sn, where p + r = n - 1. There exist dual bases {yJ, {aJ such that 0(0), Yk) = (jjk' where 0(0), Yk) is the linking coefficient of the cycles aj and Yk'

Let the form w be regular in the domain D = e\ T (T is the singular set of the form w). We compactify the space en to form the sphere s2n = en = en u {oo} by attaching a single point {oo} at infinity. Since T is closed in en, T =

T u {oo} is compact, and D = en \ f Then by Alexander-Pontryagin duality, Hp(D) ~ H 2n - p- 1 (T). Thus to find the dimension and a basis {Yj} of the group Hp(D), it is necessary and sufficient to find the dimension q an'! a basis aj of the (2n - p - I)-dimensional homology group of the singular set T. Then the coefficients kj of an expansion of an arbitrary cycle Y E Zp(D) with respect to the basis {yJj=1' the basis dual to {OJj=1' are the linking coefficients of the cycle Y with the cycles of the dual basis:

o(aj, y) = 0 (aj, t kvYv) = t kv(jvj = kj. v =1 v =1

To find the integrals with repect to the basis cycles, Sri w, it is sufficient to take q homologically independent p-cycles in D, T1 , ••• , I'q. Then the integrals over

2The first to apply Alexander-Pontryagin duality to complex analysis was Martinelli (1953), who used it to deduce a generalization of the Cauchy integral formula to en for multiple integrals of degree n + /, 0 :s; / :s; n - 1.


the basis cycles are found from this system of linear equations,

I w = ± kjv I w, j = 1, ... , q, Ij v=1 y,

where kjv = 0(0"" Ij). We observe that detllkjvll =F 0 is the condition of homological independence of the cycles F 1 , ••• , Fq •

Let p = n and let w = fez) dz be a holomorphic form in D. From the preceding this then follows:

Theorem 2.1 (On residues). Let the function f be holomorphic in the domain D c ([n, let T = en\D, and let i = T u { oo} be a subpolyhedron in the spherical compactification of en, en u {oo}. If {OJj=1 is a basis for the (n - I)-dimensional homology of the singular set f, and if {Yj}j=1 is the basis dual to this in the n-dimensional homology of D, then for any cycle Y E Zn(D) this equation holds:

I fez) dz = (2nW t kjRj, y }=1

where kj = o(O"j, y) and Rj = (2nwn Sr j f(z) dz.

The application of Alexander-Pontryagin duality is especially effective in the case when n = 2 and T is an analytic set. In this case the study of the twodimensional homology group of a domain in real four-dimensional space reduces to the study of the one-dimensional homology group of a surface (a complex curve). As an example we consider the residue of a rational function of two variables.

2.2. Residues of Rational Functions of Two Variables. We apply the approach described above to integrals of the form

II pew, z) dw A dz (10) y Q(w, z) ,

where P and Q are polynomials (or we may assume that P is an entire function), the cycle Y E Z2(([2\ T) and T = {Q(w, z) = O}. Let Q = Q~' ... Q~m, where Q~" ... , Q~m are irreducible polynomials. Then T = Uj=1 1], where 1] = {Q)w, z) = O}. Using the Euler-Poincare formula, it is not difficult to compute the dimension of the group H 1 (i) and consequently that of its dual group H 2(([2\ T).

Theorem 2.2 (Yuzhakov, see Aizenberg-Yuzhakov (1979)). The dimension of the homology groups H 1 (i) ~ H2(([2\ T) is defined by the formula

m s

q = I 2pj + I (qi - 1) + 1- m, j=1 i=1

where Pj is the genus of the surface (the genus of the Riemann surface defined by the equation Qj(w, z) = 0) and qi is the number of irreducible elements of the subset T intersecting at the point Ai; s is the number of such points of self-intersection, and I is the number of elements at infinity of the set T (the number of connected cmponents of the set Tn {lwl2 + Izl2 > R2} for R sufficiently large).


We construct the dual bases of H 1(T) and H 2 (C 2 \T). For this, on each irreducible component 1] we take 2pj canonical cycles a]., s = 1, ... , 2pj' (a basis of the one-dimensional homology of the corresponding Riemann surface) so that the cycles a},2k-l and a},2k intersect each other only at one point and do not intersect the other aJs. Moreover we assume that the curves aJs do not pass through any self-intersection point Ai' We take y]. = 15m]., where m]. is obtained from aj " r = s - (-1)', by a small perturbation, and 15 is the Leray coboundary. Further, suppose that through the point Ai pass qi elements S" r = 1, ... , qi' of the set T. On each of these, except for Sq, we construct a simple closed curve m; surrounding the point Ai' We set y; = 15m;. As the dual cycle a; we take a simple closed curve which begins as a curve on S, with initial point Ai and which ends as a curve which returns to Ai on Sq,. We obtain LI;1 (qi - 1) pairs of cycles. Analogously we construct I - m = Lj;1 (lj - 1) pairs of cycles y], = 15m], and a]" v = 1, ... , lj - l,j = 1, ... , m, in a neighborhood of the point {CI)}.

Theorem 2.3. The cycles {yIs} and {aIs}, r = 1, 2, 3, form dual bases of the homology groups H 2 (C 2 \ T) and HI (T).

We may assume that 8Qj/8w #- 0 at the points of the curves m;' Then the integral over a basis cycle is equal to

where

If P(w, z) f ij. Q(w, z) dw /\ dz = wj. Res[P dw /\ dz/QJ,

Res[P dw /\ dz/QJ = Res[F dQ, /\ dz/QJ 3 [l/(r, - l)!J

x 8,,-IF/8Q';-1 dzIQ,;o,

F = PQ:'/Q' (8Qv/8w),

if ImIsl c 7;,. Since 8,,-1 F/8Q:,-1 is a rational function, we have

(11)

Theorem 2.4 (Poincare (1887». The integral of a rational function of two variables over an arbitrary cycle y E Z2(C2\ T) can be expressed in terms of the periods of abelian integrals on the Riemann surfaces defined by the equations Qiw, z) = O,j = 1, ... , m.

It is clear that for r = 2, 3 the computation of the integral (11) reduces to the computation of the residue of an algebraic function relative to its pole.

Example 3. If p and q are relatively prime integers, the integral J L P(w, z) dw /\ dz/(zP - wq ) = 0 for any cycle y E Z2(C2\ T), where T =

{zP - wq = O}, since T is homeomorphic to the two-dimensional sphere and H2(C2\T) ~ Hl(T).

Example 4. Let Q(w, z) = wz - 1. Then T is homeomorphic to the Riemann sphere with two points identified (z = CI) and z = 0, the pole of the function w = l/z). Thus HI (T) ~ lL. The cycles Yl = {Iwl = Izl = 2} and a 1 = {w = l/t, z = t, ° ::;; t ::;; CI)} form dual bases of the homology groups H 2 (C2 \ T) and


H 1 (f). For any cycle Y E Z2(C2\ T) the integral

I l P(w, z) dw 1\ dz/(wz - 1) = (2ni)2k' R,

where k = o(y, CT d,

R = (2no-2 I l, P dw 1\ dz/(wz - 1)

co If co 1 a2mp(0 0) 2: (2no-2 P dw 1\ dz/(wz)m+1 = 2: -( ,)2 a ma' m • m=O y, m=O m. w z

13

2.3. Application of De Rham Duality. In some cases, for the study of residues it is useful to apply De Rham's Theorem, which establishes a duality between the homology groups and the cohomology of exterior differentiable forms. The theorem can be stated thus: Let X be a differentiable manifold. For any homomorphism 2: Hp(X) ~ C, there exists a unique element h = [w] E HP(X) of the De Rham cohomology group such that 2(g) = Jg h = L W for any g = [y] E H p(X). From De Rham's Theorem we get the following proposition, which is useful in applications.

Proposition 2.5. If for cycles Yj E Zp(X), j = 1, ... , q, and forms Wj E ZP(X), j = 1, ... , q the determinant det Ilajkll =1= 0, where ajk = Li Wk; and if any form WE ZP(X) can be represented in the form W ~ 2:1=1 CjWj' where the cj are complex numbers, then dim HP(X) = dim Hp(X) = q and the {Yj}J=l and {Wj}J=l are bases of the p-dimensional homology and cohomology of the manifold X.

If ajk = bjb then these bases are called dual in the sense of De Rham.

Theorem 2.6. Let {Yj} and {wJ be bases of the p-dimensional homology and cohomology of the manifold X, dual in the sense of De Rham. Then for any cycle Y E Zp(X) and any cocycle W E ZP(X), the integral

f W = f kjRj, y j=l

where the kj = L Wj are the coefficients of the expansion Y ~ 2:1=1 kjYj of the cycle Y with respect to the basis {Yj} and where the Rj = Li ware the coefficients of the expansion W ~ 2:1=1 Rjwj of the De Rham class of the form W with respect to the basis {wj}.

In the case of complex analytic manifolds (for example, domains in en) the next theorems are also useful. (See, e.g., the paper of Onishchik in Volume 10, I of this series.)

Theorem 2.7 (Serre). If X is a Stein manifold (for example a domain of holomorphy in en), then any cohomology class hE HP(X) contains a holomorphic form W E h.

Theorem 2.8 (Grothendieck). IJ either X is a Stein maniJold and S is a submaniJold oj X oj codimension 1 or iJ X is an algebraic maniJold in ClPn and S is a positive divisor on X, then any cohomology class hE HP(X\S) contains a holomorphic Jorm W E h with a pole on S or, respectively, a meromorphic Jorm WE h whose only pole is on S. In particular, iJ X = ClPn or cn and S is the zero set oj a polynomial, then any class h E HP(X\S) contains a rational Jorm.

Example 4. Let Q(z) = zi + ... + z; - 1 and let T = {z E cn: Q(z) = O} and D = cn\ T. By Theorem 2.8 any closed form W E zn(D) is cohomologous in D to a rational form P dz/Qr, where P is a polynomial. It is sufficient to consider the case when P is a homogeneous polynomial of degree p 2:: O. Computing

d(P jt1 (-lYZjdzUl/Qr).

where dZ[i] = dZ 1 1\ ... [j] ... 1\ dzn, and also computing d[(z"/z)( _ly-1 x dZ[j]/(r:t.j - l)Q], r:t. = (r:t. 1, ... , r:t.n), r:t.j 2:: 2, we obtain the recurrence formulas

P dz/Qr+1 '" [(n + p - 2r)/2r]P dz/Qr,

z" dz/Q '" (r:t.j - l)(z"/zj) dz/(n + 1r:t.1 + 2)Q,

(12)

(13)

from which it follows that P dz/Qr '" c<p, where <p = dz/Q, since z" dz/Q '" 0 if r:t.j = 1 for any j. The constant c is found from formulas (12) and (13). We take the cycle

y = {z: xi + ... + x; = 1, Y! = ... = Yn = O},

where Xj = Re Zj and Yj = 1m Zj. Then

f <p = 2ni f res[<p] = ni f dZ 1 1\ ... 1\ dzn-dzn = niLn- 1, lJy y y

where Ln -1 = 2nn/2 T(n/2) is the volume of the unit sphere in IRn. Thus according to Proposition 2.5, dim Hn(D) = dim Hn(D) = 1 and by and <p/niLn-1 are bases of the homology and cohomology of D which are De Rham dual. For any cycle T c Zn(D) the integral J r P dz/Qr = m· c· niLn- 1, where T", m· by.

As another example, we consider local residues in en. Let <P1' ... , <PN be holomorphic functions in a neighborhood V" of a point a E cn and 11" = 8 (<p" 1 , ••• , <p"J/8(z l' ... , Zn) =F 0 for any r:t. = (r:t. 1, ... , r:t.n) E {I, ... , N}. We define 1j = {z E V,,: <Pj(z) = O}. From the assertions of 2.5-2.8 and from the separation of singularities of holomorphic functions this follows:

Proposition 2.9. The dimension oj the n-dimensional homology and cohomology

groups oj the domain V" \(UJ=1 1j) is equal to (N - 1), and their De Rham-dual n - 1 bases consist oj the cycles

and forms

w" = (2ni)-n drp", 1\ ... 1\ drp"n_l 1\ drpN/rp", ... rp"n-l rpN, where e and b/e are sufficiently small positive numbers and 1".1. = (1".1. 1 , ••• , I".I.n), I".I.n = Nand {1".I. 1 , ••• , I".I.n-d E {1, ... , N - 1}.

Let the hoi om orphic form w = f dz = f dz 1 1\ ..• 1\ dzn , where f E

A(U" \ U7=1 1j). The residue of this form with respect to the basis cycle y" can be computed in this way. We perform a change of coordinates (j = L,,/z),j = 1, ... , n, where Lj(z) is the linear part of the function jj at the point a. For sufficiently small e and b/e the cycle y" - y" = {1(11 = ... = I(n-11 = b, I(nl = e}. Then the residue

R" = (2nWn f w = (2nwn f- 1(0 d(, Yll Ya

where I(() = f(L;l(O). The function 1 is hoi om orphic in a neighborhood of the cycle )I" and can be expanded there into a Laurent series /(0 = L~I=-oo cp(fJ. Thus the residue R" = Cl = C1, ... ,-1' In the case of a meromorphic function f = g/rp~' ... ¢J'; the coefficient C l can be found directly. Here ¢J"j = (j + g"j((), where g,,/O = LIPI>l C"jp(fJ, j = 1, ... , n, ¢Jv = cvn(n + gv(O, where gv(O = L;:=t Cvk(k + LIPI>l cvfJ(fJ, v # 1".1. 1 , ... , I".I.n, Cvn # O.

For e and e/b sufficiently small, these inequalities hold on )I,,: Igv(()1 < ICvn(nl, v # 1".1. 1 , ••• , I".I.n, Ig,,(OI < I(J·I, j = 1, ... ,n. Expanding the fractions 1/¢J" =

J J

1/(l1 + g"j(()/(j], 1/¢Jv = 1/cvn (n[1 + gv(()/cvn(n] into series of geometric pro-gressions and cross-multiplying the resulting series, we obtain the desired Laurent series and its coefficient Cl'

§ 3. Homological Methods for Studying Integrals that Depend upon Parameters. Application to Combinatorial Analysis

Leray (1959) applied the theory that he had developed about residues on a complex analytic manifold, along with the topological Picard-Lefschetz Theorem, to the study of integrals depending on parameters. The integrals arise in the solution of the Cauchy problem for partial differential equations in the complex domain. 3 Beyond this, homological methods were developed in connection with the study of singularities and of the character of the branching of the Feynman integrals that arise in theoretical physics (see Fotiadi et al. (1965), Pham (1967), Golubeva (1976)). Various multidimensional analogs of the Hadamard product also lead to the study of holomorphic functions defined by integrals over cycles in en. Consider how the integral of a closed form over a cycle contained in

3 Earlier topological methods were used to study integrals by Picard, Lefshetz, I.G. Petrovskij, and A.A. Borovikov.


the level surface of a holomorphic function in en depends on a parameter, the constant defining the level surface. This defines a function of the parameter; the branching of this function has been studied in detail in (Arnol'd et al. (1984)).

3.1. Analytic Continuation of Integrals Depending on Parameters. Isotopy Theorem. Let X and T be complex manifolds of dimension nand q and let wt(z) = w(t, z) be a closed differential form of degree p in z E X\St which depends holomorphically on the parameter t E T. Let St be an analytic set in X (the singular set of the form wt(z)) which also depends holomorphically on t. We will consider the problem of analytic continuation of the function defined by the integral

I(t) = L w(z, t), (14)

where T is a compact cycle in X\Sto' By the compactness of T the integral (14) is a holomorphic function in a neighborhood of the point to.

Theorem 3.1 (Analytic continuation of integrals (Fotiadi et al. (1965), Pham (1967)). If the projection

n : (X x T, ff) --+ T, (15)

where ff = {(z, t) E X x T: ZESt}, is a locally trivial fibration of the pair (X x T, ff) with fiber (X, Sto)' then for any cycle T E Zp(X\Sto)' the integral (14) can be continued holomorphically along any path .?c in T.

In fact for a path .?c: [0, IJ --+ T, .?c(0) = to, using local trivializations one can construct a covering isotopy, a homeomorphism of the pair

(16)

continuously depending on , which establishes a continuous deformation of the cycle T in X x T\!T, denoted by Tt = gt(r) E Zp(X\S'«t»)' The holomorphic elements J r; w(z, t) define an analytic continuation of the integral (14) along the path.?c, since for nearby values '1' '2 and for t close to .?c('d and .?c('2)' the cycles Ttl and Tt2 are homologous in X\St. Consequently

f w(z, t) = f w(z, t). ~1 rr;2

In applications an important case is when St is a family of algebraic sets depending algebraically on t. In this case the projection (15) is a locally trivial fibration over T\ {algebraic set}. A sufficient condition for the local triviality of the fibration (15) is given by Thorn's theorem on covering isotopies.

1. Multidimensional Residues and Applications 17

Theorem 3.2 (Thorn (Fotiadi et al. (1965), Pham (1967))4. Let n: Y -+ T be a proper differentiable mapping of a stratified 5 set Y to a connected differentiable manifold T such that the restriction of n to any stratum has rank equal to the dimension of T Then n is a locally trivial stratified map.

In the case when Y = X x T, and

S = !Y = {(z, t): t E T, ZESt},

the mapping n is proper only when X is compact.

Corollary 3.3 (Fotiadi et al. (1965)). Let T and X be differentiable COO _

manifolds, with X compact. Let St = Uj Sj(t), where {Sj(t)} is a finite family of Coo-manifolds in X depending Coo on t and in general position for any t E T Then the projection of the pair, n: (X x T,!Y) -+ T, where !Y = {(z, t) : t E T, ZESt} is a locally trivial stratified fibration.

Theorems 3.1 and 3.2 and Corollary 3.3 give conditions for the analytic continuation of the integral (14) along any path in T for an arbitrary initial cycle r E Zp(X\Sto)' However for specially-chosen cycles r these conditions can be made more precise. For example, in (14) let w(z, t) be a merom orphic differentiable form of degree n = dime X with poles on Sit), j = 1, ... , n, where the Sj(t) are analytic sets of codimension 1 (divisors) depending holomorphically on the parameter t E T Assume that a is an isolated point of the set S 1 (to) (l ..• (l

Sn(tO) and that

where jj(z, t) is a defining function of the set Sj(t) in a neighborhood U~ of the point a. In this case the integral (14) is a Grothendieck residue (see Chapter 3) depending on the parameter t. In this case the following holds:

Proposition 3.4 (Tsikh). The analytic element I(t) defined by the Grothendieck residue depending on the parameter t can be continued holomorphically along any path

y = {t = t(r), 0 ~ r ~ I},

for which there exists a lifting

y' = {z = z(r), t = t(r), 0 ~ r ~ I},

in X x T such that z(O) = a and the intersection multiplicity of the divisors S)t(r)),j = 1, ... , n, at the point z(r) does not depend on r.

4 See also IN. Mather, "Stratifications and mappings," Dynamical Systems, New York and London, 1973,195-232. sit is assumed that the stratification satisfies Whitney's Conditions A and B (see Pham (1967)).

3.2. Foliation near a Landau Singularity. Picard-Lefschetz Formula. A holomorphic function defined by the integral of (14) can have a singularity only at those points t' E T for which the cycle r cannot be deformed to one disjoint from the singular set St" In the case of a compact manifold X, according to Theorems 3.1 and 3.2, such points are the points of the Landau set L = U A 7r( {(z, t) E A : rank 7r IA < dim T}), where U A is the union over all strata A of the stratified set !Y. In this case the integral (14) can be continued holomorphically along any path in T\L. This continuation is in general a multi valued function. The Landau set L is an analytic set in T (see Pham (1967)). If the manifold X is noncom pact, then the singularities of the integral (14) are not exhausted by the Landau set. Therefore, one usually considers a compactification of X (see Fotiadi et al. (1965), Golubeva (1976)). For any closed path (loop) A in T\L, with ,1,(0) = ,1,(1) = to, the isotopy (monodromy, see Arnold et al. (1984)) (16) defines an automorphism of the homology group (monodromy transformation)

(17)

If two paths A and Ai are homotopic, the 'corresponding monodromies are also homotopic and consequently ,1,* = Ai.' In other words, the monodromy transformation (17) depends only on the class [A] E 7r 1(T\L, to). Thus (16) and (17) generate a homomorphism (a representation) of the fundamental group 7rl (T\L, to) into the automorphisms of the homology group Hp(X\Sto):

7rl(T\L, to) ~ Aut Hp(X\Sta>- (18)

The monodromy group, the image of homomorphism (18), completely describes the character of the multivalued function defined by integral (14). Its jump as it is continued around the loop A is equal to the integral

f w(z, to)· '-.r-r

We will assume that T is simply connected, i.e., 7r 1(T) == O. We choose a regular point bEL and choose a coordinate system in a neighborhood U b of the point b such that L nUb = {t 1 = O}. The loop in U b given by the equations t2 = ... = tq = 0, tl = Be i8, 0 ~ (} ~ 1, is called a simple loop in T\L.

Proposition 3.5 (Pham (1967)). If 7rl (T) == 0, then 7rl (T\L) is generated by simple loops.

In a special case, a stronger version holds:

Proposition 3.6 (pham, Zariski). Let L = Ll u··· u Lm, where

Lj = {t E U: Pit) = O},

for an irreducible polynomial Pj' If the compactification of the sets L 1 , ... , Lm in ClPn and the hyperplane at infinity CIP~-l are manifolds in general position at every point except possibly for an algebraic set of co dimension 2:: 3, the fundamental group 7rl (Cq\L) is a free abelian group with m generators.

Thus, in the case 7r l (T) ~ ° the study of the representation (18) reduces to finding the automorphisms (17) for simple loops A. In the simplest cases of these automorphisms, the branching of the integral (14) around the Landau set, are described by the Picard-Lefschetz formula (see Leray (1959), Fotiadi et al. (1965), Griffiths-King (1973), Arnold et al. (1984)).

We will show the example of the simplest singularity. Let the point bEL be the projection of a simple critical point (a, b) of some stratum which does not belong to the projection of the closure of the set of critical points of the other strata. Then in a neighborhood of the point (a, b), the set

IT = {(z, t): t E T, ZESt}

has the following form: St = Uj;l Sit), m < n, which, in suitable local coordinates, can be written as

Sj(t) = {s}z,t)=Zj=O}, j=I, ... ,m-l,

Sm(t) = {Sm(z, t) = tl - (Zl + ... + Zm-l + Z;' + ... + z~ = a}.

We define what are called the vanishing cycles:

e = {z: Xl = ... = Xm - l = Yl = ... = Yn = 0, x;' + ... + X~ = t}

where Xj = Re Zj' Yj = 1m Zj' and e = Dl ° ··",,0 Dme E Zn(X\St), where Dj is the Leray coboundary with respect to Sj(t) and also the vanishing square

e = {Yl = ... = Ym = 0, Xj ~ O,j = 1, ... , m, x;. + ... + x~ :::;; t}.

Under the given hypotheses this theorem holds:

Theorem 3.7 (Leray (1959), Fotiadi et al. (1965), Pham (1967)). A circuit around L along a simple loop A in the neighborhood of a point b induces a homomorphism (17), which for p < n is the identity but for p = n defines the following Picard-Lefschetz formula:

A*h = h + N· [e], hE HiX\St),

where [e] is the homology class of the cycle e, N = (_1)(n+1)(n+2)/2, and <e, h) is the intersection number of the chain e with any representative cycle in the class h.

Since

A _ {2. (_I)(n+1)(n+2)/2+t, ifn - m = 2k, <e, e) - ° ·f - 2k 1 , 1 n-m- +,

then

A*[e] = {[-e][,e], ifn - m = 2k, ifn - m = 2k + 1,

from which follows

Corollary 3.8. If n - m is even, then A;h = h (after a double trip around L the class h is carried into itself). This means that this multi-valued function has branching like that of a square root. If n - m is odd, then A!h = h + k· N· [e] (each circuit adds a multiple of the cycle e). In this case the multi-valued function is of logarithmic type. If <e, h) = 0, then there is no branching.

The jump of the integral (14) as one goes around L along the loop A is equal to 1 w(z, t) = (2ni)m-l 1 Resm - 1 [w].

The study of the asymptotics of the integral by the saddle-point method (Arnol'd et al. (1984), Varchenko (1983)) leads to the study of the branching behavior and asymptotics of integrals of the following type. Let f: (en, 0) --+ (e, 0) be the germ of a holomorphic function having an isolated critical point at 0, and let w be a holomorphic differential form of degree n - 1. We denote by X t = {z : f(z) = t}, t #- 0, the level surface of a non-critical value of the function f, and by (It E Zn-l (Xt ) the family of cycles obtained by a continuous deformation of the cycle (lta E Zn-l (X ta ) as t varies along some curve in e\ {o}. Let M be the matrix of the monodromy transformation Hn-1(Xta ) --+ Hn-1(Xta ) corresponding to the simple loop y = {t = e'exp(2nir), 0::; T::; I}.

Then we have

Theorem 3.9 (Arnol'd et al. (1984)). The integral

I(t) = f w c(t)

is a multivalued holomorphic function which for small values of t#-O is represented by the series

I(t) = I a",k t"· (In t)k, ",k

where r:t. is a non-negative rational number and the k are integers. All the coefficients a",k are zero when k > O. Each number r:t. has the property that exp(2nir:t.) is an eigenvalue of the matrix M. The coefficient a",k is zero whenever the matrix of the Jordan form of M has no blocks of size k + 1 or larger belonging to the eigenvalue exp(2nir:t.).

In Arnol'd et al. (1984) a more general situation was considered, where in addition f depends holomorphically on a parameter y E ek. In this case I(y, t) also depend holomorphically on y.

3.3. Some Examples of Integrals Depending on Parameters

(1) In (14) let X = en, and let w(z, t) = P(z, t) dz/Q(z, t), where P and Q are polynomials in z E en and t E eq ; let St = {z : Q(z, t) = O}. The polynomial Q factors into a product of terms of degree one in z:

Q = Q;' ... Q~~


where n

Qj(Z, t) = L ajv(t)zv + ajo(t), j = i, ... , m. v=1

Proposition 3.10. Under the hypotheses above, the analytic function defined by the integral (14) is a ratonal function whose denominator consists of powers of minors of rank n + 1, not identically zero, of the matrix Ilajv(t)II(j=I ..... m;v=o.1.. .. n)

and minors of rank n of the matrix lIajv(z)II(j=l, .... m;v=l •.... n).

(2) The Feynman integral for a one-loop graph with vertices has the form (see Golubeva (1976)):

where

z = (zo, Zl'"'' zn) E.E = {z E en+1 : z6 + zi + ... z~ = I};

P = (PI' ... , Pm), Pj = (PjO, Pj1' ... , Pjn) E en +1 ,

PjZ = PjOZO + ... + PjnZn;

Pj = {Z: PjZ = I}, r E Zn(.E\ Uj=1 Pj), n

w(z) = L (-ltzv dzo /\ ... [v] .. · /\ dZn-v=1

Theorem 3.11 (Boiling, Golubeva). The integral J(p) extends holomorphically along any path in em(n+1)\UPEB Lp, where

Lp = {p: detllpjPv - 111 j.vEP = OJ,

B = {P = {P1' ... , Pd E {l, ... , m}, 1 :s; k:s; n + I}.

The character of the branching of J(p) around the Landau set is determined by the Picard-Lefschetz formula.

One can find a survey of results and a bibliography of research on Feynman integrals in Golubeva (1976). The branching of several integrals depending on parameters is examined in the works of Varchenko (1983), Pedan6 and a series of others.

(3) Multidimensional analogs of the Hadamard product. The Hadamard product of two power series in n variables

f(z) = L aaza, g(z) = L baza, (19) a~O a~O

6 Yu.v. Pedan, "Investigation of the Riemann surfaces of some multiple integrals that depend on a complex parameter II: the Riemann surface of the element I,(t)," Izv. Vuzov Matern., 1976, No. 12, 66-76.


is defined to be the series

h(z) = L a"b"z". ,,;0,0

If the series (19) converges in a closed polydisk ofradius p, then

(20)

where

rp = {«(, 11) E C2n : "jl = l11vl = p, j, v = 1, ... , n}.

The integral representation (2) allows one to study the singularity of the Hadamard product h(z) (Odoni, Dyakovich, Haustus and Klamer, A.I. Yanashauskas, E.K. Leinartas, K.V. Safonov et al).

3.4. Application of Residues to Combinatorial Analysis (see Egorychev (1977), Aizenberg-Yuzhakov (1979)). The fundamental idea of using multidimensional residues to compute combinatorial sums and discover generating functions was developed, with many illustrative examples, by G.P. Egorychev (1977). The combinatorial expressions in a sum (series) are represented as the Taylor coefficients in integral form of their generating functions; replacing the products of integrals with multiple integrals, interchanging the integration and summation signs, then combining the terms of the sum under the integral sign, we obtain the integral of a holomorphic form over a cycle. Computing it using residues, we find the desired sum (generating function). This idea is realized in quite general circumstances by the following theorem:

Theorem 3.12 (Egorychev, Yuzhakov). Let the members of an n-multiple numerical sequence c" = C"' .. . "n be represented in the form

c" = (2nifn Lp ,p(z) }j [}j"jPj(z)/z!j("j+l)] dz,

where ,p and the jj are holomorphic functions in the closed polydisk Up =

{z: Iz) s p,j = 1, ... , n}, rp = {z: IZjl = p,j = 1, ... , n}. Then the generating function for the c" is expressed by the integral

F( ) ~ " (2 .)-n f ,p(z) dz t = L. c"t = nz TIn [Pj _ I'Pj()]'

1"1;0,0 Ip j=l Zj tjJj Z

In particular, if either jj(O) # 0 or f31 = ... f3n = 1, then

[ low] I F(t) = L ,p(z) - , I'EM oz z=z(p)(t)

where

(21)


M = {z(I')(t), /1 = (/11' ... , /1n), 1 ~ /1j ~ {3j} is the set of zeros of the mapping (21) in the polydisk Up.

From this we obtain the following corollary:

Theorem 3.13 (Main Theorem of MacMahon). Let

n

Xj = L ajkxk, j = 1, ... , n. k=l

Then the coefficient of x" = X~l ... x:" in the expansion of X~l ... x:" in powers of x is equal to the coefficient of t" in the Taylor series expansion of the function F(t) = 1/det Ilojk - ajktjll, where

{1, j = k,

Ojk = 0, j =P k.

Example 5. Compute the sum

Amn = ~ (m)(n)(m + n + p - k), L, N = max{m, n, pl· k=O k k m + n

Since

1 (1 + zr dz = {(~), k+1

Izl=e z 0,

k~m,

k>m,

then the generating function for Amn equals

F(u, v)

= f: umvn f (1 + zdm(1 + Z2t(1 + Z3)m+n+p -k dZ 1 1\ dZ2 1\ dZ3 k =0 (2ni)3 r z~ +1 Z~ +1 Z~ +n +1

1 f (1 + Z3)P dZ 1 1\ dZ2 1\ dZ3 = (2ni)3 r [Z3 - u(1 + zd(1 + Z3)] [Z3 - v(1 + z2)(1 + Z3] [ZlZ2(1 + Z3) - 1]

= [(1 - u)(1 - v)]-(P+1),

where r = {IZ11 = IZ21 = 2, IZ31 = 1/2}. Hence, after expanding F(u, v) into a

T I . b' (m + p) (n + p) ay or senes we 0 tam Amn = m n'

Chapter 2 Multidimensional Logarithmic Residues

and Their Applications

L.A. Aizenberg

§ 1. Multidimensional Logarithmic Residues

By the words "logarithmic residue formula" one usually understands an integral representation for the sum of the values of a holomorphic function at all the zeros of a holomorphic mapping in a given region (for instance, a formula for the number of these zeroes). Therefore, to begin with we must introduce the concept of the multiplicity of the zero of a holomorphic mapping. We will give the most natural, the so-called "dynamic" definition of multiplicity. We consider a mapping

W = f(z) (1)

holomorphic in a neighborhood of the point a E en, where W = (W1' ..• , wn ),

f = (f1, ... , fn)· Let the point a be a zero of this mapping, that is, f(a) = O. If the closure of a neighborhood Va contains no other zeroes of the mapping (1), then there is an G > 0 such that for almost all , = (' l' ••• , en), with I" < G, the mapping

W = f(z) -, (2)

has the property that, at each of its zeroes, the jacobian of the mapping is not zero: afjaz "# O. Such zeroes are called simple zeroes. For a small enough choice of G, the number of simple zeroes does not depend on the choice of, nor on the neighborhood Va.

This number of zeroes of the mapping (2) is called the multiplicity of the zero a of the mapping (1). For the relaton between this multiplicity and the coefficients of the Taylor expansion of the mapping (1) at the point a, see §4.

Example 1. For the mapping W l = Z1' W z = z~ + zI, the point (0,0) is a zero of multiplicity m, since the mapping W l = Z1 - '1, W z = z~ + zi - 'Z, for small

1'1 and 'Z "# n, has m simple zeroes of the form ('1' .::I'z - ,n in a neighborhood of this point.

Let D be a bounded domain in C" with piecewise smooth boundary aD. Consider a mapping (1), holomorphic on the closed domain 15 and having no zeroes on aD. The mapping (1) in this case has only a finite number of zeroes inside D. We consider the function ¢J E AAD) (holomorphic in D and continuous


in 15) and pose the problem of computing the sum

L fjJ(a), (3) aEZ,

where ZI is the set of zeroes of the mapping (1) in D and where the number of times each zero appears in the sum (3) is equal to the multiplicity of the zero. The sum (3) can be written as integrals of various dimensions, similar to the integral representation formula for holomorphic functions of n complex variables (in various circumstances the dimension of the integral can vary from n to 2n - 1). The formula for the integral representation of the sum (3) (the formula of the multidimensional logarithmic residue) is best-known and has the most applications either when the integration is over the entire (2n - 1)dimensional boundary aD or in the case when the integration is over an n-dimensional skeleton. We will present these formulas, assuming throughout that we are given a map f E A"(15) (holomorphic in the closed domain 15) and a function fjJ E Ac(D) and that multiplicities are taken into account in the sum (3).

Consider the following exterior differential form, which will be important in the sequel. The form depends on the mappingf, on a continuous vector function w(O) and on the continuously differentiable vector functions w(1), ... , W("-l);

(_1)"("-1)/2 (w(O), df) (w i1 ), df) (w("-1), df) = (2ni)" (w(O),J) /\ d (W(l), f) /\ ... /\ d (w(" 1),J)'

where

" " (w,J) = L wJ;, (w, df) = L Wi dJ;. i=l i=l

Theorem 1 (Leray-Koppelman formula (Aizenberg-Yuzhakov (1979)). If the vector functions wU),j = 0, 1, ... , n - 1 satisfy the condition

(WU)(z),J(z) =I- 0, zED, j = 0,1, ... , n - 1,

then the following formula holds:

f fjJQ(w(O), w(1), ... , w(n-l/, f) = L fjJ(a). ~ aE~

(4)

Corollary 1 (Roos (1974)). If the vector function WE C(l)(aD) is such that (w,J) =I- ° on aD, then

f fjJw(w, f) = L fjJ(a), aD aEZ,

(5)

where

w(w f) = (n - 1)' Lk=I (_I)k-IWk dw[kJ 1\ df '(2nW <w,J)n '

df = dfl 1\ ... 1\ din,

dw[kJ = dW I 1\ ..• 1\ dWk-I 1\ dWk+I 1\ ... 1\ dwn •

Corollary 2 (Aizenberg-Yuzhakov (1979), Roos (1974)).

f ¢>w(J, f) = L ¢>(a). aD aEZJ

(6)

Formulas (5) and (6) follow from (4), since

Q(w, w, ... , w, f) = w(w, f).

On the other hand, the general formula (4) is obtained from formulas (6) and (5), which were already known. This is so because one can show that if the vector functions w(l), ... , wen -1), p(l), ... , pen-I) are in the class C<2)(oD), then the difference

Q(w(O), w(l), ... , w(n-l), f) - Q(p(O), p(1), ... , p(n-I), f)

is a a-exact form. Consequently, it is orthogonal to holomorphic functions ¢> with respect to integration over oD.

If f = z - a, then the logarithmic residue formulas (4)-(6) reduce to, respectively, the general integral representation formula of Leray-Koppelman, the Leray formula, and the Bochner-Martinelli formula.

We observe further that the form Q(w(O), wei), ... , w(n-l), f) does not depend on w(O). Thus, the multidimensional logarithmic residue formula for n > 1 contains arbitrary choices (the choice of the vector function w(l), ... , wen-I) in (4) or the vector functions win (5)). In some cases it is useful to pick the formula for the multidimensional logarithmic residue according to the nature of the problem at hand (see § 3).

To formulate another important multidimensional logarithmic residue formula we will consider special analytic polyhedra

D = {z: z E G, l.fjl < Pj,j = 1, ... , n},

where f E An(G), 15 c G c en. Theorem 2 (Caccioppoli (1949), Martinelli (1955), Bishop (1961), Sorani

(1962)). If D is a special analytic polyhedron, then

1 f df(z) (2 ')n ¢>(z) f( ) = L ¢>(a),

nl r z aEZJ (7)

where r = {z: z E 15, l.fj(z) I = Pj,j = 1, ... , n} is the skeleton of this polyhedron.

Using the Stokes Theorem, one can lower the dimension of integration and deduce formula (7) from formula (6), and conversely. This is similar to the way in which the Martinelli-Bochner integral representation is obtained from the Bergmann-Weil integral representation for functions holomorphic in analytic polyhedra (see Volume 7, Chapter II of this series).

For applications the following result is useful; it includes a variant of Rouche's principle and a multidimensional logarithmic residue formula.

Proposition 1 (Yuzhakov (Aizenberg-Yuzhakov (1979)). Let D and F = Ff be as in Theorem 2 and let the mappings f, g E An(.D) satisfy the inequalities I g/z) I < 1.fj(z)l,j = 1, ... , n, on F.

Then 1) the cycles Ff and

Ff +g = {z: Z E G, l.fj(z) + gj(z) I = B,j = 1, ... , n}

are homologous in the domain G\ {z: OJ;1 [.fj(z) + gj(z)] = O}; 2) the mappings f and f + g have the same number of zeroes (counting multi

plicity) and

~ f ¢J(z) d(f + g) = L: ¢J(a). (2m) r f+g aeZ/+g

(8)

We observe that in formula (8) the integration is taken over the skeleton of a special analytic polyhedron corresponding to the mapping f (and not to f + g).

A number of multidimensional variants of Rouch6's Theorem are also known in which the conditions on the mappings are given for the entire boundary aD of the domain D. For example, when anyone of the following conditions are satisfied, the mappings f and ¢J E An(i» have the same number of zeroes in D (counting multiplicity):

1) on aD the inequality If - ¢JI < If I + I¢JI holds; 2) on aD the real part, Re(f1 ~1 + ... + f,,~n) > -lfll¢Jl; 3) the set

{ .,1 f1(Z) fn(z) a}

O::O:EIG ,0:= ¢J1(Z)="'= ¢In(z),ZE D

does not separate the points 0 and 00.

The previous theorem includes the classical case of a discrete set of zeroes for the mapping (1). Let us consider the more general case of a holomorphic mapping of a complex manifold X of complex dimension n to CP, 1 ~ p ~ n. For such a mapping

we set

and introduce the Martinelli form

W (f) = (p - 1)! L~=l ( -l)'x-1L dJ[ctJ /\ df p,p-1 (2nW Ifl 2p ,

where Ifl2 = Ifl12 + ... + Ifnl2. Let us consider the case 1 ~ p ~ n - 1 under two conditions:

(i) If CJ is the critical set of f, then the analytic set ZJ n CJ has at each point complex dimension no greater than n - p - 1.

We write iJ = ZJ \ CJ. If D is a relatively compact open subset of X, then the integral of a :,.ontinuous 2(n - p)-form over ZJ n D can be defined to be the integral over ZJ n D.

We assume that the boundary aD is piecewise smooth and (ii) the set iJ n aD has measure 0 in i J. Now we can formulate a result generalizing formula (6).

Theorem 3 (Lupacciolu (1979)). Assuming (i) and (ii) are true, then for any a-closed form ¢In-p,n-p of type (n - p, n - p) which is smooth on X,

I Wp,P-1(f) /\ ¢In-p,n-p = f ¢In-p,n-p' oD Z/nD

(9)

For p = n the a-closed form ¢In-p,n-p is actually a holomorphic function and formula (9) reduces to formula (6).

If X is a Kahler manifold with Kahler form D, the form ¢In-p,n-p can be taken

to be the form ( . 1 ) Dn-p; we obtain the following corollary. n - p!

Corollary 1. If (X, D) is a Kahler manifold and if f and D satisfy conditions (i) and (ii), then

( 1 )1 I Wp,P-1(f) /\ Dn-p = V2n - 2p(ZJ n D), n - p. cD

where V2n - 2p denotes the (2n - 2p)-dimensional volume.

(10)

Example 2. Let X be an open manifold in en and let D be the standard Kahler form

then (10) reduces to the following equation

(~)n-p I Wp,p-1(f) /\ L dza, /\ dza, /\ ... /\ dzan_p /\ dzan_p

aD 1 ::;a1 < ... <an- p

= V2n - 2p(ZJ n D),

which for the case p = 1 is contained in the work of Wirtinger (1937).


Now we introduce a generalization of formula (7) in the case of a holomorphic mapping f = Ul' ... ,fp) of an open set D c en to U. We recall (Dolbeault (1985), Sect. 3.4) the inductive definition of an essential intersection CeUl, ... , fp):

CeUd = fl- 1 (0), CeUl'···' Is) = CeUl, ... , Is-d n Is-I (0),

where CeUl, ... , Is-I) is the sum of those irreducible components of the cycle CeUl, ... ,Is-I) on which Is is not identically equal to O. We consider the following residue current RJ[tfoJ of Coleff and Herrera (Coleff-Herrera (1978), see also (Dolbeault (1985), Sect. 3.5)); for the form tfo E COO with compact support in [)

RJ[tfoJ = lim f ~ " tfo, b-+O T.(f) fl .. .jp

where YoU) = {z: Ijj(z)1 = bj,j = 1, ... , p} under the condition that (b)bj\l) ~ o for all k > 0, 1 ~ j ~ p - 1.

Theorem 4. (Coleff-Herrera (1978), Colomin (1977)) For any 2(n - p)-form tfo E Coo with compact support in D,

RJ[tfoJ = (2ni)P f tfo· Ce(J,,···,J p)

We observe that in this theorem there are no conditions on the intersection of the zero sets of the functions Is, s = 1, ... , p (the intersections are possibly not complete).

Besides the formulas given above for the multidimensional logarithmic residue, there are formulas of another type,but with a similar right-hand side of the equation. They are written in terms of currents, and the corresponding integration is over the whole complex manifold. We will use the following notation.

- i -d = J + J, de = 4n (J - J)

and D J is the divisor of the merom orphic functon f. Theorem 5 (Poincare-Lelong formula (Lelong (1968))). Let F be a meromorphic

functon on a complex manifold X. Then this equation of currents is true:

(10)

Formula (10) means that for any C.o-form tfo = tfon-1.n-l with compact support in X, there is an equality

f log Ifl2 dde tfo = f tfo· x Df

In the case of a holomorphic function f, the Poincare-Lelong formula can be generalized to the case when f is a holomorphic mapping from an ndimensional complex manifold X to CPo We denote

f*(), = (dde log IfI2)'.

Theorem 6 (Poincare-Martinelli formula (King (1971), Griffiths-King (1973)). The form f*()l and the form log Ifl 2f*()1 are locally integrable on X for alii.

If Zf = f- 1(0) has dimension n - p, then for I < p,

ddC(log IfI 2f*()I-I) = f*()l

and

ddC(log IfI 2f*()p-l) = Zf

where Zf is counted with multiplicity. In other words,

f log IfI 2f*()p-l 1\ ddC tP = f tP x ~

for any Coo-form tP = tPn-p.n-p with compact support in X.

(11)

From the Poincare-Lelong and the Poincare-Martinelli formulas, one can deduce the usual formulas for the multidimensional logarithmic residue. For example, we show how to obtain formula (6) from (11) for p = n and discrete Zf' In this case the right-hand sides of (6) and (11) are identical. We consider the left side of (11) for tPo.o = tPX, where tP E A(15) and X is a function of class COO with support in a domain Dl ::::> 15 and xlD = 1 (assuming the boundary oD is sufficiently smooth), with (15 1 \D) n Zf = 0. By the Stokes Formula

f 2 i-if 2 -log If I f*()n-l 1\ -2 ootPo.o = --2 _ 0 log If I 1\ f*()n-l 1\ otPo.o D, 1C 1C D, \D

= - 2i f _ d(o log Ifl2 1\ f*()n-l tPo.o) 1C D, \D

Using f*()n = 0 and applying the Stokes Theorem again, we obtain

- 2i f _ 0 log Ifl2 1\ f*()n -1 tPo. 0 = f tPOJ(J, f). 1C a(D, \D) aD

To conclude, we will demonstrate versions of Theorems 5 and 6 generalized to the case of line bundles L ~ M, where M is a complex manifold (for the generalization ofthe concept of residue to such a case, see Chapter 3, Subsection 3.5). Suppose that the transition functions of L are {gjk}; a metric on L is given by positive COO functions Pj on neighborhoods Vj with relationships

pj=lgjklpk inVjnVk·

Given this, the (1, I)-form OJ given by the equation

OJlu j = ddc log Pj'

is defined globally and is called the curvature form of the line bundle L ~ M.


Now take any global holomorphic section u E HO(M, L) with divisor D. The function log lul 2 is locally integrable on M.

Theorem 5' (Griffiths-King (1973)). On M the following equation of currents is true:

ddC log lul 2 = D - w.

Intersection in homology is dual to exterior product in de Rham cohomology, so using Theorem 5, we find that Theorem 6 has the following natural analog for holomophic sections u l' ... , u, of the line bundle L --+ M.

Theorem 6' (Griffiths-King (1973)). If the divisors Du, intersect in a set of complex codimension r, then this equation of currents is true,

where A is the locally integrable form

1 ,-1

A = log -2 I W~-1-k /\ W\ W o = W + ddc log IUI2 . lui k=O

Moreover, if W 2 0, and lui ~ 1, then A 20.

We mention yet another set of directions, related to the generalization or the further study of the multidimensional logarithmic residue (formulas (6) and (7)) (see Aizenberg, Yuzhakov (1979)):

1. In formulas (6) and (7) the inegration is taken over cycles of dimension 2n - 1 or n. One can also find formulas for the case of cycles of intermediate dimension (Yuzhakov, Kuprikov).

2. The cycle r in formula (7) can be replaced by a cycle of a more general nature, called a "separating" cycle. These cycles were studied by Martinelli, Sorani, and Tsikh.

3. One can construct the multidimensional logarithmic residue formula based not on the Bochner-Martinelli integral representation but rather on the more general integral representation of Andreotti-Norguet. In this direction we mention the results of Norguet, Aizenberg and Bolotov.

§ 2. Series Expansion of Implicit Functions

The easiest corollaries of the Cauchy formula for holomorphic functions of one complex variable are the Taylor expansion and the Laurent expansion for functions holomorphic in a disk or, respectively, in an annulus. Analogously, the easiest corollaries of the logarithmic residue formula for holomorphic functions of one variable are the Lagrange expansion and the Biirmann-Lagrange series. This allows one to represent one holomorphic function in the form of a series in the powers of a second holomorphic function, e.g., the formula for inverting a holomorphic function, etc. In this section, using the multidimensional loga-

rithmic residue (mainly formula (8)), we will introduce some generalizations of these expansions.

Let cJ>(w, z) and Fiw, z),j = 1, ... , n, be holomorphic functons of the variables w = (WI' ... , wm ) and z = (ZI"'" zn) in a neighborhood of a point (0, 0) E em+n

such that Fio, 0) = 0, for j = 1, ... , n, with (8F/8z)l(o,o) =I 0. The system of equations

Fiw, z) = 0, j = 1, ... , n,

defines a system of functions

Z = tjJ(w) : Zj = (h(w), j = 1, ... , n,

holomorphic in a neighborhood of the point ° E em. The problem is to expand the function cJ>(w, tjJ(w)) in a series. Without loss of generality, we may assume that

Theorem 7 (Yuzhakov (1975), Aizenberg-Yuzhakov (1979), § 20)). The function (w, qj(w)) is represented by the following series of functions, which converges absolutely and uniformly in a neighborhood of the origin.

( -1)IPI [ 8FJ I cJ>(w, qj(w)) = L -p,-D: cJ>(w, z)gP(w, z)-8 ' P'2:.0 • Z z=h(w)

(10)

where h = (hI, ... , hn) is an arbitrary vector-valued function, holomorphic in this neighborhood, with the condition that h(O) = 0;

gp - gP, gPn P' - P' f3' IPI - P + ... + f3 . - 1 ... n' . - 1···· no' - 1 n'

gj(W, Z) = Fj(w, Z) - Zj + hj(w), j = 1, ... , n;

81PI D: = 8 p, Pn;

ZI .. 'Zn

the notation P ?: ° means that all the Pj ?: O,j = 1, ... , n.

Corollary. Let the mapping (1) be holomorphic in a neighborhood of the point ° and satisfy the condition

8jj(0) . jj(O) = 0, -8- = bjk , j, k = 1, ... , n,

Zk

and let the function cJ>(z) also be holomorphic at 0. Then in some neighborhood of the point ° the following expansion is valid

cJ>(qj(w)) = p~o (_p1t l

DP [ cJ>(z)(}p(z) izJ Iz=w' (11)

where (}j(z) = jj(z) - Zj' j = 1, ... , n. For cJ>(z) = Zj' j = 1, ... , n, formula (11) represents the inverse of the holomorphic mapping (1).

If the functions <D(w, z) and Fj(w, z), for j = 1, ... , n, are given in a neighborhood of the origin by their Taylor series, then one can also express the Taylor series of the function <D(w, tfo(w)). As a corollary of this exapnsion, one obtains the result of Cayley-Sylvester-Sack (see Ajzenberg-Yuzhakov (1983), § 20)). For the series (10) and (11), one can give estimates for the domains of convergence and the remainders (loc. cit.).

Finally, the multidimensional logarithmic residue can also be applied to systems of equations where oFjozl(o,o) = 0. This permits one in some cases to separate a holomorphic single-valued branch of an implicit vector-valued function.

Example. We separate the holomorphic branches of the complex curve Z3 -

3wz + w3 = ° in a neighborhood of the point (0, 0) E ([2. Applying the method described above, one can show that the branch tangent to the complex line {(z, w): z = o} has the form

00

z = I CkWk, k=O

where

§ 3. Application of the Multidimensional Logarithmic Residue to Systems of Nonlinear Equations

We will investigate the system of algebraic equations

Q/z) + Pj(z) = 0, j = 1, ... , n (12)

where the Q/z) are homogeneous polynomials with kj being the highest degree in the variables jointly. We further assume that the only common zero of the polynomials is the origin and that the degree of each Pj is less than kj,j = 1, ... , n. It is easy to show that system (12) has a finite number of solutions equal to N = kl k2 ••• kn- Let m1 , •.. , mn be natural numbers such that the equation

(13)

has a solution of the form

n

Wj = I ajk(z)z;:'\ j = 1, ... , n, (14) k=l

where the ajk are polynomials in z. According to the well-known theorem of Macauley (see Chapter 3, Theorem 4.1), this condition is automatically satisfied for mj = Ikl + 1 - n, for j = 1, ... , nand Ikl = kl + ... + kn ; but sometimes mj

can be chosen smaller. The solution (14) of equation (13), which in general is not unique, can be found by the method of undetermined coefficients. Now one can apply formula (5) to the vector-valued function w in the domain

Dp = {z: IZI12ml + ... + IZnl2mn < p},

which results in the following assertion.

Theorem 8 (Aizenberg (Aizenberg-Yuzhakov (1979), §21)). Let R(z) be a polynomial of degree /1; then

(15)

where Al is the jacobian of system (12) and A2 = detllajkll. The summation in the left side of (15) is taken over all the zeroes Z<e) (counting multiplicity) of system (12), and A is the linear functional on the polynomials in ZI' ..• , zn' z~', ... , z::'n defined by the equation

Corollary 1.

if {3j = mp'j + mj - 1,j = 1, ... , n, otherwise.

t '\' R(z(e») = -l..J jl+IX' e S

where t is a polynomial in the coefficients of system (12) and the polynomial R, while s is a polynomial only in the coefficients of the polynomials QI' ... , Qn.

Formula (15) takes an especially simple form in the case of the system

Z;i + lj(z) = 0, j = 1, ... , n (16)

where the degree lj is less than kj for j = 1, ... , n. Here one can consider mi = ki' i = 1, ... , n. Then Wi = z~;, the determinant A2 = 1 and we arrive at the following proposition.

Corollary 2. For system (16) and a polynomial R(z) of degree /1, the following formula is true:

L R(z(e») = JV [RAI ~~ ... Zt t (_1)1"'1 (~I,)"" ... (~)"'nJ, (17) e ZI ... zn 1"'1=0 ZI zn

where JV is the linear functional on the polynomials in ZI' ... , zn' 1/zl' ... , 1/zn which associates to any such polynomial its free term.

Using formulas (15) and (17) one can compute power series, for example the first coordinates of the roots of

N

L (zle)y = Sj' j = 1, ... , N. e=1


The coefficients of the polynomial Q(zd = zf + bIzf-1 + '" + bN-IZ I + b N ,

having roots z\1), ... , Z\N) given by Waring's formula or the recurrence formula of Newton

(18)

are expressed in terms Sj.

Thus formulas (15) and (17) lead to a new method for eliminating unknowns, which does not add extra roots and which does not omit any roots. This method seems to us to be simpler than the classical methods of elimination using the resultants of polynomials.

Formula (17) leads to a particularly simple computation in the case when the degree of the polynomial R(z) is small.

Example 1. Consider in 1R3 the three surfaces of third order

xi + L aijkXiIX~X~ = 0, i+j+k,,;2

X~ + L bijkX~X~X~ = 0, (19) i+j+k,,;2

where a ijk , b ijk , and Cijk are real. Let the surfaces in (19) be in "general position" in the sense that they have 27 points in common in 1R3, the maximum possible number. We fix a point (A, B, C) E 1R3 and compute using formula (17) the sum of the squares of the distances from this point to the 27 common points of the surfaces (19). We find that the sum we seek is equal to

9(a~00 + b620 + C602) - 18(a loo + bOlO + cood + 6(a 101 C002 + allob020

+ a200 bII0 + bOIIC002 + a200cIOI + b020COII) + 12(a002cIOI + a020 b II0

+ allob200 + b002COll + bOll C020 + a lOI c200 ) + 27(A2 + B2 + C2)

+ 18(Aa200 + Bb020 + CC002 )'

It is curious that the answer does not depend on 12 of the 30 coefficients of the equations of the surfaces (19).

Formula (15) has found an application in the determination of all stationary solutions of certain chemical kinetic equations (see Aizenberg et al. (1983)).

Example 2. In the study of the reaction of the oxidation of hydrogen appear the following equations of stationarity with respect to intermediate substances:

2kIZ2 - 2LIX2 - k4 xy + L 4 zu - k3X - L3Z = 0,

2k2Z2 - 2L2y2 - k4 xy + L 4 zu - ksyu = 0,

k4 xy - L 4 zu - ksyu - 2k6U2 = 0

x+y+z+u=l.

(20)


After eliminating u = 1 - x - y - Z and making the change of variable Z = ty, we apply formula (15) to the resulting nonlinear system of equations. This allows us to eliminate in the general form all the unknowns except t. One obtains a particularly simple expression when k6 = O. In this case the desired polynomial is

where

Po = k~E4(2Ll + k3),

Pi = kzL4(4kzk5k_l + 2kz k3k5 + kZ k3k4)'

pz = 2k~k;Ll + k~k3k; + k~k3k4k5 + kZ k3k4k5k-4 - kzk4kSL3L4'

and so forth. Moreover, applying the classical methods of Descartes and BudanFourier, one can investigate the number of positive roots of system (20) and write down a condition on the parameters (coefficients of system (20)) guaranteeing the uniqueness of the stationary condition or of a certain number of stationary conditions; and this in its turn leads to some information about the chemical reaction in question.

The results cited above have developed in the following directions: 1. Instead of system (16) on can consider a system of more general form

i-l

Z~i + L Zj¢Jij(z) + Pi(Z) = 0, i = 1, ... , n, j=l

(21)

where the ¢Jij(z) are homogeneous polynomials of degree ki - 1 and the Pi(Z) are polynomials of degree no greater than k i - 1. Under these conditions, formula (17) remains essentially true. Moreover, instead of system (16) or (21), one can take analogous systems in which the ordinary homogeneous polynomials are replaced by weighted-homogeneous polynomials (Aizenberg-Tsikh, (AizenbergYuzhakov (1983), § 21)). More general systems were investigated by Yuzhakov (in the collection Some Problems of Multidimensional complex Analysis, Krasnoyarsk, 1980, 197-214).

2. A multidimensional analog of Waring's formula for systems of the form (21) was found by Bolotov (Aizenberg-Yuzhakov, 1979, §21).

3. A multidimensional analog of Newton's formula for systems of nonlinear algebraic equations of a different type was proved by Aizenberg and Kytmanov (Siberian Math. Journal 22, No.2, 19-30 (1981)).

4. On the basis of formulas (15) and (17), Aizenberg proved a relatively simple algorithm which provides an answer to the question of how many real roots the system (12) or (16) has, about the number of roots in a given ellipsoid, etc. (see loco cit.).

5. The problem of computing the remaining coordinates of the roots if the first coordinates have already been determined, and also the problem of solving nonlinear algebraic systems in radicals has been considered in the work of L.A. Aizenberg, B.A. Bolotov, and A.K. Tsikh (Dokl. Akad. Nauk USSR 252, No.1, 11-14 (1980)).


§ 4. Computation of the Zero-Multiplicity of a Holomorphic Mapping

37

We consider a mapping (1), holomorphic in a neighborhood of a point a E en. Suppose that a is an isolated zero of this mapping with multiplicity {ta(f). When n = 1 the multiplicity {ta(f) only depends on the index of the first non-zero Taylor coefficient of the function f at the point a. For n > 1 the situation is significantly more complicated. For example, the systems

(Zl + Zz + zf, Zl - Zz + z~) and (Zl + Zz + zi, Zl + Zz + z~) are made up of the same collection of monomials, but the multiplicity of the zero at (0,0) differs for the two maps. But for almost all systems (1) the multiplicity {tA!) is nonetheless determined only by the monomials which appear in /; with non-zero coefficients, in other words, the sets

Si = supp /; = {IX E r\jn, Ci~ # O},

where

/; = L Ci~(Z - a)~. ~

Next we will study an important concept introduced by Tsikh, the resultant of a function !/J with respect to a system (1), denoted by R(f, !/J). If system (1) is holomorphic in a domain D c en, with a finite set of zeroes {z(e)}, then R(f, !/J) = De !/J(z(e»), where the number of times each zero appears in the product is equal to its multiplicity. With the help of formula (7), one obtains

Theorem 9. (Tsikh (Aizenberg-Yuzhakov (1979), § 22)) The multiplicity of an isolated zero z = 0 of system (1) is equal to the multiplicity of the zero Zn = 0 of the resultant of the function f" relative to the system 'f = (f1' ... , fn -1):

{to(f) = {to(R('f, fn))'

From this result and Rouche's principle, it is easy to show the theorem of Bezout concerning the number of isolated zeroes of a system of algebraic equations in IClP'n.

Moreover, let

Jj(z) = I Ci",z"', j = 1, ... , n. 1~I:?:kj

We denote by do(fi) = kj the order of the zero at z = 0 of the function fi at the point 0 and

(22)

The most important corollary of Theorem 9 is

Corollary (Tsikh-Yuzhakov). The multiplicity of an isolated zero z = 0 of system (1) is equal to the product of the orders of the zero at z = 0 of the functions of the system, {to(f) = do(f1)' .. do(fI ), if and only if 0 is an isolated zero of the


system (22) of homogeneous polynomials. It is always true that

}loU) ~ doUd .. ·doUf )·

This result is easy to generalize to the case when the polynomials are weighted homogeneous polynomials rather than polynomials homogeneous in the ordinary sense (Tsikh, see (Aizenberg-Yuzhakov (1979)).

One can introduce a further generalization of the homogeneous principal part (22) of the system (1) by employing the concept of the Newton polyhedron r+Ul' ... ,fn) of the system (1) at the zero. This is the convex hull in [R~ of the set

{IX: IX E [R~, IX E (supp fl u'" U supp f,,)\{O}}.

Let rUl' ... ,f,,) be the Newton polygon, the union of the closed faces of the polyhedron r+Ul"'" f,,) and let LUI"'" fn) be the union of all the segments with beginning point at 0 and end point on rUl,"" f,,). We will call the number n!v(r_) the Newton number of the system (1) at zero; here v is the ndimensional volume. We include in jjr only those terms of the form caza in the Taylor series of jj for which IX E r,j = 1, ... , n. The polynomials jjr,j = 1, ... , n, are called the principal parts of system (1) at zero. System (1) is called nondegenerate at zero if and only if for any face (J of the Newton polygon the polynomial jja is not zero on the set (C\Ot. Using the methods of § 1 we obtain

Theorem 10 (Kushnirenko (1975)). 1) The multiplicity of the isolated zero z = 0 of system (1) is no less than the Newton number of the system at zero; if the system is nondegenerate at zero, then equality holds. 2) The set of principal parts of systems nondegenerate at zero and having a given Newton polygon, is an open dense subset of the manifold of all principal parts with given Newton polygon.

§ 5. Application of the Multidimensional Logarithmic Residue to the Theory of Numbers

Using formula (6) one can obtain an integral formula for the difference between the number of integer points in a domain in the space [Rn and its volume. A whole series of classical problems of number theory are problems of computing the asymptotic behavior of this difference. Thus, these problems reduce to the study ofthe asymptotics of the integrals below. We will restrict our attention to the cases n = 2, 3, which are important in classical problems.

Theorem 11 (Aizenberg (1983)). 1) Let Q be a bounded domain in [R2 with piece-wise smooth boundary oQ, and let s(Q) be the area of Q. If there are no integer points on oQ, then

N(Q) - s(Q)

_ 1 foo foo d d f t z sin 2nxl dx z - tl sin 2nxz dX 1 -- t 1 " t2 Z 2 2'

n 0 0 iJQ (tl - 2tl cos 2nxl + t2 - 2tz cos 2nx2 + 2) (23)


2) Let Q be a bounded domain in [R3 with piecewise smooth boundary. If there are no integer points on aQ, then

N(Q)-v(Q)

4 fro fro fOO =- dt l A dt2 A dt3 n 0 0 0

If t2t3 sin 2nxi dX2 Adx3-t 1 t3 sin 2nx2 dX 1 Adx3+t1 t2 sin 2nx3 dX1 Adx2

x oQ (ti - 2t I cos 2nxi + t~ - 2t2 cos 2nx2 + t5 - 2t3 cos 2nx3 + 3)2 (24)

where v(Q) is the volume of the domain Q.

The proofs of formulas (23) and (24) reduce to embedding the domain Q in en and considering a suitable extension to a domain Da c en and a suitable mapping which has zeroes only at the integer points of the real subspace, and finally considering the limiting behavior with respect to a in formula (6), applied to this domain and this mapping. If we use the more general formula (5), then we can obtain generalizations of formulas (23) and (24). As the first application of formulas (23) and (24) we mention the following. If in the integrals (23) or (24) the kernel is expanded into a series for the case when Q is a disk (respectively, a ball) of radius t, then after integrating one obtains an expansion of the difference N(Q) - s(Q) (respectively, N(Q) - v(Q)) into a series of Bessel functions II (respectively, 13/2 ), This series has several advantages in comparison with previously-known expansions of this kind: its coefficients can clearly be written in terms of Gamma functions and do not contain rk(t), the number of representations of t as the sum of k squares of integers, k = 2, 3, in contrast with the expansion ofVoronoy-Hardy (n = 2) and Oppenheim (n = 3).

Chapter 3 The Grothendieck Residue and its Applications to

Algebraic Geometry

A.K. Tsikh

Introduction

From the point of view of the theory of residues in complex analysis, the Grothendieck residue is represented as the integral of a merom orphic form of an extremely general type over a special cycle. It is the most precise multivariable analog of the Cauchy residue at a point for an arbitrary meromorphic function


of one complex variable. The Grothendieck residue is defined locally: it is actually associated to what is called a regular sequence of germs f = (f1' ... , fn) of holomorphic functions at some point a E IC". It can be interpreted as a homomorphism

res: f9a/Ia(!) ~ C,

where f9a is the ring of germs offunctions hoI om orphic at a, and IA!) is the ideal in this ring generated by the sequence f In the light of this algebraic interpretation, the Grothendieck residue is a useful and effective instrument in algebraic geometry. In addition, it generalizes the logarithmic residue for a mapping w = f(z). Instead of the terminology "Grothendieck residue," we will also use the phrase "local residue" with respect to the mapping f

§ 1. Integral Definition and Fundamental Properties of the Local Residue

1.1. Definitions. Let hand f1' ... , f" be holomorphic functions in a neighborhood Va of a point a E IC" such that the mapping f = (f1' ... ,fn) has an isolated zero at the point a:f-1(O)n Va = {a}. The "local residue"7 or "Grothendieck residue" of the merom orphic form w = h dz/ f ... f" at the point a is defined to be the integral (Griffiths-Harris (1978), Tong (1973))

where

1 f h(z) dZ 1 1\ ..• 1\ dZn resw =-a 2ni ra f1(Z) ... f,,(z)

Fa = {z EVa: l.fj(z) I = B,j = 1, ... , n}.

(1)

Here B is taken sufficiently small and the cycle Fa is oriented by the condition

d(arg fd 1\ ••• 1\ d(arg f,,) ~ O.

We will also call the residue (1) the local residue of the function h with respect to the mapping f at the point a and denote it by resf (h).

a

We note that in the case h = rjJaf/az, where af/az is the jacobian of the mapping J, the local residue (1) reduces to the logarithmic residue (see Chapter 2) and is equal to the product fJ.a(f)· rjJ(a), where fJ.a(f) is the multiplicity of the zero at a of the mapping f; this coincides (see Palamodov (1967)) with the dimension of the vector space f9a/Ia(f).

The following proposition then follows from Stokes Formula.

Proposition 1.1 (Griffiths-Harris (1978), Tong (1973)). The local residue vanishes for functions h which belong to the ideal Ia(f).

7 In (Dolbeault (1985), Sect. 04) the term "point residue" is used instead of "local residue".


Thus, taking account of the linearity of the local residue with respect to functions h E (!)a, we obtain a well-defined homomorphism

resf: (!)aI1a(f) -+ C, a

which is the residue mapping of Grothendieck (see Lomadze (1981), Beauville (1971), Carrell (1978), Hartshorne (1966)).

1.2. Representation of the Local Residue by an Integral over the Boundary of a Domain. Analogously to the way that the integral of h dflf, the logarithmic differential with weight h, can be expressed as an integral over the boundary of a domain of hf3(f, J), where f3 is the Bochner-Martinelli kernel (see Chapter 2), the local residue (1) can be represented in the form ofan integral over a (2n - 1)dimensional cycle.

Proposition 1.2 (Griffiths-Harris (1978)). Let the mapping f be holomorphic in the closure of a neighborhood Va with piecewise smooth boundary oVa, where f- 1 (0) n Va = {a}. Then the local residue

res OJ = f I'/w, a aUa

where

(n - 1)1 "'n (_1)k-17 d7 /\ ... [kJ··· /\ d7 /\ dz /\ ... /\ dz -h' ·.L..k=l Jk J1 In 1 n

1'/", - (2ni)n (lfll 2 + ... + (lfnI 2 )n

We remark that the form I'/w differs from h· f3(f, J) by the absence of the jacobian J = ofloz just as the form OJ differs from the weighted logarithmic differential h dflf. Instead of the form '1w one can take another representative of the Dolbeault cohomology class [l'/wJ E H~·n-1(Vn, V: = Va\ {a}, for example the form hQjJ, where Q is the form in Theorem 1 of Chapter 2.

1.3. Transformation Formula for the Local Residue. When a system of germs f is changed to a system g, contained in the ideal Ia(f), the local residue is transformed in the following manner. Let

n

gj = I (hdk' j = 1, ... , n, rPjk E (!)a, (2) k=l

where the system g = (g 1, ... , gn), like f, has an isolated zero at the point a. Then the local residue satisfies a transformation formula (Griffiths-Harris (1978), Tong (1973)):

resf (h) = resg (h' det IlrPjkll). (3) a a

The transformation formula provides a means of computing local residues. For example, inserting the monomial (Zj - ali +1 into (2) as gj (by Hilbert's Nullstellensatz any monomial of sufficiently high degree lies in the ideal Ia(f)),

we arrive at the formula

From this formula it is evident that

resf = L C~D~(j(z - a) a [~[,,;m

is an analytic functional which equals a finite linear combination of derivatives of delta functions. (It would be interesting to know if the converse is true: can every such analytic functional be realized as the local residue relative to some mapping?)

We remark that if the only zero of the mapping f in the neighborhood Va is at the point a, then the mapping g, related to f by formula (2), can have zeroes in Va in addition to the isolated zero z = a. It is not difficult to check that for any such zero b =1= a the determinant det II~jkll E Ib(f). Therefore, according to Proposition 1.1, the local residue resg = (h·detll~jkll) = 0. A special case of this fact

b

explains the nature of the Bergman-Weil integral representation in special analytic polyhedra (see Ajzenberg Yuzhakov (1979)); it is related to the Cauchy integral representation by the transformation formula (3). To be concrete, let the polyhedron II be defined by the conditions I ltj(z) I < B, j = 1, ... , n, and let a E 11. In (3) we set f = z - a and g = W(z) - W(a). Then for the ~jk in (3) we can take the coefficients of the Hefer expansion for the functions ltj. Consequently, taking account of what was said above and recalling Proposition 1 from Chapter 2, we find that in (3) the residue on the left is equal to the Cauchy integral of the function h while the one on the right is equal to the BergmanWei I integral of this function (see Volume 7, article 2).

1.4. Local Duality Theorem. According to Proposition 1.1, the symmetric pairing resf : (!)a/1a(f) ® (!)a/1a(f) --+ C is well-defined, if we set

a

resf (h, g) = resf (h· g). a a

One of the most remarkable properties of the local residue is reflected in the following:

Theorem 1.3 (Local duality (see Griffiths-Harris (1978), Tong (1973))). The pairing resf is nondegenerate, i.e., if for all g E (!)a the local residue resf (h· g) = 0,

a a

The nondegeneracy property of the local residue is applied effectively in § 4 to the question of whether or not a polynomial or germ h E (!)a belongs to a given ideal.


§ 2. Using the Trace to Express the Local Residue

2.1. Definition of the Trace and its Fundamental Properties. Let D c c~ be a bounded domain in the space of the variables z and let f: Ii - C::, be a holomorphic mapping having no zeroes on the boundary aD. Then this mapping has only a finite set Z J of zeroes in D. We assume that Z f is not empty. Let G be the connected component of the point 0 in C::, \f(oD). Then the systems of equations f - w = 0 are homotopic for any w E G, and the number of roots in the domain D is the same for each of these systems. In this case one says that f has finite type.

For every merom orphic function H(z) = rjJ(z)/t/J(z) in the domain D we associate its trace

(Tr H)(w) = L H(z(Vl(w)), WE G\f( {t/J = O}), (4)

where the summation is taken over all roots z(Vl( w) in D of the system f - w = O. The trace of the function H is a meromorphic function in G and is holomorphic if H is holomorphic.

Let 8 be a noncritical value of the mapping Ifl2 : D - IRn, z ~ (lf1 (zW, ... , Ifn(zW). In this case the skeleton rJ •t , defined by the condition IJjl2 = 8, is a smooth manifold. From the definition of the integral comes the following change of variables formula: if a polydisk, defined by the condition Iwl ~ 8, is contained in the domain G and its skeleton rw • t does not touch the image under f of the poles of the meromorphic function H, then

(2nifn Lj" H(z) dz = (2nifn Lw" (Tr H)(w) dw, (5)

where df = df1 A ..• A dfn and dw = dW 1 A ... A dwn .

We apply formula (5) to the case when the left-hand side is the Grothendieck residue, i.e., when D = Va and H = hjJ· f1 .. , J", with J = of/oz, the jacobian of the mapping f. To do this, we begin by observing that the following assertion is true; it follows directly from Proposition 1 of Chapter 2.

Proposition 2.1 (Khovanskij in Arnold et al. (1982». In the polydisk of radius j;., the trace of the function hjJ, meromorphic at the point a, has an integral representation

. -n f h(z) dz (Tr hjJ)(w) = (2m) TI~ [f.() - T

Fj " J=l J Z wJ

(6)

Consequently it is holomorphic.

Thus (Tr H)(w) = (Tr hjJ)· wI 1 ., . W;;-l and we obtain

Proposition 2.2 (Khovanskij in Arnold et al. (1982)). The local residue can be expressed using the trace by the formula

resJ (h) = (Tr hjJ)(O), (7) a

where J = of/ oz is the jacobian.

2.2. Algebraic Interpretation. The concept of the trace, which has just been introduced, has an algebraic interpretation in terms of extensions of fields. From this point of view, formulas (5) and (7), which we obtained using the transcendental definition of the residue as an integral over a manifold, are at the foundations of the definition of the residue in algebraic geometry (see Lomadze (1981), Beauville (1971)). Thus if K is a finite extension over the field of formal series E then we set ResK = ResE TrK/E' where ResK is the residue of an object (in our case, of a merom orphic function or form) associated with K while ResE(t/I) is the residue of the formal power series t/I = L c(a)wa, which is equal to c(-1, ... -1).

In our case the algebraic interpretation of the trace Tr hjJ allows us to give an improvement of an integral representation of Bishop (Bishop (1961)), which we present here. This concerns an integral representation of the merom orphic function hjJ in a polyhedron Il, c C Va, defined by the condition Ihl < E.

To do this we observe that if we set w = f(O in (6), then the residue on the right gives the value of the function hjJ at the point z = ( plus the sum of the values of this function at the preimages (V)(f(m, v = 2, ... , J1 of the point w = f( 0 (here J1 is the multiplicity of the zero a of the mapping f). In order to avoid summing at these J1 - 1 points, we introduce a nontrivial weight Q((, z) into the integrand in (6). This weight has the property that for any fixed (E Il, the function Q is equal to zero at all of the points z = (V)(f(m which are not equal to (. Since the analytic set {((, z) E Il, x Il, :f(O = f(z)} contains the set g = z} as an irreducible component, such a function Q exists. Its construction goes like this. Let w(O) be a noncritical value of the mapping f and let t/I(z) be a holomorphic function in lie for which all the values t/I(z(v)(w(O»)) are different. Then

Q((, z) = {vUz [t/I(z(v)(f(z))) - t/I(O] } {S>D;;'1 [t/I(Z(k)(f(Z))) - t/I(Z(S)(f(Z)))]}' s#k

Theorem 2.3. At each point z E Il, at which the jacobian J of the system f is different from zero, there is an integral formula for the meromorphic function hjJ

h(z) . -n f h(OQ(z, 0 d( J(z) . Q(z, z) = (2m) Ft., nj=1 [h(O - h(z)] , (8)

where rI " is the skeleton of the polyhedron Il,.

We remark that Q(z, 0 is a polynomial of degree J1 - 1 in t/I(z), the coefficients of which depend holomorphically on (. Besides this, Q(z, z) is a function of the form c(f(z)), where c(w) is holomorphic in the polydisk of radius E. Since the integral in (8) is holomorphic with respect to f(z), then from this formula we obtain the result that for any h in the ring of germs lDAz) in the variable z there is a representation

h(z) 0 -1 J(z) = co(f(z))t/I (z) + ... cr1 (f(Z))t/l1l (z), (9)


where Ck(W) E Ato(w) is merom orphic in a neighborhood of zero. Thus the family of merom orphic germs {hjJ, h E (9a(z)} is contained in the finite extension of the field At o(w) generated by the basis {I/Io, ... , I/I1'-1}. Therefore, the trace of the germ H = hi] is defined as the sum of the diagonal elements of the matrix Ilakj(w)1I which represents H in terms of the basis {I/I k}:

This trace coincides with the trace defined by formula (4). It is appropriate to compare Formula (9) with the Weierstrass Preparation

Theorem (see (Arnol'd et al. 1982)) according to which, for any h E (9a(z), there is a unique representation

h(z) = a1 (f(z))e 1 (z) + ... + ak(f(z))ek(z), aj E (9o(w), (10)

where {e)z)} is a basis of the local algebra (9aIIa(f) and k is its dimension. From (10) it is easy to conclude that k ;::: J1. (J1. is the geometric multiplicity, i.e., the degree of the mapping f at the point a). From (9) one derives the opposite inequality. Thus we obtain J1. = k (see Palamodov, 1967), where this equality is obtained from the properties of analytic sheaves; also see Arnol'd et al. (1982).

§ 3. The Total Sum of Local Residues

3.1. The Total Sum of Residues on a Compact Manifold. The Euler-Jacobi Formula. The local residue is a generalization of the residue of a merom orphic function hlf of one complex variable. We now present the analog of the theorem that the sum of all the residues of a meromorphic function equals zero (the sum is taken over the Riemann sphere or a compact Riemann surface). This analog is a special case of Theorem 6.2 of (Dolbeault (1985)), the case of a complete intersection. Let X be a complex manifold of dimension n and let D 1, ... , Dn be effective divisors on X, i.e., each Dj can be represented locally as a finite linear combination of irreducible analytic hypersurfaces with nonnegative coefficients. We assume that the intersection Z = D1 n··· n Dn is discrete. Consider a mero-morphic form w on X of degree n with pole divisor D = D1 + ... + Dn. This means that the form can be represented locally as w = h dzlf1 ... fn' where Jj is a function defining the divisor Dj • Thus for every a E Z a local residue is defined res w = res! (h). Then the following theorem on total sum of residues

a a

is true.

Theorem 3.1 (Griffiths (1976)). If X is compact, then

L res w = o. aeZ a


This theorem is proved by representing the local residue as an integral over the boundary oVa (see Proposition 1.2). In fact the result of theorem is a consequence of the homological dependence LaEZ Fa - ° of the cycles Fa that appear in the definition of the local residues.

We give some corollaries of Theorem 3.1. Let X = IClPn be projective space and let z = (z l' ... , zn) be local coordinates on one of the affine pieces of this space, which we will consider the "finite" part.

Corollary 3.2 (Griffiths (1976)). Let the system of algebraic equations ij(z) = 0, j = 1, ... , n have no root at infinity and let h(z) be a polynomial of degree deg h ::; degf1 + ... + degfn - n + 1. Then

L resf (h) = 0, aEZj a

where Z f is the set of roots in en of the system f = U1' ... ,fn) = 0.

The truth of this assertion follows from Theorem 3.1, for the condition on the degree of the polynomial h ensures that the form OJ = h dzjf1 .. .fn has no poles along the hyperplane at infinity.

Corollary 3.3 (Euler-Jacobi formula Griffiths-Harris (1978)). If under the hypotheses of Corollary 3.2 the system of equations f = ° has only simple roots (at each of the roots of this system the jacobian ofjoz is different from zero), then

L h(a) = 0. aEZj of (a)

oz

For n = 1 the Euler-Jacobi formula reduces to the Lagrange interpolation formula

h(a) L f'() = 0, deg h ::; deg f - 2. aEZj a

The Euler-Jacobi formula admits the following generalization. Let f =

U1' ... ,f,,), ij = L cjaza, be a system of Laurent polynomials with Newton polyhedra LJUd, . .. , LJUn) (see Chapter 2 or Khovanskij (1978)). Each face (fjt with normal vector t E [Rn of the polyhedron LJ(ij) corresponds to the following piece of the polynomial ij:

The system f is called nondegenerate if for any t E [Rn\ {a} the system J; = Ult' ... , fnt) have only simple zeroes in (IC\ {a} t.

Corollary 3.4 (Generalized Euler-Jacobi formula (Khovanskij (1978))). Let f = (f1' ... , fn) be a nondegenerate system of Laurent polynomials and let Zf be the set of zeroes of this system in (IC\ {a} t. If all the zeroes z E Zf are simple and if the Newton polyhedron of the polynomial h lies strictly inside the sum

Here the inclusion condition ,1(h) c ,1(f1) + ... + ,1(fn) is analogous to the inequality of degrees in Corollary 3.2. This condition permits one to extend a differential form meromorphic on (C\ {O})",

h dZ 1 dZn w=-----I\···I\-

f1··.fn Zl Zn '

to a suitable toroidal compactification without adding any poles.

3.2. Applications to Plane Projective Geometry. The Euler-Jacobi formula show that, in contrast to the one-dimensional case, not every set of points in the projective space of dimension n > 1 is the set of solutions of a system of n algebraic equations. As a corollary of the Euler-Jacobi formula, we will introduce some geometric illustrations of this phenomenon in the case of the plane (i.e., n = 2).

Theorem 3.5 (Cayley-Bacharach (see Griffiths-Harris (1978))). Let C and D be curves in CIP 2 of degrees I and k; and suppose the curves intersect in I· k points. Then any curve E of degree I + k - 3 which passes through I· k - 1 of the points of intersection enD must pass through alii· k of these points.

Theorem 3.6 (Pascal (see Griffiths-Harris (1978))). The pairs of opposite sides of a hexagon inscribed in a smooth conic Q, a curve of second order, intersect in three collinear points.

The proof of Pascal's theorem follows from the Cayley-Bacharach theorem: Let L l' ... , L6 be the lines containing the sides of the hexagon. We set C = L1 + L3 + L s, D = L2 + L4 + Ls and E = Q + P14P36 , where Pij = Li n Lj •

Then E passes through PS2 , a point of the intersection enD. The converses of these two theorems are also true (see Griffiths-Harris).

3.3. The Converse of the Theorem on Total Sum of Residues. Assertion 6.4 of (Dolbeault (1985)) shows that if the divisors Dj = lj are positive in the sense of Kodaira (see Griffiths-Harris (1978)), then Theorem 3.1 can be reversed. In the case X = ClPn we give a constructive proof of this fact. For this we take an affine piece of ClPn with coordinates Z such that this affine set contains the discrete set Z = Y1 n ... n Y". Then in order to construct the desired form t/J it is sufficient for any pair of points ai' aj E Z to find a polynomial h such that the following condition is satisfied for the form t/J = h dz/f1 ... f,. (lj = {fj = O}):

rest/J= -rest/J= 1; rest/J=O, aEZ\{ai,aj }; aj aj a

deg h + n < deg f1 + ... + deg in-

48 L.A. Aizenberg, AX. Tsikh, A.P. Yuzhakov

Using the Bergman-Weil integral representation, it is not difficult to convince oneself that for h one can take h(z) = Q(z, a;) - Q(z, aj), where Q(z, () is the determinant formed from the coefficients of the Heffer expansion of the polynomials/;,j = 1, ... , n.

3.4. Abel's Theorem and its Converse. It is known that the integral of a rational function r(x) of a variable x E 1C 1 is a rational logarithmic function, that is,

f r(x) dx = R(x) + L ak log(x - Xk),

where R(x) is a rational function. At the same time, an arbitrary abelian integral

u(P) = fP r(x, y) dx, Po

related to the curve V = U(x, y) = O} and the rational function r(x, y) is not of this type. If one considers the intersections Dt ' V = {Pv(t)} of the curve V with a family of curves Dt = {O(x, y, t) = O}, which depend rationally on t, then according to the classical theorem of Abel, the sum u(t) = La u(P.(t» is rational logarithmic. Thus the differential du(t) is equal to the sum

L t/I(Pv(t», t/I = r(x, y) dxl v , (11)

is rational. We will consider sum (11) in the multidimensional case, when V =

{f(x 1 , ••• , xn , y) = O} is a hypersurface in IClPn+1, defined in local coordinates x I' ... , xn , Y as the zeroes of a polynomial f without multiple factors. In this case t/I is a meromorphic n-form on V (for a precise definition of a merom orphic form on an analytic set, see Griffiths (1976); here, for simplicity, we may assume that V is smooth). The family Dt is the family {L} of all complex lines in IClPn+I, in other words the Grassmannian G(I, n + 1).

Consider in V x G(I, n + 1) the submanifold of flags (the pairs "point and a line passing through the point"):

I={(P,L):PcL}.

If the line L is described by an equation, L = {li = Xl - aly - bi = 0, ... , In = Xn - anY - bn = O}, then local coordinates are defined on the product IClPn+I x G(l, n + l)by x = (Xl' ... , Xn),Y, a = (aI' ... , an), b = (b I , ... , bn). Then we have

I = {x = ay + b,f(x, y) = O}.

Let n i : I ~ V and n2: I ~ G(1, n + 1) be the projections (such a pair of projections in integral geometry is called a "dual fibration"). Then the analog of the sum (11) is the following concept of the trace of a form

Tr t/I = (n2)*(n*t/I),


which for a line L in "general position" (that is, for which L· V consists of d different points Pv , where d is the degree of f) is written in the form,

Tr IjJ = IjJ(P1(L)) + ... + IjJ(PiL)).

We will show a formula for the trace of a holomorphic form IjJ on V. According to Chapter 1, Subsection 1.2, every such form is the Poincare residue of some rational form, i.e.,

p(X, y) I ljJ=of dX1 1\"'l\dxnv'

oy (x, y)

(12)

For a general line L = L(a, b) we set

L· V = {P1(a, b), ... , Pia, b)},

where the Pv(a, b) = (Xv (a, b), yv(a, b)) have distinct coordinates Yv = yv(a, b). Let

P(y) = p(ay + b, y), F(y) = f(ay + b, y);

then by a direct computation we obtain

{ y~IP(Yv)} Tr IjJ = ~ ± ~ F'(y.) daA 1\ dbAc,

where A = (i 1, ... , ik ) c {I, ... , n}, daA = dail 1\ ... 1\ daik; AC is the complement to A in the set of indices, and I A I = k is the cardinality of A.

Since on L the form di1 1\ '" 1\ din 1\ df = F'(y) dX1 1\ ... 1\ dXn 1\ dy, then from the Euler-Jacobi formula applied to the system (i1, ... , in,f) we have

ytA1p(yv) L '() =0, IAI=O,I, ... ,n, v F Yv

if deg P ~ d - n - 2. Consequently, for the given bounds on the degree, the trace Tr IjJ == O. One can show (Griffiths (1976)) that every merom orphic form IjJ on V whose trace is holomorphic on G(I, n + 1) has the form (12), where deg p ~ degf - n - 2 (the trace equals zero, since on G(I, n + 1) there are no nontrivial forms). Forms of this type are called forms of the first kind. Thus, Abel's Theorem for forms of the first kind becomes the statement that

IjJ(P 1 (L)) + ... + IjJ(PiL)) == O.

We remark that the left side of this identity has a local character; that is, it can be defined for arbitrarily small pieces V. of a hypersurface V, with each piece intersecting some fixed line at a single point. Moreover, this identity characterizes those collections of pieces of analytic sets v which extend to a global algebraic hypersurface V. Namely, let some given pieces Vi> ... , Y.J be irreducible n-dimensional complex analytic sets in CJP>n+l, along with merom orphic forms IjJv =1= 0 on Vv' We assume that there exists a line Lo intersecting each V. in a single simple point which is not a pole for IjJv' In such a case, for lines L in a


neighborhood U(Lo) c G(I, n + 1), one can define the trace

Then the following is true:

Theorem 3.7 (Griffiths (1976)). If the trace Tr{ljI} == 0, then there exists an n-dimensional algebraic set V c CIPn+1 and a rational n-form ljI of the first kind such that v" c V and ljIlv, = ljIv'

Here is an outline of the proof of this theorem. Let fv be the defining functions for v,,; then ljIv = Resv If'v is the Poincare residue of the form If'v = gv dx A

dylfv. We denote by I~ = {(P, L): PeL c W} the part of the flag manifold lying over the neighborhood We CIPn+1 of the line L o. On Iw we consider the form

where Pv = (xv (a, b), yv(a, b)) are the points of intersection L(a, b)' v", lk = Xk -akY - bk, and res OJv is the local residue of the form OJv = If'vill" .In.

P,

The condition Tr{ ljIv} == 0 implies that cP descends to W under the projection n1 : Iw -+ W; this means that cP = nflf', where If'is a meromorphic (n + I)-form on W. Moreover, using the fact that every function which is meromorphic in a neighborhood of a line in CIPn+1 is a rational function, we conclude that If'is rational. Its pole set is the desired hypersurface V, and the form ljI is the Poincare residue Resv If'.

3.5. Residue Theorem for Vector Bundles. It is known that to every divisor D on X with local data fa E At*(Ua ) there corresponds a line bundle L = [D] with transition functions {gaP = falfp}. The local residue res OJ of the form OJ with

a pole divisor D = Dl + ... + Dn at the point a E Z = D = Dl n'" n Dn can be viewed as the residue with respect to the holomorphic section f = fl (z)e 1 + ... + fn(z)en of the vector bundle E = [D 1 ] EB'" EB [Dn], where {ed is a frame. Proceeding with this analogy, we define the local residue relative to a holomorphic section with an isolated zero for any vector bundle of rank n. Let X be a complex analytic manifold of dimension n and let E -+ X be a vector bundle of rank n. Consider the vector space HO(X, (9(K ® det E)), where K = /\ n Tx*' is the highest exterior power of the holomorphic cotangent bundle. An element of this space can be written in terms of a local holomorphic frame e1 ,

... , en for E and local holomorphic coordinates z = (z 1, ... ,zn) in the form ljI = h(z)(dZI A ..• A dzn) ® (e 1 A •.• A en). To the element ljI and the section f = fl(z)e 1 + ... + fn(z)en we associate the differential form

ljI h(z) dZ I A •.. A dZn f fl(Z) ... f,,(z)

(13)


Under the assumption that the section f has an isolated zero at the point a E X, we define the residue

(14)

Although the right side of (13) depends on the choice of frame and coordinates, according to the transformation formula (3), the residue (14) does not depend on these choices. A theorem about the total sum of residues is also true.

Theorem 3.8 (Griffiths-Harris (1978)). Let E ~ X be a holomorphic vector bundle of rank n over a compact complex analytic manifold of dimension n, and let f E HO(X, (I)(E)) be a holomorphic section haVing a discrete set of zeroes Z.Jf for IjI E HO(X, (I)(K ® det E)) and a E Z we define the residue res (1jI/f) by formula (14), then a

L res (-fiji) = o. aeZ a

Griffiths and Harris proved that for residues relative to hoi om orphic sections the converse of the theorem on total sums of residues is also true (Ann. Math. 108, No.3, 461-508 (1978)).

3.6. The Total Sum of Residues Relative to a Polynomial Mapping in cn. Let h, f1' ... , fn be polynomials in the ring C [z] = C [z l' ... , Zn] such that the set ZJ = {z E en :f1(z) = ... = fn(z) = o} is discrete. We consider the total sum of residues (the global residue) of the polynomial h relative to the polynomial mapping f = (f1' ... , fn):

ResJ(h) = L resJ (h). aEZf a

For the global residue ResJ the transformation formula and duality remain true.

Proposition 3.9 (Transformation formula for the global residue (Tsikh, Yuzhakov (1984))). If g = (g1' ... , gn), with gj = L tPjdk, tPjk E C[z], and if g-1 (0) is discrete, then

(15)

Proposition 3.10 (Global duality (Tsikh, Yuzhakov (1984))). Let I(f) be the ideal in the ring C[z] generated by a system of polynomials f = (fl' ... , In) with a discrete set of zeroes in en. Then the polynomial F belong to the ideal I(f) if and only if ResJ(FtP) = 0 for every tP E C[z].

We remark that from the transformation formula for the global residue one can deduce formula (15) of Chapter 2, by applying it to a system f with no zeroes at infinity. In Tsikh Yuzhakov (1984) Yuzhakov used the transformation formula (15) and elimination theory to produce an algorithm for the computation of an arbitrary global residue. Global duality is applied in § 4 to the problem of the inclusion of F E I (f).


§ 4. Application of the Grothendieck Residue to the Algebra of Polynomials and to the Local Ring (!)a

In this section we will apply the Grothendieck residue (principally local and global duality) to the problem of determining whether or not a given polynomial F(z) belongs to the ideal I(!) generated in the ring of polynomials C [zJ =

C[Zl' ... , znJ by the system of polynomials f = (fl (z), ... , fn(z)). In other words, when can F be represented in the form

(16)

where the qj are polynomials? We will also consider the analogous question in the local ring (9a. In the polynomial case we will examine two approaches: in the first approach it will be assumed that the zeroes of the system f are known (Subsection 4.1, Subsection 4.2) and in the second we will not assume they are known (Subsection 4.4).

4.1. Macauley's Theorem. The application of local duality to the question of equation (16) is most transparent in the case when the Jj are forms (homogeneous polynomials).

Theorem 4.1 (Macauley). If the system of forms f ;= (fl (z), ... , fn(z)) has an isolated zero at the coordinate origin, then every form of degree greater than deg fl + ... + deg fn - n belongs to the ideal I(f).

The proof follows immediately from local duality if we observe that every differential n-form P dz/Q, where P and Q are homogeneous polynomials, is exact ifdeg P + n of. deg Q.

4.2. Noether-Lasker Theorem in cpn. The well-known Noether-Lasker theorem in IC" asserts that for a polynomial FequatioQ(16) is satisfied with polynomial coefficients qj (i.e., F E I(f)) if this equation is satiSfied at every point a E Zf =

f- 1 (0) for some holomorphic germs qj = qja (that is, if the local Noether condition is fulfilled: F E Ia(f), a E Zf' where IA!) is the ideal generated by the system f in the ring of germs (9a). It turns out that in the case when the system f has no zeroes at infinity in the projective space, then the local Noether condition ensures that equation (16) is valid for qj with deg qj + deg Jj s deg F,j = 1, ... , n. This assertion can be formulated in the language of forms in homogeneous coordinates on projective space. Namely, let ,= ('0' '1' ... , 'n) be homogeneous coordinates on the projective space cpn and let 1 = (11 (0, ... , J..(O) be a system of homogeneous forms which has in this space a discrete set of zeroes Z J. For each a E Z J (let a = (ao, a1 , ••• , an) and for definiteness, assume that ao of. 0) denote by Ia(f) the ideal in (9a' the ring of germs in the variables Zj = 'j/(o generated by the system fez) = 1(1, z 1, ... , zn). In this case, the following theorem is true.


Theorem 4.2 (Noether-Lasker in CiP>n (see Griffiths, Harris (1978) and Tsikh (1984)). If a homogeneous form F(O satisfies the local Noether condition, F(z) E

IaU), at each point a E Z j in the local coordinates z, then F E I(f).

We remark that the system 1 does not have isolated zeroes in the space cn+1 of the variables ,. Therefore, local duality does not apply to it directly. However, the local Noether conditions ensure that the global residue is equal to zero (we assume that z is a local coordinate system in an affine piece that contains Z j)

ResJ(F·g)= L resJ(F·g), gEC[Z]. aeZj a

This residue admits "localization" (see Tsikh (1984), which explains the link between arbitrary residues in CiP>n and local residues in cn+1) in the form of a local residue in cn+ 1 with respect to the extended system (/> = ('0',11 (0, ... , ],,(0):

ResJ(F· g) = res(/> (F· g), o

where m is some number depending on the degree of the polynomials F and g. Thus the proof of Theorem 4.2 reduces to local duality. In Griffiths, Harris (1978), in the proof of this theorem in the case n = 2 this "localization" is replaced by the Kodaira Vanishing Theorem and Kodaira-Serre duality.

4.3. Verification of the Local Noether Condition. In order to apply the Noether-Lasker Theorem, it is useful to have criteria for the fulfillment of the local Noether condition. We will now state several criteria in terms of the multiplicity )laU) at the zero a of the mapping f and dAF), the order ofthe zero at the point a of the function F under consideration. This number is defined to be the order of the lowest-order derivative of the function F which does not vanish at a.

The crudest of these criteria is the following:

Proposition 4.3 (see Arnold et al. (1982)). If da(F) 2 )lAf), then F E IaU).

A more precise criterion is the following generalization of a result of Bertini (1889).

Theorem 4.4 (Tsikh (1984)). Let jj = Pj + ... , j = 1, ... , n, where Pj is the homogeneous polynomial of lowest degree in the Taylor expansion of the function jj at the point a, with dj = deg Pj the order of the zero of the function jj at a. If for some j E {1, ... , n} the set of solutions of the system of equations

P1 (z) = ... = Pj-dz) = Pj+1(z) = ... = Pn(z) = 0

consists of a finite number of rays in cn and the order

then F E IAf).


As a corollary of this theorem we state the following assertion:

Proposition 4.5. Just as in Theorem 4.4, let

jj = Pj + "', j = 1, ... , n,

where the system P = (PI' ... , Pn ) has a unique zero z = a. If dAF) > d l + ... + dn - n, then F E Ia(f).

Proposition 4.5 follows from Theorem 4.4, since in this case it is a consequence of Theorem 9 of Chapter 2 that the multiplicity Ila(f) is equal to the product of the orders d l ... dn • In turn, this proposition generalizes Macauley's Theorem.

The proof of Theorem 4.4 is obtained by using local duality and a sigma process at the point a, as a result of which the system f is transformed into a system f* with a finite number of zeroes {a.} on the "blown-up" hyperplane. Then one applies Proposition 4.1 to the system f* at each zero a. for which the multiplicity llaJf*) ::;; Ila(f) - d l ... dn·

4.4. A Consequence of Global Duality. Now let us consider the problem of representing F in the form (16) in the case when the zeroes of the system fare unknown. In this case it is appropriate to use global duality (see Section 3.6). It turns out that to verify whether the polynomial F belongs to the ideal I(f) it is sufficient to compute the global residue for only a finite set of polynomials rjJ E lC[z]. The following assertion explains how to choose these polynomials. Let

n

jj(O - jj(z) = I Pjk(" z)· ('k - zd, j = 1, ... , n, k=l

be the Hefer expansion of the polynomials jj and let Q(" z) = detllPjkll. The determinant Q can be represented in the form

L

Q(" z) = I gk(O' hk(z). (17) k=l

Theorem 4.6. If the system of polynomials f = (fl' ... , fn) has a discrete set of zeroes, then the polynomial F belongs to the ideal I(f) if and only if Resf(F' hk) =

0, k = 1, ... , L, where the hk are the polynomials in the expansion (17). If the mapping w = f(z) is proper, then the family of polynomials {hk(z) h= I,L generates the ring lC[z] viewed as a module over lC[w], i.e., for every hE 1C[z] there is a representation

moreover, this family forms a basis of the IC [w J-module IC [z] if and only if


Bibliography

The results of § 5 of the second chapter are contained in Aizenberg (1983). The monograph by Aizenberg-Yuzhakov (1979) contains an exposition, from one point of view, of integral representations of holomorphic functions of several complex variables and the theory of multidimensional residues, including the logarithmic residue and its applications, in particular the results of§§ 1, 2 and subsections 3 and 4 of Chapter 1 and §§ 2-4 of Chapter 2 of this survey. In the first volume of the book by Arnol'd et al. (1982, 1984) there is a proof of the equality of the algebraic multiplicity of a holomorphic mapping and its topological degree. This fact was proved earlier in Palamodov (1967) using the theory of analytic sheaves. Morever, in Arnold et al. (op. cit.) local duality for the Grothendieck residue is proved; propositions 2.1 and 2.2 of Chapter 3 and also proposition 4.5 of Chapter 3 are stated.

The second volume of the book Arnol'd et al. (op. cit.), examines integrals depending on a parameter and works out a technique for computing monodromy and the asymptotics of integrals. In Varchenko (1983) the asymptotics of integrals are investigated by the saddle-point method. This is equivalent to the study of the asymptotics of integrals over vanishing cycles of holomorphic forms which depend continuously upon parameters; it gives a definition of the mixed Hodge structure on the cohomology vanishing at a critical point of a holomorphic function. The survey by Golubeva (1976) is devoted to the study of the analytic properties of Feynmann integrals, their singularities (Landau manifolds) and their branching properties. The monograph by Egorychev (1977) develops a method of determining generating functions and the computation of combinatorial sums using integral representations and residues. The work by Krasnov (1972) studies the cohomology of complexes of meromorphic forms on a Kahler manifold, forms whose poles lie on a fixed submanifold; here one investigates the link between the order of the pole of these forms and the type of the class-residue depending of the infinitesimal neighborhood of the submanifold. Theorem 1 of Chapter 2 was taken from Kushnirenko (1975), which also includes other results about the multiplicity of a zero of a holomorphic mapping and the Newton polyhedron. In Lomadze (1981) and Beauville (1971) the algebraic concept of residue is introduced and examined. The generalized EulerJacobi formula of Chapter 3 is stated in Khovanskij (1978). Theorems 4.2 and 4.4 in Chapter 3 are proved in Tsikh (1984) where, also, more general local residues are studied than in Chapter 3. The algorithm for computing the total sum of local residues relative to a polynomial mapping is stated in Tsikh-Yuzhakov (1984).

In Yuzhakov (1975), using the multiple logarithmic residue, formulas are given for the expansion of implicit functions into power series; this generalizes the expansion of Biirmann-Lagrange. The article by Aizenberg et al. (1983) is devoted to applying the results of § 3 of Chapter 2 to some mathematical problems of chemical kinetics. In Andreotti-Norguet (1964) there is a formula generalizing the Bochner-Martinelli integral representation, the so-called formula of AndreottiNorguet. The work by Bertini (1889) contains Theorem 4.4 of Chapter 3 in the case n = 2. The article by Bishop (1961) is about holomorphic mappings of analytic spaces; it uses several variants of formula (8) of Chapter 3 and also formula (7) of Chapter 2. The latter formula was also in (Caccioppoly (1949); Martinelli (1955); Sorani (1962).) The goal of the work by Carrell (1978) is to obtain a direct proof of the representability of several integrals on a compact complex manifold X by means of Grothendieck residues with respect to a merom orphic vector field with isolated zeroes on X and then to apply this to prove the theorem of Bott and Baum on residues. The monograph by ColelT-Herrera (1978) is devoted to residue-currents generalizing the Grothendieck residue considered in Chapter 3. Moreover, in particular, there is an assertion close to Theorem 4 of Chapter 2, though the stated formulation of the theorem was taken from Solomin (1977). In the work by Fotiadi et al. (1965) the theorem of Froissard on expansions is presented (see Chapter 1) as well as an application of the Thorn isotopy theorem to the study of integrals depending on parameters. The Thorn theorem itself is proved in the work of Mather (see the footnote on page 17). In Gordon (1974) the concept of geometrical residue is introduced as an element of the kernel


where S is an analytic subset of arbitrary codimension of a manifold X; this generalizes the exact sequence of Leray. The article by Griffiths (1969) is the fundamental work on the residues of rational functions; it describes a procedure for bringing these integrals to canonical form, and it studies problems of algebraic cycles on projective manifolds and inversion of generalized abelian integrals. In Griffiths (1976) the generalized theorem on the total sum of residues is stated (see Chapter 3) and this is applied to the multidimensional Abel's theorem. The book by Griffiths-Harris (1978) contains a systematic exposition of local residues and their application to algebraic geometry. In particular, it contains the algebraization of local and global duality in the language of the Ext functor. As a result of this, the analytic definition of the local residue appears in invariant form (not depending on local coordinates and the choice of generators of the ideal). Moreover, in this book are described many of the conceptual and technical methods useful in working with multidimensional residues.

Theorem 6 of Chapter 2 is contained in the paper by King (1971), and Theorem 5 is contained in Le10ng (1968). All these results and their generalizations (Theorems 5, 6, 5', 6' of Chapter 2) are found in the book by Griffiths-King (1973), which is about Nevanlinna theory for holomorphic mappings of algebraic manifolds.

The work by Hartshorne (1966) is devoted to the algebraic computation of residues as morphisms of the ring (i)x associated to an algebraic set X. It contains a proof of (Grothendieck) duality for proper schemes over fields, with arbitrary singularities; this generalizes the local and global duality of Chapter 3 and also Serre duality.

Leray (1959) developed the theory of residues on a complex analytic manifold; he introduced the concept of form-residue and the exact (co)homological sequence associated with a submanifold of codimension 1 and proved the existence of the class-residue and the residue formula (simple and multiple).

Theorem 4 of Chapter 2 is in Lupacciolu (1979); formulas (5) and (6) of this chapter are in Roos (1974); example 3 of § 1, Chapter 2, is taken from Wirtinger (1937).

Norguet (1959 and 1971) generalized the theory of residues of Leray to the case of sub manifolds of codimension k > 1 and also to closed subsets of a locally compact space. He introduced the concept of partial derivatives for differential forms.

In the book by Pham (1967), integrals depending on a parameter are investigated. The majority of the results of § 3 of Chapter 2 were taken from this book: the isotopy theorem, the concept of a Landau set, the Picard-Lefschetz theorem, etc.

Although individual fragments relating to multiple residues are encountered in works by Jacobi and Picard, one should consider the true beginning of the theory of multidimensional residues to be the memoir of Poincare (1887), where he generalized the Cauchy integral theorem to functions of two complex variables and investigated several cases of the residues of rational functions of two variables and proved that they can be expressed as the periods of abelian integrals. Moreover, using double residues he established a generalization of the Lagrange expansion for functions of two variables that had been stated by Stieltjes.

In Tong (1973), apparently for the first time, the Grothendieck residue was considered as the integral (1) of Chapter 3 and an integral proof was given for the transformation formula and local duality.

References *

Aizenberg, L.A. (1983): Application of multidimensional logarithmic residue to represent the difference between the number of integer points in a domain and its volume in the form of an integral. Dokl. Akad. Nauk SSSR 270, No.3, 521-523. Engl. transl.: Sov. Math., Dokl. 27, 615-617 (1983), Zb1.535.l0047

* For the convenience of the reader, references to reviews in Zentralblatt fUr Mathematik (Zbl.), compiled using the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (Jbuch) have, as Jar as possible, been included in this bibliography.


Aizenberg, L.A., Yuzhakov, A.P. (1979): Integral Representations and Residues in Multidimensional Complex Analysis. Novosibirsk: Nauka, 335 pp. Engl. trans!': Transl. Math. Monogr. 58. Providence: Am. Math. Soc., 283 pp. 1983, Zbl.445.32002

Aizenberg, L.A., Bykov, V.I., Kytmanov, A.M., Yablonskij, G.S., (1983): Search for all steady-states chemical kinetic equations with the modified method of elimination. Chern. Eng. Sci. 38, No.9, 1555-1567

Andreotti, A., Norguet, F. (1964): Probleme de Levi pour les classes de cohomologie. C. R. Acad. Sci., Paris, Ser. A 258, No.3, 778-781, Zb1.l24,388

Arnol'd, V.I., Varchenko, A.N. Gusein-Zade S.M. (1982, 1984): Singularities of Differentiable Mappings. I, Moscow: Nauka. Engl. transl.: Boston: Birkhiiuser. 382 pp (1985). Zb1.513.58001 II, Moscow: Nauka. 336 pp. Engl. transl. Boston: Birkhiiuser 1988, Zb1.545.58001

Beauville, A. (1971): Une Notion de Residue en Geometrie Analytique. Lect. Notes Math. 205, 183-203, ZbI.236.32004

Bertini, E. (1889): Zum Fundamentalsatz aus der Theorie der algebraischen Funktionen. Math. Ann. 34,447-449, Jbuch21,426

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II. Plurisubharmonic Functions

A. Sad ullaev

Translated from the Russian by P.M. Gauthier

Contents

Introduction ................................................... 61

Chapter 1. Elementary Theory of Plurisubharmonic Functions ....... 62

§ 1. Subharmonic Functions ..................................... 62 1.1. Definition and Basic Properties ........................... 62 1.2. The Poisson Integral .................................... 63 1.3. Polar Sets ............................................. 64 1.4. Approximation of Subharmonic Functions ................. 64 1.5. The Riesz Representation ................................ 65 1.6. Hartog's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

§ 2. Plurisubharmonic Functions and Their Elementary Properties .... 66 2.1. Approximation ......................................... 67 2.2. The Operator dd c ••.•••.•••••••••••••••••••••••••••••••• 68 2.3. The Upper Envelope of psh Functions ..................... 68 2.4. Connection with Holomorphic Functions .................. 69

Chapter 2. Complex Potential Theory ............................ 70

§ 1. The Complex Monge-Ampere Operator ....................... 71 1.1. The Operator (ddc)" ••••••••••••••.••••••••••••••••••••.• 71 1.2. Integral Estimates ...................................... 72 1.3. The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

§ 2. Extremal Functions and Capacity ............................. 75 2.1. .'?J!-measure ............................................. 75 2.2. Condenser Capacity ..................................... 77 2.3. Solution of the First Lelong Problem ...................... 79

§ 3. Capacitary Properties of psh Functions ........................ 80 3.1. C-Property of psh Functions ............................. 80 3.2. Convergence of (ddC)k .................................... 82

60 A. Sadullaev

3.3. The Structure of the Irregular Points ...................... 83 3.4. Capacitability of Borel Sets .............................. 85

Chapter 3. Applications of Complex Potential Theory ............... 87

§ 1. Rational Approximation and Pluripolar Sets ................... 87 1.1. The Maximum Principle for Pseudo concave Sets ........... 88 1.2. Pluripolarity of Pseudoconcave Sets ...................... 88 1.3. Some Properties of the Class RO .......................... 89 1.4. Further Properties of Pseudoconcave Sets ................. 90

§ 2. Holomorphic Extension in a Fixed Direction .... . . . . . . . . . . . . . . 92 2.1. Analyticity of the Singularity Set ......................... 92 2.2. Pseudo concave Sets and Analytic Tubes ................... 94

§3. Multidimensional Analog of the Bernstein-Walsh Theorem and Separately Analytic Functions ............................... 94 3.1. The Generalized Green Function ......................... 94 3.2. The Main Result ....................................... 96 3.3. Green Functions for Circled Sets. Projective Capacity ....... 97 3.4. Separately Analytic Functions ........................... 100

References ..................................................... 104

II. Plurisubharmonic Functions 61

Introduction

Plurisubharmonic functions playa major role in the theory of functions of several complex variables. The extensiveness of plurisubharmonic functions, the simplicity of their definition together with the richness of their properties and. most importantly, their close connection with holomorphic functions have assured plurisubharmonic functions a lasting place in multidimensional complex analysis. (Pluri)subharmonic functions first made their appearance in the works of Hartogs at the beginning of the century. They figure in an essential way, for example, in the proof of the famous theorem of Hartogs (1906) on joint holomorphicity.

Defined at first on the complex plane IC, the class of subharmonic functions became thereafter one of the most fundamental tools in the investigation of analytic functions of one or several variables. The theory of subharmonic functions was developed and generalized in various directions: subharmonic functions in Euclidean space IRn, plurisubharmonic functions in complex space en and others.

Subharmonic functions and the foundations ofthe associated classical potential theory are sufficiently well exposed in the literature, and so we introduce here only a few fundamental results which we require. More detailed expositions can be found in the monographs of Privalov (1937), Brelot (1961), and Landkof (1966). See also Brelot (1972), where a history of the development of the theory of subharmonic functions is given.

The theory of plurisubharmonic functions is connected with analytic functions of several variables. The foundations of the theory were laid in the 40's and 50's while studying properties of domains of holomorphy and entire and meromorphic functions. Of primary importance were the works of Oka (1942, 1953), Lelong (1941, 1957, 1966), and Bremermann (1956, 1959). These results are presented fairly completely in the monograph of Vladimirov (1964).

The subsequent development of the theory of plurisubharmonic functions was connected with the elaboration of a complex potential theory encompassing capacitary properties of plurisubharmonic functions. Interest in this direction has increased particularly in recent years thanks to the connection, established by Bedford and Taylor, between extremal plurisubharmonic functions and solutions of the complex Monge-Ampere equation (ddcut = O. At the present time there are already a large number of papers devoted to both complex potential theory itself as well as to its applications.

The aim of the present paper is to, firstly set forth the foundations of the theory of plurisubharmonic functions and, in particular, the complex potential theory (Chap. 2), and secondly, to exhibit some of its applications to multidimensional complex analysis (Chap. 3). Our exposition of complex potential theory is, as much as possible, self-contained. To this end, we, first of all in Chap. 1, introduce some necessary results from the theory of subharmonic functions. The further development proceeds without invoking more profound theorems of classical potential theory.

62 A. SaduJlaev

Fundamental applications of complex potential theory are related to rational and polynomial approximation and problems of analytic continuation. As in the classical case n = 1, at the core of such applications lie estimates connected with extremal plurisubharmonic functions. The multidimensional problems which are considered in this context have deep connections with the classical works of Bernstein, Walsh, Keldysh-Lavrentiev (1937), Gonchar (1972, 1974) and others.

Chapter 1 Elementary Theory of Plurisubharmonic Functions

§ 1. Subharmonic Functions

1.1. Definition and Basic Properties. A function u, -00 :::;; u(x) < +00, defined in a domain D of Euclidean space ~n, is said to be subharmonic in D if

a) u is upper semicontinuous in D, i.e. lim u(x) :::;; u(XO) for an arbitrary point x~xo

XO ED; b) for each point XO ED, for sufficiently small r > ° (r :::;; r(xO)), the value

u(XO) does not exceed the average value of u on the sphere S(XO, r):

u(XO) :::;; -k f u(x) d(J, (1) (Jn r S(xO,r)

where (In is the area of the unit sphere S(O, 1) in ~n. From condition a) it follows that the set {x ED: u(x) < t} is open for each

t E ~. Moreover, on each compact subset KeD, the function u attains its maximum. From this together with condition b), we easily conclude the following:

Theorem (maximum principle). Suppose u is subharmonic in a domain D c ~n

and attains a maximum at some point XO ED: u(XO) = sup u(x). Then u == const in XED

D.

The following properties follow immediately from the definition of subharmonic functions:

1) A linear combination a1 U 1 + ... + arnUm of subharmonic functions uj with nonnegative constant coefficients (aj ~ 0) is a subharmonic function.

2) The maximum of finitely many sub harmonic functions, u(x) = sup{ U 1 (x), ... , urn (x) }, is also subharmonic.

3) The limit of a uniformly convergent or monotonically decreasing sequence of subharmonic functions is subharmonic.

We remark that for a harmonic function v, the functions v and -v are both sub harmonic. The converse is also true: if v and - v are both subharmonic

functions, then v is necessarily harmonic. From this and from the maximum principle, it follows that if u is subharmonic and v is harmonic, then if on the boundary of the domain D, we have ulw ::; VI"D' then we have the same inequality, u(x) ::; vex), everywhere in D. This relationship to harmonic functions actually characterizes subharmonic functions and in some textbooks is given as the definition of a subharmonic function.

1.2. The Poisson Integral. If u(x) is harmonic in the ball

B(xO, r): Ix - xOI < r

and continuous in the closure Ii, then the values of the function u in the interior B can be expressed in terms of the values of u on the boundary vB = S(XO, r) via the Poisson integral

u(x) = f u(y).o/'(x, y) drr(y) S(xO.r)

(2)

where

r2 -Ix - xOl 2

~(x, y) = I In rrnr x - y

is the Poisson kernel. Formula (2) can also be used for the harmonic extension into the ball, of a continuous function qJ given on the boundary aB. The integral

u(x) = f qJ(x, y)~(x, y) drr(y) S(xO.r)

is the solution to the classical Dirichlet problem in the ball B, since u is harmonic in B and continuous on Ii and UIr'B = qJ.

In the sequel we shall repeatedly make use of the following technique for studying subharmonic (and also plurisubharmonic) functions. Consider, in a domain D, a subharmonic function u(x) and take its trace uls on the boundary S of some ball BcD. Since uls is upper semicontinuous (on S), there exists a decreasing sequence of continuous functions qJj Luis. For each j, let Vj be the Poisson integral of the function qJj. Then the vj , being a decreasing sequence (as j ...... CfJ), converge to the Poisson integral of u,

vex) = Is u(y).o/'(x, y) drr(y).

The function vex) is harmonic in B and its boundary values agree with ul s almost everywhere on S.

From this follows the interesting consequence that for an arbitrary subharmonic function u, the average

1 f --..=t u(x) drr rrn r S(xO.r)

is an increasing function of r (for r < dist(xO, aD).

64 A. Sadullaev

Indeed, if v is the Poisson integral for uIS(xo,r) and r 1 < r, then

-k f u(x) da ::; -k f v(x) da = v(XO) anr 1 S(xO,rd anr 1 S(xO,r,)

1 f 1 f = ------,;=r v ( x) da = ------,;=r u (x) d a. anr S(xO,r) anr S(xO,r)

From this monotonicity it is not difficult to obtain also the monotonicity of the averages over balls

1 f - u(x) dV, ~rn B(xO,r)

where ~ is the volume of the unit ball B(O, 1). Thus, from condition b) on subharmonic functions, we have the inequality

u(XO) ::; ~ f u(x) dV ::; -k f u(x) da, (3) Vnr B(xO,r) anr S(xO,r)

and moreover, in view of the semicontinuity, both integrals in (3) converge to u(XO) as r ~ 0.

1.3. Polar Sets. A set E c D, on which some function u, subharmonic with u ¢ -00 in D, takes the value -00, is called a polar set. In classical potential theory, it is shown that a set E is polar if and only if it has outer Newtonian capacity zero. In particular, for n > 2, metric dimension (n - 2) separates polar and non-polar sets (for smaller dimension, polarity; for larger, non-polarity). The proofs of these facts are non-elementary and we shall neither present them nor refer to them. From (3) easily follows the weaker (but sufficient for us) assertion:

Theorem. Polar sets have zero Lebesgue measure.

Indeed, from (3) it follows that if u ¢ -00 is subharmonic in a domain D, then u is locally integrable. Consequently, the set E: u(x) = -00 has zero Lebesgue measure.

1.4. Approximation of Subharmonic Functions. For approximation, it is usual to make use of some kernel K(x) ~ ° of class C'Xl1, whose support lies in the unit ball B(O, 1), normalized such that J K(x) dV = 1. As K(x) it is convenient for us to take, for example, the function

K(x) = n , - , {ct e1/(jxj2_1) for Ixl < 1

0, for Ixl > 1

1 Here and subsequently, Ck, 0 :s; k :s; 00, denotes the class of k times continuously differentiable functions (Co = C is the class of continuous functions).


with an appropriate constant an' For a function u subharmonic in D, we define

uo(x) = f u(x + (jy)K(y) dV = ;n f u(y)K (y ~ x) dJl, (j > O. The first integral

clearly represents a subharmonic, and the second, an infinitely smooth function. Consequently, uo(x) is a subharmonic function of class COO.

In contrast to the situation for an arbitrary summable function, for a subharmonic function u, the sequence Un monotonically decreases as (j! 0, and converges to u at each point XED. This follows easily from what was said at the end of 1.2.

1.5. The Riesz Representation. For functions of class C2, subharmonicity is

. I . .. f h I a2 u a2 u C2() . eqUlva ent to pOSItivIty 0 t e Lap acian Llu = ~ + ... +~; u E D IS uX 1 uXn

subharmonic in a domain D if and only if Llu :;::: 0 in D. Indeed, substituting

the Taylor formula for u(x) (about XO) in the integral ~ f u(x) da, we anr S(xO,r)

obtain

where C is a constant independent of r. The validity of the criterion Llu :;::: 0 clearly follows from this formula and property b).

If u =1= -00 is an arbitrary (not necessarily twice smooth) subharmonic function in a domain D, then u is locally sum mabie in D (see 1.3) and, consequently, is a distribution. Let Llu be its Laplacian in the distributional sense. By approximating u by infinitely smooth subharmonic functions, we see that L1u :;::: O. Since every positive distribution is a measure, the distributional Laplacian L1u is a (positive) measure on D. This measure is called the measure associated with u.

The associated measure Llu allows us to associate, with a subharmonic function u, a corresponding potential with respect to the Newtonian (Logarithmic for n = 2) kernel

j21n lnlxl, K(x) =

r(n/2 - 1) 1 - 4nn/2 'lxln - 2 '

for n = 2,

for n > 2.

We remark that if /1 is any finite Borel measure in [Rn, then its potential

UIl(X) = f K(x - y) d/1(Y)

is a subharmonic function with Laplacian Llu = /1. Consequently, if we consider the restriction /1G of the associated measure /1 = Llu to some subdomain Gee

66 A. Sadullaev

D, we obtain that the difference

u(x) - f K(x - y) d/lG(y)

is a harmonic function in G. Thus, we arrive at the following Riesz Representation Theorem.

Theorem. Let u be a function sub harmonic in a domain D c Ik£n. Then on any relatively compact domain Gee D, u has a representation of the form

u(x) = UI'(x) + h(x),

where UI' is the potential of some measure /l and h is a function harmonic in G.

Corollary. If a distribution u is such that Au ~ 0, then it is given by a subharmonic function.

1.6. Hartog's Lemma. Suppose in a domain D that g(x) is a continuous function and uj(x) is a sequence of locally uniformly upper-bounded subharmonic functions such that

lim uj(x) :::; g(x) (4) j-OJ

at each point XED. Then, on any compact set KeD, inequality (4) holds uniformly; that is, for each B > 0 there exists an integer jo such that uix) :::; g(x) + B, for each x E K and each j ~ jo.

The proof of this very important lemma is not difficult (cr. e.g. Shabat (1976». It fol1ows easily from the fol1owing inequalities which result from (3) and (4):

lim uixO) ~ lim ~ f uj(x) dV :::; ~ f g(x) dV, XO E K, r > O. j~CX) j~CX) v"r B(xO.r) v"r B(xO,r)

§ 2. Plurisubharmonic Functions and Their Elementary Properties

The concept of a plurisubharmonic (psh) function is connected with the complex structure of e: a function u(z) is said to be plurisubharmonic in a domain Dc eif

a) it is upper semicontinuous in D and b) the restriction u II is subharmonic on 1 (\ D for every complex line 1. It fol1ows immediately from the definition that a psh function is simultane

ously a subharmonic function. Hence, properties 1), 2), and 3) from 1.1 for subharmonic functions are valid also for psh functions. Rather than repeat these properties, we pass directly to properties which are inherent only to psh functions.

2.1. Approximation. If u is a plurisubharmonic function in a domain D c en, then the functions U,j, (j > 0, constructed in 1.4, are also plurisubharmonic. Hence a function U E Psh(D) can always be approximated by psh functions of class C'X) on compact subsets of D. The functions U,j will be defined on all of D only in the case D = en.

As the following example of J.E. Fornaess and J. Wiegerinck (1989) shows, for an arbitrary domain D, it is not in general possible to approximate on all of D.

Example. On the plane ew consider the subharmonic function

v(w) = kt21~->n I w - ~I· Note, v(l/k) = 00, but v(O) = -1/2.

We surround the point w = 11k by a disc Uk: Iw - 11kl < rk of radius rk > 0 so small that v(w) < -1 in Uk' k = 2, 3, .... For such a choice, the function

( ) {max { v(w), -l}, for Izl < 1, U z w = '-1, for Izl > 1,

is plurisubharmonic in the disconnected set e2(z, w)\{lzl = 1, w E q and extends to be psh through the holes

00

Q = U {{ I z I = I} x Ud k=2

by setting u(z, w) = - 1 on Q. In other words, the extended function u(z, w) is psh in the domain Q = {(z, w) : Izl i= I} u Q. This function u(z, w) cannot be globally approximated by a sequence of continuous psh functions uj t u. Indeed, if it could, then the sequence uj would converge to -Ion the circles {Izl = 2, w = O} and {Izl = 2, w = 11k}, k = 2,3, .... Since the discs {Izl < 2, w = 11k} belong entirely to the domain Q, by the maximum principle, the convergence uj t - 1 also holds (uniformly) on the union of these discs. But this is impossible since u(z, 0) = -1/2, Izl < 1, and the uj are continuous.

The domain Q in the preceding example is not a domain of holomorphy in e2 . It turns out that on domains of holomorphy, global approximation by psh functions of class Coo is nevertheless possible. This assertion easily follows from the following theorem on continuation of psh functions from submanifolds.

Theorem (Sadullaev (1982)). If Me en is a closed complex submanifold, then any function U psh on M2 psh-extends to all of en; i.e. there exists a pshfunction w in en such that w 1M = U.

To construct global approximations uj t U on a Stein manifold M embedded in en, we approximate the extension w in en and consider the restrictions of these approximating functions on M.

2 Psh on a manifold is defined by local coordinates.

68 A. Sadullaev

2.2. The Operator ddc• We suppose that the reader is acquainted with the theory of currents and, in particular, with the notion of a positive current. The necessary facts are well exposed in Harvey (1977) (see the supplement in the Russian translation by Chirka).

We recall that d = 0 + 8, dC = i(8 - 0), and hence ddc = 2i08. Let u be a psh function of class C2 in a neighborhood of O. Then if 1: Zj = ajw, j = 1, 2, ... , n is any complex line, with parameter WEe, the restriction ul1 = u(aw) is subharmonic in w.

Therefore A",u11 ~ 0 which means that the quadratic form

02U

L 0 0- aiik , j,k Zj Zk

is positive for all a E en, i.e. this form is positive definite. From this it follows that ddcu is a positive differential form of bidegree (1, 1). For an arbitrary psh function u we use smooth approximations Uj 1 u to see that ddcu is also a positive current, i.e.

f uddcrx ~ 0 (5)

for any positive differential test form rx ofbidegree (n - 1, n - 1). Formula (5) may be taken as the definition of a psh function: an upper

semicontinuous function u is psh if and only if ddcu ~ 0 as a current of bidegree (1, 1).

We remark that every positive current is a current of measure type, i.e. is a generalized differential form whose coefficients are Borel measures (cf. Harvey (1977)).

2.3. The Upper Envelope of psh Functions. It was mentioned above that the maximum of finitely many psh functions is again plurisubharmonic. In practice, we often encounter the upper envelope u = sup Ua of an infinite number of psh functions ua , rx E A. In this situation, we require, first of all, that u be locally upper-bounded (i.e. the family {ua } be locally uniformly upper-bounded). However, this is insufficient for plurisubharmonicity of u; u could turn out not to be a semicontinuous function. For this reason, we employ the regularization u* of a function u (the least upper semicontinuous majorant):

u*(z) = lim sup u(w) = lim u(w). £-0 Iw-zl<e w"""'z

The function u* is plurisubharmonic. This follows easily from the definition of a psh function.

The situation is analogous for the upper limit

u(z) = lim uiz) j-oo

of a locally uniformly upper-bounded sequence {uJ: the regularization u* is psh in the domain of plurisubharmonicity of the functions uj • Later (see Chap. 2) it

will be shown that the set {z : u(z) < u*(z)}, where u differs from its regularization u*, is negligible: it is a pluripolar set.

2.4. Connection with Holomorphic Functions. If a function f is holomorphic in a domain D c en, then In If I is a psh function. Conversely, every psh function can be represented by holomorphic functions in the following way.

Theorem (Bremermann (1956)). Any pshfunction u on a Stein manifold D can be represented in the form:

U(z) = [~im rxj In IfiIJ*, rxj 2 0, fi E (9(D). )-00

(6)

F or a plane domain DeC, this theorem was proved by Lelong (1941). His proof is based on methods of classical potential theory and is rather long. However, if we invoke a famous theorem of Oka on holomorphic nonextendability for pseudoconvex domains, then the proof becomes simpler and carries over to the general case.

Indeed, consider the domain

D* = {(z, w) E D x IC: In I wi + u(z) < O}

in the product D x Cw ' Since In Iwl + U is a plurisubharmonic function in D x C and D is pseudoconvex, the domain D* is also pseudoconvex. By a theorem of Oka (1942, 1953), D* is a domain of holomorphy. Consequently, there exists a function F E (9(D*) which is nonextendable holomorphically out-

00

side of D*. Expanding F in a Hartog's series: F(z, w) = L fi(z)w j , we find that j=l

the radius of convergence of this series

. [-. In IflJ* almost everywhere agrees WIth e-U • Thus, u(z) = ~Im -.-) . )-00 ]

Remark. Bremermann showed the uniform proximity offunctions of the type rx In If I to continuous psh functions. More precisely, if u is a continuous psh function in D, then for each compact KeD and each 8 > 0, there exist functions fl' f2' ... , fN and positive constants rx l' rx 2 , ••• , rxN such that, on K,

max{rxl Inlfll, ... , rxN InlfNI}:-:;; u:-:;; max{rxllnlfll, ... , rxN In IfNI} + 8. (7)

Thus, using an approximation uj ! u, we get that for any psh function u, we may omit the regularization symbol (*) in (6).

Now, we introduce an example showing that the requirement that D be holomorphically convex (a Stein manifold) is essential in the theorem of Bremermann.

70 A. Sadullaev

Example. Let D = C2\S+, where S+ is the half-sphere

S+ = {z = (Zl' Z2): Izl = 1, 1m Z2 ~ O}.

Then the function

{O, for Izl > 1 or 1m Z2 < 0, u(z) = I

m Z2' for Izl < 1 and 1m Z2 ~ ° (8)

is plurisubharmonic in D. However u does not have a representation in the form (6), because every function J; would be hoi om orphic in C2 , but the function u is not psh in C2 .

Remark. The theory of subharmonic functions and potential theory are so intertwined in their contemporary state of development that they are often identified. However, previous to the connection between subharmonic functions and potentials of Borel measures (F. Riesz (1926), cf. 1.5), these two theories developed in parallel: subharmonic functions, as a part of function theory, and potential theory in connection with the Dirichlet problem for equations of mathematical physics. In the theory of psh functions of classical potential theory, certain capacitary properties (continuity almost everywhere in capacity, structure of the set {u < u*} for the upper envelope etc.) correspond to these functions. We now turn to such properties.

Chapter 2 Complex Potential Theory

The following investigation of psh functions is connected with extremal problems and capacitary properties. In these questions, an important role is played by the complex Monge-Ampere operator (ddct, where n is the dimension of the given complex manifold. The operator (ddct will playa role here similar to the role played by the Laplacian in classical potential theory.

In contrast to the classical situation, for n > 1, this equation is not linear. For this reason, the associated "complex potential theory" is also called non-linear potential theory. Research in this area was initiated by Chern, Levin, and Nirenberg, who showed that the operator (ddT is bounded in the mean for the class of uniformly bounded psh functions of class C 2 , and by the fundamental paper of Bedford and Taylor (1976), in which the operator (ddc)n is defined for bounded (not necessarily smooth) psh functions as a certain positive measure.

In the years 1976-1983, interesting studies in multi-dimensional complex potential theory were carried out. These studies laid the foundations for the central concepts of this theory (Bedford-Taylor (1976, 1982), Sadullaev (1981». During this same period, methods were developed for applying this theory to problems of several complex variables, especially to questions of approximation


(Zakharyuta (1974, 1976), Siciak (1962, 1969), Sadullaev (1982b, 1984), Nguyen Thanh Van et Zeriahi Ahmed (1983, 1985)). At the present time the elaboration of the foundations of complex potential theory is essentially completed. A remarkable result of this theory has been the solution of two problems of Lelong concerning pluripolar sets posed by him in the early 60's. The first of these problems, global pluripolarity of a locally pi uri polar set, was solved by J osefson (1978). The solution of the second problem concluded a series of studies by Bedford-Taylor (1982) and the author (1982b) (see 3.3 below); the first solution to the second problem is due to Bedford-Taylor (see also Cegrell (1978)).

In this chapter we give a brief presentation of the foundations of complex potential theory.

Our approach is to construct this theory based on certain integral estimates and somewhat differs from the path taken by Bedford-Taylor.

§ 1. The Complex Monge-Ampere Operator

1.1. The Operator (dd")". For a function u of class C2 , we define (ddcut = ddcu /\ ... /\ ddcu (k times). This represents a differential form ofbidegree (k, k). It is easy to see that

(ddCu)n = nnn! det(a::a~Jpn,

where Pn = (~y )] dZj /\ dZj is the volume form on e. The operator (ddcut for an arbitrary bounded (not necessarily twice smooth)

psh function can be defined in the distributional sense. Let u be a bounded psh function in a domain G c en and denote by D(n-k,n-k) the space of differential forms of bidegree (n - k, n - k), of class COO and of compact support in G. Then, the recurrence relation

f (ddCu)k /\ cp = f u(ddCu)k-l /\ ddccp, cp E D(n-k,n-k)

defines (ddCu)k as a positive current ofbidegree (k, k).

(1)

Indeed, for k = 1, this was shown in 2.2 of Chap. 1. Suppose, by induction, that the assertion is true for k - 1 and let us prove it for k. Since by assumption (ddCu)k-l is positive, it is a current of measure type. Consequently, u(ddCu)k-l is also a current of measure type and hence the right side of (1) is well-defined (recall that u is bounded). Thus, (ddCu)k is a current. In order to show the positivity of (ddCu)\ we use an approximation uj ! u. We have

f U(ddCU)k-l /\ ddccp = lim fUiddCu)k-l /\ ddccp. J~OO

72 A. Sadullaev

By definition of the current (ddCu)k-l,

f (ddCu)k-l /\ ujddc<p = f (ddCu)k-2 /\ ddCuj /\ ddc<p = f (ddCu)k-l /\ ddCuj /\ <p.

Hence, J uj(ddCu)k-l /\ ddc <p ~ 0 for all positive forms <p, since the current (ddCu)k-l is positive by the inductive hypothesis and the form ddCuj /\ <p is positive, being the exterior product of two positive forms. It follows that the current (ddCu)k is positive.

1.2. Integral Estimates. The basic difficulty in making practical use of the operator (ddCu)k consists in establishing the convergence of the currents (ddCuj)k for a monotone sequence of psh functions uj • In this section we show such a convergence only for a uniformly convergent sequence uj • If u is a continuous psh function, then its approximating sequence uj ! U of COO-functions converges uniformly on compacta. In this case, it follows that (ddcul--+ (ddCu)k in the sense of currents, for k = 1, 2, ... , n. The convergence of (ddCul for an arbitrary monotone sequence uj will be shown in 3.2. Till then, it will be necessary for us to work only with continuous functions.

Lemma. If G = {p(z) < O} is a strictly pseudo convex domain, p E C2 (G), (J = min p(z), and u is a continuous psh function in G, then for each rand k,

G

(J < r < 0, 1 :::;; k :::;; n,

fr dt f (ddC prk /\ (ddcut :::;; (M - m) f (ddC prk+l /\ (ddCu)k-l, (2) (1 p(z) ';1 p(z) ';r

where M = max u(z), m = min u(z). G G

For a C2-function, the lemma follows from Stokes' Theorem and Fubini's Theorem

fr dt f (ddC p)n-k /\ (ddcut = f dp /\ dCu /\ (ddC p)n-k /\ (ddCu)k-l (J p(z) ';1 p';r

= I=r udc p /\ (ddC prk /\ (ddCu)k-l

-I,;r u(ddCprk+l /\ (ddCu)k-l. (3)

Since dC p /\ (ddC pt-k /\ (ddcut- 1 = AS, where A ~ 0 and s is the Euclidean volume form on {p = r} and since the form (ddcpt-k+l /\ (ddCu)k-l is positive, we obtain (2) by estimating u and invoking once again Stokes' Theorem. In case u is an arbitrary continuous psh function, we use an approximation uj ! u. In order to pass to the limit, it is sufficient to establish the compactness of the family (ddCul (cf. Theorems 1 and 2 below).

Let u be a psh function of class C2 in G. Applying the inequality (2) succes-sively for k = p, p - 1, ... , 1, we obtain:

f o dtl ft! dt 2•·· f tp-! dtp f (ddCpr p /\ (ddCu)P::s; (M - m)P f (ddcpt (J (J U P '5::tp P =f,0

(4)

For fixed r, (J < r < 0, the left side of (4) can be estimated from below by the quantity

Ir ilP f (ddCpr p /\ (ddCu)p.

p. p "r Consequently, for each r < 0, the following mean estimate holds

I"r (ddC prp /\ (ddCu)P ::s; Cr(M - m)P(u), (5)

where Cr is a constant independent of u. Since (ddcu)P is a strictly positive current and p is strictly psh, the left member of (5) is greater than C· II(ddcu)PII, where the constant C > 0 depends only on p and II' II denotes the mass of a current of measure type (see Harvey (1977)). Consequently, for any test form IX E D(n-p.n- p), we have the estimate

I"r a /\ (ddCu)P ::s; Cr' IIallG(M - m)P(u), (6)

expressing the weak compactness of the family (ddcu)P for the class of bounded (above and below) psh functions u. From this estimate and the ones shown earlier, the assertion of the lemma follows easily for the general case.

Let us now state separately the compactness result and the passage to the limit which we have obtained (see Bedford-Taylor (1976)).

Theorem 1. Let {u} be a locally uniformly bounded family of psh functions of class C2 in a domain Gee. Then, the family of currents {(ddcu)P} is weakly compact in G for any p, 1 ::s; p ::s; n.

, Theorem 2. If a sequence of psh functions Uj E C2 (G) converges uniformly to U

in a domain G, then (ddcuY"""""* (ddCu)P, 1 ::s; p ::s; n, weakly.

Indeed, for p = 1, the theorem is obvious. We suppose the theorem true for p - 1 and show it for p. By definition, for any test form a E D(n-p.n- p),

L (ddCu)P /\ a = L uiddCuj)P-l /\ (ddCa) = L u(ddcuy-1 /\ ddca

+ L (uj - u)(ddCuj)P-l /\ (ddCa).

By the inductive hypothesis, the first integral on the right side tends to

L u(ddcu)P-l /\ ddca:= L (ddCu)P /\ a,

74 A. Sadullaev

and the second integral to zero, in view of the boundedness of (ddCuj)p-l and the uniform convergence of uj - U to zero.

Remark. The estimate (6), which was shown for psh functions of class C2 , is valid also for continuous functions. Thus Theorem 1, and consequently Theorem 2, are also true for the class of continuous psh functions (see also below, 3.2).

1.3. The Dirichlet Problem. Plurisubharmonic functions satisfying the Monge-Ampere equation (ddcu)" = 0 (for n = 1 we have the classical harmonic functions) play an important role for n > 1. These functions, in general, are not COO-smooth; they can even be discontinuous.

Consider the following Dirichlet problem. Given a continuous function cp on the boundary aG of a domain G c en, we seek a solution w the homogeneous Monge-Ampere equation with boundary data cpo

(7)

For an arbitrary domain G, problem (7) in general has no solution. However, if G is strictly pseudoconvex, then a solution exists and can be found, for instance, by the Perron method (see Brelot (1961)). Let Olt(cp) be the family of all psh functions U in G such that the boundary values along the normal are dominated by the boundary data: u*18G :::;; cpo Set

w(z) = sup u(z). UE -fI

Then the regularization w* satisfies the condition w*laG = cp (Bremermann) and is continuous in G (Walsh). Bedford and Taylor (1976) showed that w* is a solution of (7), i.e.

(ddcw)" = 0 and w*laG = cpo

Just as for harmonic functions, the solutions u of the homogeneous MongeAmpere equation satisfy a maximum principle: for any psh function v in G and for any domain Dec G, vlcD :::;; ulaD implies v :::;; u in all of D. The proof of this statement is connected with the following proposition.

Theorem. If u, VEL 00 n Psh(G) and the set

U = {z E G : u(z) < v(z)}

is precompact in G, then

fa (ddcu)" ~ fa (ddcv)".

F or infinitely smooth functions, the proof is not difficult. Using Coo -approximations and Theorem 2, 1.2, it is easy to prove the theorem for continuous functions u, V. The general case can be proved as a consequence of the convergence property of (ddcu)n for a monotone sequence uj, which will be given in 3.2 of Chap. 2.

§ 2. Extremal Functions and Capacity

In this section we introduce the notions of ~-measure and capacity of a set E with respec~ to a domain G ::> E. These are plurisubharmonic analogs of the classical notions of harmonic measure and capacity of condensers. For simplicity, we suppose that G = {p(z) < O} is a strictly pseudoconvex domain in en, though all results hold in the more general situation.

2.1. &l-measure. Let E be a subset of G and ill/(E, G) the class of psh functions u in G such that ul G ::;; 0, ulE ::;; -1. Set

w(z, E, G) = sup{ u(z) : u E ill/(E, G)}.

The regularization w*(z, E, G) is called the ~-measure of the set E with respect to the domain G.

Note that the topological lemma of Choquet (cf. Brelot (1961)) implies that there is a countable family ill/' c ill/(E, G) such that

[sup {u(z) : u E ill/'}]* = w*(z, E, G).

As a corollary, it follows that the ~-measure w* can be represented as a limit

w* = [~im UiZ)]*' where uj E ill/(E, G) )""00

and {uj } is an increasing sequence. We state the following simple properties of ~-measure (cf. Sadullaev (1981)). a) Monotonicity. If E1 c E2 , then

w(z, E1, G) ;;:: w(z, E2 , G).

00

b) Approximation. If V eGis an open set and V = U Kj, where Kj c Kj+1 j=1

are compact, then

w*(z, K j , G)! w*(z, V, G).

If E is an arbitrary set, then there exists a decreasing sequence of open sets ~::> E, ~::> ~+1 (j = 1,2, ... ) such that

( ~im w(z, ~,G))* = w*(z, E, G). J~OO

In analogy to polar sets (Chap. 1, 1.3), a subset E eGis said to be pluripolar in G if there is a psh function u =1= -00 in G such that ulE == -00. Since psh functions are also subharmonic, every pluripolar set is polar as well. In particular, pluripolar sets have Lebesgue measure zero. It is easy to prove that a countable union of pluripolar sets is pluripolar.

c) &l-measure and pluripolarity. w*(z, E, G) is either nowhere ° or identically O. The latter holds if and only if E is pluripolar in G.

76 A.Sadullaev

In fact, if w* = ° at some point Zo E G, then by the maximum principle, w* = ° in G. In this case, there is a sequence zi E G(zi -? Zo E G) such that w(zi, E, G) -? ° as j -? 00. It follows that w(z, E, G) = ° almost everywhere in a neighborhood of zo, because w satisfies the inequality (see 1.2):

. 1 f w(zJ, E, G) ~ --2n w(z, E, G) dV. V2n r B(zi,r)

Fix a point z' such that w(z', E, G) = ° and take a sequence uj E OlI(E, G) with the 00

property uiz') ~ - r j. The sum u(z) = L uj(z) is psh in G; moreover, u(z') ~ j=l

-1 and ul E == -00. On the other hand, if u is a psh function in G such that u ~ 0, u =1= -00 in G, but ulE == -00, then eu E OlI(E, G) for every e > 0. This means that w(z, E, G) = ° for every z E G such that u(z) #- -00, i.e., for almost every z E G. Hence, w*(z, E, G) == ° in G.

Although the function w is equal to -1 on E, the regularization w* is not necessarily -1 on E. In the process of regularization, it canjump at some points at which E is sufficiently thin. A compact set KeG is said to be pluriregular if w*(z, K, G)IK == -1. For more information about pluriregular sets, see Sadullaev (1981).

d) Continuity of £1}J-measure (Zakharyuta (1974)). If a compact set K is pluriregular, then the function w*(z, K, G) is continuous in G. In fact, using the function p defining the domain G, we can continue w* to some neighborhood D :::J G as a psh function. Then, in the neighborhood D, we construct approximations uj ! w* by functions uj of class COO. Applying twice the Hartogs Lemma (see 1.6, Chap. 1) to the sequence Uj (first for K in some neighborhood, and then for G and D), we find, for every e > 0, ajo such that

UjIK~-1+e and ujIG~e, j~jo.

The functions uj - e belong to OlI(K, G) and thus uj - e ~ w*(z, K, G),j ~ jo. On the other hand, uj ~ w*(z, K, G), which means that uj converges uniformly to w*(z, K, G). Thus, w* is continuous.

e) For a pluriregular compact set KeG, the function w*(z, K, G) satisfies the homogeneous Monge-Ampere equation in G\K. In fact, consider a ball Be G\K and let v be the solution of the Dirichlet problem in B:

(ddcvt = 0, VlilB = W*lilB'

By the maximum principle (1.3), v(z) ~ w*(z, K, G), z E B. On the other hand, the function

w(z) = {V(Z), z E B, w*(z, K, G), z E G\B,

is in OlI(K, G) and consequently, w(z) ~ w*(z, K, G). In particular, v ~ w* in B and thus w* = v, q.e.d.

In conclusion, we state a two-constants theorem which easily follows from the definition of £1}J-measure.

Theorem. If u is a plurisubharmonic function in G, u ::;; M and ulE ::;; m on some subset E c G, then we have the estimate

u(z) ::;; M(l + w*(z, E, G)) - mw*(z, E, G), z E G.

2.2. Condenser Capacity. The integral

qK, G) = L (ddcw*(z, K, G)t

is called the capacity of the compact set KeG with respect to the domain G. For n = 1 the capacity qK, G) is the well known capacity of the condenser with plates K and aGo For an open set U c G, we set

qu, G) = sup{ qK, G): K c U, K pluriregular and compact}.

The constraint of pluriregularity of the compact sets in the definition of qu, G) is needed for technical reasons. At the end of this section, this constraint will be dropped. We recall that the 9-measure of a pluriregular compact set is continuous and Theorems 1 and 2 of 1.2 as well as Theorem 3 of 1.3, so important in the theory of capacity, were proved for continuous functions.

For an arbitrary set E c G, we can define its outer capacity as

C*(E) = C*(E, G) = inf{ qu, G): U open, E cUe G}.

We state several important properties of capacity (see Sadullaev (1980, 1982b) and Bedford-Taylor (1982)).

a) For any increasing sequence of open sets ~ c ~+1 ,j = 1,2, ... , we have 3

c(O ~) = ~im q~)(obvious). J=l )-00

b) For any open set U c G,

qU) = sup {Iv (ddCut: u E qG) (\ Psh(G), -1 ::;; u ::;; o}. (8)

Indeed, taking as psh function u the 9-measure of a pluriregular compact set K c U, we have that the left side of (8) is no greater than the right side. On the other hand, fix a number t: > 0 and a function u E qG) (\ Psh(G), -1 ::;; u ::;; O. Then there exists a pi uri regular compact set K c U with

Iv (ddcut - L (ddcut < t:. (9)

It is clear that the set

D = {(I + 2t:)w*(z, K, G) + t: < u(z)}

3 When the containing domain G does not play an important role, we shall drop it from the notation.

78 A.SaduJlaev

is precompact in G. Thus, using Theorem 1.3, and noting that w* = w*(z, K, G) is continuous in G, we have

In {ddC[(1 + 26)W* + 6]Y ~ In (ddcut ~ L (ddcut·

By property d) of &>-measure,

In (ddcw*)" = L (ddcw*)" ~ qU).

From this and from (9), we have that the right side of (8) does not exceed qU). o

If the open set U in b) is precompact in G, U c c G, then we can replace the functions u in (8) by max {u(z), Mp(z)}, where M = M(U) is a constant, such that M· plu < -1. Such a function can be continued plurisubharmonically to some neighborhood D:::J G, such that lui < 1 in D. Approximating such a u by infinitely smooth functions, we see that in (8) we may replace the class qG) n Psh(G) by the class

L = {u E C'()(D) n Psh(D): -1 ~ u ~ 0 in G} (10)

where lui ~ 1 in D, and D :::J G is a fixed neighborhood. c) The set function qE) is countably subadditive, i.e., for any collection of

sets Ej c G,j = 1,2, ... , we have 00 00

C*(E) ~ L C*(Ej ), E = L Ej • j=l j=l

For open sets E, this property follows from (8). In the case of arbitrary E, for 6

~ny 6 > 0 we may construct open sets ~ :::J Ej such that q~) - C*(E) < P' ] = 1, 2, .... Thus

C*(E) ~ C CQ ~) ~ jt C*(Ej ) + 6.

d) C*(E) = 0 if and only if E is pluripolar in G. Moreover, we have the uniform estimate:

oc(r)' C*(E) ~ fa Iw*(z, E, G)I(ddCp)" ~ fJ(r) [C*(E)] 1/", (11)

where E is an arbitrary subset of Gr = {p(z) ~ r}, r < 0, and oc(r), fJ(r) - are constants depending only on r.

It is sufficient to show (11) for an arbitrary pluriregular compact set E. We invoke the integral estimate (2). Applying (2) (n - 1) times to the function w*(z, E, G), we obtain

f o dtl fll dt z '" fln-

l dt" f (ddcw*)" ~ fO dtl f ddcw* A (ddCp)"-l. (J U U P =:;tn (1 P :::;;t1

(12)

According to the middle estimate in (3), the right side of (12) is no greater than JP';o Iw*1 (dd C pt. As in 1.2 of Chap. 2, the left side of (12) is estimated below by the quantity

Irl,n f (ddcw*t· n. p,;r

Thus,

C*(E) :::;; 1;I'n L Iw*l(ddcp)n,

which is the first inequality in (11). To prove the second inequality, we set

Then there exists a constant y > 0 such that inf Iw*1 ~ yM. Thus, the open set Gr

D = {w* < ~~ (p + O-)}, where, as above, 0- = i~f p(z), is compact in G and

contains the compact set Gr. Consequently (cf. 1.3),

Since JD(ddcw*t = JE(ddcw*)n = C*(E), the second inequality in (11) follows.

2.3. Solution of the First Lelong Problem. An important role in the study of pluripolar sets is played by the following problem of Lelong (1957). Let E be a pluripolar set in a domain G c en. Is E pluripolar in all of en?

The properties of the capacity C*(E) given in 2.2 yield a simple solution to this problem (first solved by 10sefson (1978)). It is based on the following reasoning: without loss of generality, in place of G we may consider the unit ball Bl = B(O, 1). If E is pluripolar in B(O, 1), then it has zero capacity, C*(E, Bd = 0 (property d)). It is not difficult to show that if R > 1, then C*(E, B1 ) ~

C*(E, BR)4. Thus, E is pluripolar with respect to any ball BR of radius R > 1. But then it is easy to construct a psh function u =1= -00 in all of en such that UIE == -00 (for further details, see Sadullaev (1981)).

4We remark that if a compact set K is pluriregular with respect to the ball HI' then it is also pluriregular with respect to the ball HR , R > 1. Thus in the definitions of the outer capactities C*(E, HI) and C*(E, HR ), we may use the same compacta.

80 A. Sadullaev

§ 3. Capacitary Properties of psh Functions

3.1. C-Property of psh Functions. Every measurable function is "almost continuous" (C-property of Luzin). The capacitary analog of this property for subharmonic functions in IRn was proved by H. Cartan. For psh functions we have the following variant.

Theorem. If u is plurisubharmonic in a domain G, then for each e > 0 there exists an open set U c: G such that C(U) < e and the restriction of u to G\ U is continuous.

It is sufficient to prove the theorem in the simple case when G = B, the unit ball. We establish first an integral inequality for the difference of psh functions in the ball. Let Y - be the class of functions (10) for G = B. Consider functions u, v, ({Jl' ({J2' ... , ({In E Y such that ({Jo = u - v ~ 0 in Band ({Jo == const on the sphere S = vB: Izl = 1. Then

The surface integral here is no greater than

where the constant on the right side is independent of ({Jl' ({J2' ... , ({In (cf. the argument in 1.2). The last integral in (13) can be estimated with the help of the following Cauchy-Bunyakovskij inequality. Let IY. - be a positive (n - 1, n - 1)form and let ({J, 1/1 be smooth real-valued functions in the neighborhood of B. Then the expression

defines a scalar product for which the following inequality holds:

I(({J, I/IW S (({J, ((J)' (1/1,1/1).

Applying this inequality, we obtain:

II. Plurisubharmonic Functions

I L d<po /\ dC <PI /\ ddc <P2 /\ ... /\ ddc <Pn I

::;; [L d<PI /\ dC <PI /\ ddc <P2 /\ ... /\ ddc <Pn T/2

X [L d<po /\ dC <Po /\ ddc <P2 /\ ... /\ ddc <Pn T/2

::;; c[L <PodC<po /\ ddC<P2 /\ ... /\ ddC<pn

-L <Po ddC <Po /\ ddC <P2 /\ ..• /\ ddC <Pn T/2

::;; CI (11<Polls + L <PoddC<pri /\ ddc<P2 /\ ... /\ ddc<Pny/2,

81

where <Pri = u ; V E!l' (since <Po ~ 0, the form 2<Poddc<pri + <Poddc<po = 2<Poddcu

is positive). Repeating this procedure n times, we obtain the inequality

where a > 0, ')' > 0 are some constants independent of <Po, <PI' ... , <Pn· Using this estimate, we sketch the proof of the theorem. Involving the

countable subadditivity of C*(E) and the local character of the theorem, it is sufficient to show that for each e > 0, there exists an open set U c B such that qu n B') < e and the restriction of u is continuous on B'\U, where B' is the balllzi < 1/2.

Since qFM) tends to zero, for the set FM = {z: u(z) < -M}, as M -+ 00

(cf. 2.2), then, replacing u by an appropriate combination a· max {u, - M} + b, where a > 0 and M > 0, we may assuame that - 1 ::;; u ::;; O. Then, replacing u by max {u(z), v(z)}, where v(z) = 2(lz1 2 - 3/4), we render u == v in some neighborhood of the sphere S.

Consequently, the average (see 1.4)

decreases to u, and, moreover, up == vp in a neighborhood of S, where vp is the average of v. Passing to a subsequence, if necessary, we may suppose that the sequence of numbers SB up(ddcup)n (bounded, by Theorem 1, 1.2) has a limit.

Set

82 A. Sadullaev

In view of the monotonicity, we have

Up,m(b) c Up,m+l (b), m = 1,2, ... ,

and

00

U Up,m(b) = Up(b) = {z E B': uiz) > u(z) + b}. m=l

From the properties of capacity, it follows that

qUp(b)) = lim qUp,m(b)) m"'oo

Since the set Up,m(b) is open,

qUp,m(b)) = sup { r (ddcw)" : w E If} J Up.m(li)

::;; sup {~ r (up - up+m)(ddcw)" : w E If} J Up.m(b)

::;; sup {~ L (up - up+m)(ddcw)" : w E If}. By (14), it follows, then, that

qUp,m(b))::;; i[IIVp - viis + L IPp,m(ddcIP:.m)"T,

where IPp,m = up - up+m'

(15)

(16)

From the convergence of Sa up(ddcup)" as p ~ 00, it follows that the integral on the right side of (16) tends to zero. Thus, qUp,m) ~ ° as p ~ 00, and the convergence is uniform in m. From this and from (15), lim qUp(b)) = 0, from

which the assertion of the theorem follows easily. D

3.2. Convergence of (ddC)k

Theorem. If {uj} is a montonically decreasing sequence of functions psh in G, such that the limit u(z) = lim uiz) is locally bounded (from below), then, weakly,

j-oo

uj(ddCu)k ~ u(ddCu)\ asj ~ 00, 0::;; k ::;; n.

We prove this by induction on k. For k = 0, it is obvious. If the assertion is valid for k - 1, then, weakly,

(ddCul ~ (ddCu)\

since, by definition,

for an arbitrary fundamental form IX E Dn-k.n-k(G). Thus, it is sufficient to prove the theorem for k under the hypotheses that it is true for k - 1 and that (ddCujl-+ (ddCu)k weakly.

Let us fix an 6 > 0 and a fundamental form IX. By Theorem 3.1 there exists an open set U c G, C(U) < 6, outside of which the restrictions of all Uj and u are continuous. If ii is a continuous function in G coinciding with u outside of U, and Ea is the support of IX, then

It uj(ddcu/ /\ IX - t u(ddCu)k /\ IXI

~ I L, \U (uj - u)(ddCuj)k /\ IX I + It ii[(ddCuj)k - (ddCu)k] /\ IX I

+ If (Uj - u)(ddCu/ /\ IX I + If (u - ii) [(ddCul- (ddCu)k] /\ IX I· Ea:nV Er,r.r.U

(17)

The integrals on the sets Ea \ U and G tend to zero as j -+ 00: the first, because Uj -+ U uniformly on Ea\U, and the second, because (ddCu)k -+ (ddCu)k and ii is continuous on G. The last two integrals in (17) are small because of the smallness of 6> 0 (since C(Ea n U) < 6) and the uniform boundedness of uj on Ea· 0

Remark. If Uj is a locally uniformly bounded monotonically increasing sequence of psh functions and u = lim uj , then in order to prove the weak convergence j-oo

uiddcu/ -+ u*(ddcu*)k,

we need the pluripolarity of the set (J: {u < u*}. If (J is pluripolar, then for any 6 > 0, there exists an open set U c G such that C(U) < 6 and Uk converges uniformly in G\ U. Precisely this property, for a decreasing sequence, was used in proving the theorem of this section. The pluripolarity of the set (J will be shown in the next section.

3.3. The Structure of the Irregular Points. Let KeG be compact. A point ZO E K is said to be irregular for K, if w*(ZO, K, G) > -1 (cf. 2.1). The collection of such points is denoted by I K • The following theorem of Bedford and Taylor (1982) is an analog of the famous theorem of Kellogg.

Theorem 1. The collection I K of irregular points of a compact set K has zero capacity: C(IK ) = o.

It is clear that the proof of the theorem reduces to the assertion: a) If a compact set KeG is such that w*(z, K, G) ~ 15, b > -1 on K then

w*(z, K, G) = O. In turn, a) follows from the following assertion: b) For each compact set K, its £3P-measure w*(z, K, G) satisfies, the Monge

Ampere equation (ddcw*)" = 0 in G\K.

84 A. Sadullaev

The following theorem, which is related to Theorem 1 gives a positive solution to the second problem of Lelong for psh functions.

Theorem 2. Let {uj} be an increasing sequence of locally uniformly bounded psh functions in G and u(z) = lim uj(z). Then the set (J = {z: u(z) < u*(z)} is pluri-

j-oo

polar in G.

The proofs of Theorems 1 and 2 as well as assertions a) and b) proceed here simultaneously, by induction on n, according to the following scheme, introduced by Bedford and Taylor.

Step I. From b) follows a), and hence, Theorem 1. Step II. From Theorem 1 follows Theorem 2. Step III. From Theorem 2 in en follows b) in en+1. The correctness of b) for n = 1, i.e., the harmonicity of the function

w*(z, K, G) in G\K, follows easily from property e) of 2.1 of Chap. 2 and from the Harnack Theorem for harmonic functions: if Kj,j = 1,2, ... are approximations of K by regular compacta (cf. 2.1), then w*(z, Kj, G) i w*(z, K, G) on G\K.

Step I. Suppose, for the sake of contradiction, that w*(z, K, G) 2 () > - 1 on K but w* =1= o. If

w*(z, K, G) 2 ()

everywhere in the domain G, then in the class

{.jfJI u(z) : U E OJi(K, G)},

(18)

there is a function v(z) such that v > w* at some point ZO E G\K. But this contradicts the fact that v ::; w* on the boundary o(G\K) and (ddcw*)" = 0 in G\K (cf. 1.3). A priori, the inequality (18) may not hold on all of G, although it holds on K. In this case we use the same argument, only for the chosen function v satisfying the condition v ::; w* on the boundary o(G\K); we consider the class

{.jfJIu(z) + w*(z, U" G): U E OJi(K, G), e > O},

where U, is the open set from the theorem in 3.1 : C(U,) < e and the restriction of w*(z, K, G) is continuous outside of U,.

Step II. Theorem 1 => Theorem 2. Suppose, first of all, that Uj are continuous in G. Then u is lower semicontinuous and the set u ::; IX is closed in G for any number IX. From this it follows that for any IX < [3 the set KaP: u(z) ::; IX < [3 ::; u*(z) is also closed in G. From Theorem 1 it follows that KaP has zero capacity in any ball B «; G, since

u(z) - SUPB u(z) w(z, Kap n B, B) 2 ( ) ,

suPBuz -IX

and, consequently, Kap c iK,p. The union U KaP over all rationals IX, [3, and hence the set (J C U Kap also have zero capacity.

In the case where the uj are arbitrary, there exists, for any e > 0, according to the C-property of psh functions, an open set U c G such that C(U) < e and the restrictions of all uj are continuous on G\U. From the previous reasoning, it

follows that C*(a\U) = 0 and consequently, C*(a) ~ C*(a\U) + C*(a n U) < B.

Since B > 0 is arbitrary, it follows that C*(a) = O. Step III. Theorem 2 in e => b) in e+1. Let KeG c Cn+1 and Kj be

00

pluriregular compacta such that Kj :::> Kj+1 , n Kj = K. Then wj = w*(z, Kj, G) j=l

satisfies the Monge-Ampere equation in G\Kj (cf. 2.1), is monotonically increas-

ing, and w* = w*(z, K, G) = (~im Wj)*. Hence for the proof of b) it is sufficient r->oo

to show that (ddcwj)n+l -+ (ddcw*)"+1 weakly asj -+ 00. For this we must establish the convergence of

~im f wj(ddCwj)" /\ ddcrx = f w*(ddcw*)" /\ ddcrx, (19) }-oo G G

for any test function rx. Using the obvious identity

1 dzp /\ dZq = 2 [d(zp + Zq) /\ d(zp + Zq) + id(zp + iZq) /\ d(zp + iZq)]

- (1 + i) dz p /\ dzp - (1 + i) dZq /\ dzq ,

we obtain that the differential form ddcrx in (19) can be represented as the sum of forms {3 dcp /\ dqi, where cp is a linear function of Z l' ... , Zn+l. Thus it is sufficient to show (19) for ddcrx = fJ dZn+1 /\ dZn+1- In this situation, by Fubini's Theorem, the integrals in (19) are transformed into iterated integrals of the form

fdZn+1 /\ dZn+1 f G[Zn+,]

on the section G[z~+1J = G n {Zn+l = Z~+1}. For (almost) every section G[zn+1J

~im f {3wj(ddCwj)" = f {3w*(ddcw*)", }-OO G[Zn+,] G[Zn+,]

by Theorem 2 and Remark 3.2. From this, the convergence in (19) follows. D

3.4. Capacitability of Borel Sets. From Theorem 2 of 3.3 it follows that for any compact set KeG its capacity C(K, G) coincides with its outer capacity C*(K, G). Moreover, the capacity introduced above sastifies the capacitability axiom of Choquet: for any increasing sequence Ej c Ej+1' we have

lim C*(Ej, G) = C*(E, G), j-oo

00

where E = U Ej • This property follows from the analogous identity for f!J>j=l

measure:

lim w*(z, Ej , G) = w*(z, E, G). (20) j-+oo

86 A. Sadullaev

Let us prove (20). By Theorem 2 of 3.3, the sets

~ : w(z, Ej , G) < w*(z, Ej , G), j = 1,2, ... ,

ao

are pI uri polar in G. Consequently, their union P = U ~ is also pluripolar, and j;l

hence, there exists in G a psh function u =1= -00, u ::;; 0, such that ul p == -00.

Then, the function w(z) + 8U(Z), where w is the limit on the left side of (20), belongs to the class Ol/(E, G) for each 8 > O. Thus, w(z) + 8U(Z) ::;; w*(z, E, G) and consequently, w(z) ::;; w*(z, E, G). From this and the obvious inequality w(z) ~ w*(z, E, G), we obtain (20).

From the preceding result, Choquet's Theorem (cf. Landkoff (1966)) yields

Theorem. For any Borel set E c G, its outer capacity C*(E, G) agrees with its inner capacity C*(E, G) = sup{ C(K, G): K c E, K compact}.

Corollary. Let {Ua} - be a locally uniformly bounded family of psh functions in a domain G and set u(z) = sup Ua(z). Then the set (J: u(z) < u*(z) is pluripolar in G.

Remarks. The method of integral estimates (1.3) for the elaboration of a complex potential theory is due to the author. Bedford and Taylor (1982) have used, for this purpose, the convergence of the sequence of currents (ddcuj1») /\ ... /\ ddc ujP) for a monotone bounded sequence {UJk)}, k = 1, 2, ... , p.

For an unbounded psh function the operator (ddcut may not be a bounded Borel measure. To see this, consider, for example, the function

U(Zl' Z2) = (IZlI2 - 1) Inl/2(l/lz212),

which is plurisubharmonic in a neighborhood of the origin. For an arbitrary function u E Psh(G) Kiselman has proposed the following definition of (ddcut in the domain G x {w E IC: 11m wi < 1/2} consider the auxilary function F(z, w) = (u(z) - Re w)+. Then the measure (ddC Ft+1 is concentrated on the graph Re w = u(z), and we call its projection on G the measure (ddCut. This measure is, in general, not finite, but for a bounded function, it coincides with the well-known measure (ddCu)" of Bedford-Taylor (1.1).

Y'-measures (2.1) were first employed in the works of Zakharyuta (1974, 1976) and the author (1976, 1980). With the help of Y'-measures, a capacitary quantity has also been constructed which is close to condenser capacity.

The capacity of a condenser in IC" in terms of (ddcw*t (2.2) was introduced practically simultaneously by Bedford and the author.

The C-property ofpsh functions and the structure of irregular points (3.1, 3.2) were established by Bedford and Taylor (1982).

If u is a twice continuously differentiable function in G and (ddCu)" = 0, then through each point ZO E G there is an analytic curve on which u is harmonic. Lempert (1982*)5 (see also Slodkowski (1990)) has established an interesting

5 Translator's note. A date with an asterisk refers to a reference added (by the author) in translation


connection between such analytic curves and extremal surfaces forthe Kobayashi and Caratheodory metrics. It turns out that the extremal surfaces for these metrics coincide with the analytic curves on which the solutions of the MongeAmpere equation, with logarithmic singularity in the given fixed point, are harmonic (see Poletskij and Shabat, in Vol. 9 of this series). For more information on the Monge-Ampere equation see also Cegrell (1986, 1990), Demailly (1987), Levenberg (1985).

Chapter 3 Applications of Complex Potential Theory

In this chapter we give applications of extremal plurisubharmonic functions and capacity to various problems of multidimensional complex analysis. In Section 1, we present the results of the author on the structure of the singularity set of functions admitting rapid approximation by rational functions. In Section 2, we present the results ofChirka and Kazaryan (1983) on holomorphic extension in a fixed direction. In the third and last section, we present a multidimensional analog of the Bernstein-Walsh Theorem (Siciak (1962)) and the holomorphic extension of separately analytic functions (Zakharyuta (1976), see also Siciak (1969)). Besides, the third section contains results on hoi om orphic extension from circled sets (Forelli, cf. Rudin (1980), Alexander (1981)).

§ 1. Rational Approximation and Pluripolar Sets

Gonchar (1974) showed that if the complement of a domain Dc en is pluripolar, then, in D, any holomorphic function f can be rapidly approximated with respect to Lebesgue measure in D by rational functions, i.e., If - rkI1/k ..... 0 in measure on D, for some sequence of rational functions rk' deg rk ::; k (denoted rk ~ f in D). Such rapid approximations have the following transmission property (Gonchar (1972, 1974)): if rk ~ f in a domain U, then the function f is single valued in its entire Weierstrass domain of existence »j 6, moreover rk ~ f in »j.

Let us denote by RO the class of all functions which admit such a rapid approximation in the vicinity of 0 in en. Gonchar's theorem may be formulated as follows:

OED, en\D pluripolar => (!)(D) c RO•

6 Let f be holomorphic in a domain U c IC". The set of holomorphic elements (g, V), which can be obtained from the element (f, U) by holomorphic continuation, when appropriately glued together, is called the Weierstrass domain of existence of the element (1, U).

88 A. SaduJlaev

From the preceding, it follows that the envelope of holomorphy D of the domain D is such that (!)(D) c RO without ramification, that is, lies in cn. Below, it will be shown that

(!)(D) c RO => en\D is pluripolar.

(The set Cn\D for n > 1 can be very thick, but all function in (!)(D) can be holomorphically continued to D and en\D is the "natural" set of nonremoveable singularities for (!)(D).) Thus, we obtain the following criterion:

(!)(D) c RO <=> 0 E D, Cn\D is pluripolar,

which is well known in the classical case (for n = 1, D = D) The complement of a pseudoconvex domain is said to be pseudoconcave; we

shall begin the proof of the pluripolarity of Cn\D by a study of the general properties of pseudoconcave sets, which are themselves of independent interest.

1.1. The Maximum Principle for Pseudoconcave Sets. A set Seen is said to be pseudo concave if for each point ZO E S there is a neighbourhood U :3 ZO such that the set U\S is open and pseudo convex in cn.

Lemma. Let D be a bounded convex 7 domain in Cn and SeD a closed pseudoconcave subset of D. Then the Shilov boundary JS of the compact set S (with respect to polynomials) is contained in S noD.

Suppose, to obtain a contradiction, that we can find a polynomial P such that 1IPIIs = P(ZO) = 1 at some point ZO E S, but IIPII as < 1 (here as = S\S =

S n aD). Let D' rg; D" rg; D be convex domains such that IPI < 1 also on S\D'. The domain D"\S is pseudoconvex (being locally pseudoconvex) and the algebraic sets

A k ={ZED':P(z)=1+1/k}, k=1,2, ... ,

which belong to D"\S along with their boundaries, are separated from S uniformly in k. Since the limit surface and S have the common point zO, we obtain a contradiction to the continuity principle (cf. Vladimirov (1964), Shabat (1976)). 0

1.2. Pluripolarity of Pseudoconcave Sets

Theorem. Let S be a pseudo concave set in Cn such that 0 ~ S and for almost each complex line I passing through 0, the section InS is polar (in I). Then S is a pluripolar subset of en.

We outline the method of proof in the case n = 2 for simplicity. The theorem is local and so, with the aid of a fractional linear transformation mapping the family of lines passing through 0 (in the neighborhood of a fixed direction) to

7 The convexity plays no role.

a family of parallel lines, we may view the problem in the following form: S is a pseudo concave closed subset of the unit polydisc U = U1 X U2 with S n {VI x aU2 } = 0 and such that the intersection of S with {ZI = zn is polar for almost every z? Shrinking U1 , if necessary, we may, without loss of generality, assume that S n {ZI = zn, is polar for almost all z?, Iz?1 = 1, with respect to linear measure on the circle.

According to the lemma in 1.1 and Bremermann's approximation theorem (2.4, Chap. 1), the &>-measure w* = w*(z, s, U) = w*(z, as, U) and hence (dd C W*)2 = 0 in U. Moreover, from the conditions of the theorem, it follow that the boundary values of w* are almost everywhere equal to zero on au, and on that part of the boundary F2 = VI X cU2, where there are no points of S, w* is identically zero. From the maximality of w* in U,

fr dt f (ddcw*f = 0, ° liz II :S;t

where IIzll = max{lzll, IZ21}, r < 1. On the other hand, such an integral can be transformed into

f w*dclizil 1\ ddcw* - f w*ddclizil 1\ ddcw* liz II =r liz II :s;r

(cf. 1.2, Chap. 2). As r --+ 1 the surface integral, here, tends to zero. From this we obtain that

w*ddClizll 1\ ddcw* = 0 in U

from which it easily follows that w* == O. 0 For further details, cf. Sadullaev (1982b).

1.3. Some Properties of the Class RO. We formulate a criterion for a function f to belong to the class RO at a point 0 E e, in terms of the Taylor coefficients of this function. The first such criterion was obtained by Gonchar (in terms of the Pade table of the function).

00

Lemma. Let f(z) = L akzk be holomorphic in a neighborhood of the closed k=O

unit disc V: Izl ~ 1 and let Aj" ... ,jk be the absolute value of the determinant (aj., aj.+1 , ... , aj.+k-l), v = 1, 2, ... , k. We set

Then

v,. = sup Aj' ..... jk· il'···.ik

f E RO .;:> lim v,.1/k2 = O. k-+oo

From this lemma, the following assertion follows in en. Theorem 1. If a function f, holomorphic in a neighborhood of 0 E en, belongs to

the class RO, then for each complex line 13 0, the restriction fl, also belongs to RO.

90 A. Sadullaev

Indeed, with the help of an appropriate linear transformation of en, we may assume that f is holomorphic in a neighborhood of the unit polydisc V = 'V X Vn and 1= {'z = O}, where I Z = (Zl"'" Zn-l)' Let us expandfin a Hartogs

00

series f(z) = L aj('z)z~. Corresponding to the coefficients ai'z),j = 0, 1, ... , we

obtain a seq~;~ce JIk('Z ). The functions :2 JIk('z) are then plurisubharmonic in 'V.

From the holomorphy of f in the vicinity of the closure of 'V, it follows that the coefficients aj('z) are bounded by some constant C, i.e., lai'z)1 :::;; C for all

j ~ O. From this it follows easily that the sequence :2 In JIk(,z) is upper bounded in 'V.

Moreover, from the condition f E RO it follows that

lim f k\ In JIk('Z) dV = -00. k-oo U

From the property of plurisubharmonicity, it follows, then, that

k12 In JIk(O) :::;; _1_ r k\ In JIk('Z) dV --+ -00, mes'V Ju

as k --+ 00. Thus lim Jlk1/k2 (0) = 0, and therefore, by the lemma, f(O, zn) belongs k~oo

to the class RO. 0 With the help of Theorem 1, we establish the fundamental assertion men

tioned at the beginning of this section.

Theorem 2. Let Dc en be a domain such that (9(D) c RO. Then the set C"\15 is pluripolar in C".

Indeed, let us fix a complex line I 3 0 and consider the function f E (9(1 (l 15). Since 15 is a domain of holomorphy, the function f extends holomorphically to 15, i.e., there exists a function j E (9(15) such that jlz == f From the hypotheses of the theorem, f E RO. But then by Theorem 1, jlz == f also belongs to RO. Thus, each function holomorphic in the plane domain I (l 15 belongs to the class RO and, consequently, the set 1\15 is polar in en. By the theorem in 1.2, en\15 is a pluripolar set in C". D

1.4. Further Properties of Pseudoconcave Sets. Let S be a closed pseudoconcave set in the unit polydisc V = 'V X Vn such that the closure (in C")S does not intersect the face 'V x aVn. If the intersection S (l {'z = IZO} is polar for almost every 'zo E 'V, then S is pi uri polar in V (Theorem in 1.2). However, an even stronger assertion is valid, which characterizes pseudo concavity of a set via its secti ons.

Theorem 1. Let S c V be a pseudoconcave set such that S (l {'V x aVn} = 0, and let E be a set of positive capacity in 'V. If the intersections l,zo (l S are polar for all lines /.zo = {'z = 'ZO}, Izo E E, then they are polar for all lines I,z, 'z E 'V, and consequently, S is pluripolar in V.


Indeed, since S n {'V x aVn} = 0, it follows that S is also pseudo concave in the domain 'V x C. Supposing, without loss of generality, that S does not meet the plane {zn = O}, we prove the theorem by contradiction. Suppose for some point 'a E 'V the intersection l'a n S is not polar. Then there exists, in l'a \S, a function f which does not belong to the class RO. Now, as in Theorem 1 of the previous section, we extend f from l'a \S to the domain ('V x C)\S, expand it in a Hartogs series and construct the sequence l-k('Z) corresponding to the coefficients. Then

lim k12 In l-k(,z) = -00 k-oo

(1)

for each fixed I Z E E. Since E is of positive capacity, (1) holds throughout 'V. In

particular, :2 In l-k(/a) = -00, and this contradicts the fact that f(/a, zn), f/: RO.

o Remark. We may also give a different proof of Theorem 1, without using the

class R O, by invoking a theorem of P6lya. If, for some point 'a E 'V the intersection l'a n S is not polar, then, there exists, in l'a \S, a holomorphic function f, such that lim Idk l1/k2 :f. 0, where d k are the Hankel determinants (cf., e.g.,

Goluzin (1966)). Extending f to the domain ('V x C)\S, we obtain, by the theorem of P6lya, that

lim k12 In I dk('z) I = -00, Z E E, k-oo

where dk(' z) are the1Hanke1 determinants for the function j, lll'a == f. Since E is not pluripolar, lim k21n I dk('a) I = -00. This contradicts the fact that Idkl 1/k2 =

k-oo

Idk ('a)1 1/k2 does not tend to zero as k ~ 00. Properties of pseudoconcave sets were studied also in the works of Oka,

Nishino, and Slodkowski. The following theorem is due to Oka (cf. Nishino (1962)) and Levi-Hartogs (cf., e.g., Shabat (1976)).

Theorem 2. Vnder the hypotheses of Theorem 1, if the intersections l,zo n S are finite (or discrete) for all I ZO E E, then they are also finite (respectively, discrete) for all lines l,z, I Z E I V, and S itself is an analytic subset of V.

We present also, without proof, the following connection between polar and pluripolar sets. A pluripolar set in cn is polar in 1R2n ~ cn (Sect. 2, Chap. 2), but the converse is, in general, not true. The metric dimension of non-pluripolar sets can range over the interval [0, 2n], while the metric dimension of non-polar sets is necessarily ~ 2n - 2. However, in the class of pseudoconcave sets, these two notions coincide (cf. Sadullaev (1982b)).

Theorem 3. If S is a pseudoconcave polar set in en ~ 1R2n, then it is pluripolar in cn.

92 A. Sadullaev

For other properties and applications of pseudo concave sets cf. Alexander and Wermer (1985, 1989), Berndtsson and Ransford (1986), Aupetit (1984) and Slodkowski (1986, 1990).

§ 2. Holomorphic Extension in a Fixed Direction

In this section we shall prove the following

Theorem 1. Let f be a function holomorphic in the polydisc

V ='V x Vn='V x {Iznl <r}

which, for each fixed ZO in some non-pluripolar set E c 'V, extends to a function holomorphic in Czn except for a finite number of singularities. Then f extends holomorphically to ('V x q\A where A is some analytic set.

The difficulty in this theorem consists in showing that the union of singular sets of f(,z, zn) is closed (under the additional assumption that this set is closed, we have the theorem of Ok a, stated in 1.4, Chap. 3). This difficulty is overcome with the help of Jakobi series.

2.1. Analyticity of the Singularity Set. Let us consider, in the plane C, a rational lemniscate G" more precisely, a connected component of the set Ig(z)1 < r, determined by some rational function g. Let f be holomorphic in some neighborhood of Gr. Then the integral

~ f f(e). g(e) - g(z) de 2ni iJG. g(e) - w e - z

defines a holomorphic function F(z, w) in the domain Gr x {Iwl < r}, and F(z, g(z)) == f(z) by Cauchy's integral formula. Expanding F(z, w) in its Taylor series in wand setting w = f(z), we obtain an expansion of f in a Jacobi series (cf. Chirka (1976)):

00

f(z) = I Ck(Z)gk(Z), (2) k=O

where the coefficients Ck are defined by the formula

1 f g(e) - g(z) Ck(z) = 2ni iJG. f(e) gk+l(eHe - z) (3)

and, consequently, are rational functions having poles at the poles of g, and deg Ck ~ deg g.

We remark that the series (2) converges uniformly in the interior of the lemniscate Ig(z)1 < R, where

(4)

and ro > ° is arbitrary, and diverges almost everywhere outside of this lemniscate (analog of the Cauchy-Hadamard formula).

Proof of Theorem 1. Let f be holomorphic in the polydisc 'V x {Iznl < r} and extendable to {'z} x (C\E('z)), as in the hypotheses, for arbitrary 'z E 'V. For fixed 'z E 'V, we develop this function in a Jacobi series with respect to the rational function

where at, a2, ... , am are "rational" numbers such that lajl ~ r,j = 1,2, ... , m. In the domain'V x {lg(zn)1 < 2e}, which is contained in'V x {Iznl < r}, for small e < 0, we have

00

f(,z, zn) = I Ck('z, Zn)gk(Zn)· (5) k=O

If R(' z) is the radius of convergence of (4), then the function ( -In R(' z))* is psh in 'V, and the series (4) converges uniformly in the domain

{'Z E 'V, Ig(zn)1 < R*('z)},

where R*('z) = exp{ -(-In R('z))*} is the regularization. Consequently, the function f is holomorphic in the union G of domains {'z E 'V, Ig(zn)1 < R*('z)} over all functions 9 corresponding to "rationals" at, a2, ... , am, lajl ~ 1,j = 1,2, ... , m, m = 1,2, ....

For a fixed point 'z E E, the union oflemniscates Ig(zn)1 < R*('z), over such g, is the plane minus a discrete set. Thus, for points 'zo E E such that R('zO) = R*('zO) for all g, the set {'z = 'ZO}\G is discrete. In view of the plurisubharmonicity of the functions

In sup Ick('z, znW/k Ig(zn)I,;,

the set of points 'z, for which R('z) #- R*('z) is pluripolar (cf. 3.3, Chap. 2). Since E is not pluripolar, it follows, then, that there exists a non-pluripolar subset E' c E such that the set {' z = ' ZO} \ G is discrete for all ' ZO E E'.

Using the class RO (cf. 1.3), it is easy to show that the envelope ofholomorphy G is single sheeted (lies in cn). Then, the set ('V x C)\ Gis pseudoconcave and {'z = 'ZO} \G is discrete for all 'zo E E'. By Oka's Theorem (1.4) ('V x C)\G is an analytic subset of V.

Remark. With the help of the rational functions

lajl ~ r,j = 1, ... , m; N = 1,2, ... , we can show the following result.

94 A. Sadullaev

Theorem 2. Suppose f is holomorphic in the polydisc 'U x {Iznl < r}, r > 0, and for each fixed' Z E 'U the function fC z, zn) has a finite number of singularities in the disc IZnl < R, R > r. Then the singularity set of f in the polydisc 'U x {Iznl < R} is analytic.

2.2. Pseudoconcave Sets and Analytic Tubes. The following theorem is due to Alexander and Wermer (1983).

Theorem. Let U1 : I Z 11 < 1 be the unit disc and S a nun-empty pseudoconcave subset of the domain U1 x C such that S c {IZ2 - a(z1)1 < r}, where a(zd is a function continuous in U1. Then there exists a function f(z 1) holomorphic in the disc U such that If(zd - a(zdl < 4r.

In other words, if a pseudoconcave set lies in some tube of constant radius, then it lies in a slightly larger tube with analytic center. A number of questions arise in connection with this theorem:

1. It is true that in any neighborhood U of a pseudo concave set S, there exists an analytic set of positive dimension A c U (Alexander-Wermer (1983»?

2. In the proof of the theorem, essential use is made of the fact that U1 is a plane disc. Is it possible to replace U1 by an arbitrary domain in IC"?

3. Is it possible to replace the disc-tube IZ2 - a(zdl < r in the theorem by a lemniscate-tube IP(Z1' z2)1 < r, where P is a polynomial in Z2?

It would be interesting also to study these questions with variable radius r(' z). The solutions to these problems might turn out to be useful, for example, in studying the capacitary properties of arbitrary pseudo concave sets.

§3. Multidimensional Analog of the Bernstein-Walsh Theorem and Separately Analytic Functions

The classical theorem of Bernstein-Walsh establishes a connection between the speed of approximation by polynomials, of a function f given on a compact set K c C, and the holomorphic extension of f to certain standard neighborhoods of K. In 3.2 of this section, an analogous theorem will be proved in IC".

3.1. The Generalized Green Function. For a compact set K c IC", the generalized Green function V(z, K) is most easily defined with the help of the class L, consisting of all functions u E Psh(lC") such that u(z) ::; rx + In(1 + Izl), z E IC", where rx = rx(u) is constant.

We set V(z, K) = sup {u(z) E L : u IK ::; o}; the regularization V*(z, K) is called the generalized Green function (or simply, Green function) for the compact set K.

F or a non-pi uri polar compact set K, the function V* exists (V* =1= + 00) and belongs to the class L; V* == +00 if and only if K is pluripolar.

As for .?J-measures (2.1, Chap. 2), we have

Theorem 1. If a compact set K is such that V*IK == 0,8 then the function V* is continuous in en and V*(z, K) == V(z, K).

If P is a polynomial of degree s, then the function ~ In IPI belongs to L. Thus, we have the Bernstein- Walsh inequality s

1 1 -In IP(z)1 :s; -In IIPIIK + V(z, K), z E en. s s

(6)

In the classical case when n = 1, the proof of the above mentioned BernsteinWalsh theorem is based on the construction of a sequence of polynomials Ps

such that IlPsIIK:S; 1 and ~ In IPs I converges to V(z, K) as s -t 00. There is a s

similar connection between the function V(z, K) and polynomials for n > 1 also.

Theorem 2. For any compact set Keen, we have the inequality

V(z, K) = sup {de~ P In IP(z)1 : IIPIIK :s; I}. (7)

Proof. Any compact set K in en can be approximated by pluriregular compacta

K/j : dist(z, K) :s;~, ~ > O.

Thus, it is sufficient to prove the theorem for pluriregular compacta. Consider the auxiliary psh function

h(z, w) = Iwl exp V(;, K), which is homogeneous and continuous in en+!.

Fix a point (ZO, 0)0) =f. 0 and a number 8, 0 < 8 < h(zO, 0)0). From Bremermann's Theorem (cf. 2.4, Chap. 1), it follows that the circled compact set K = {(z, 0)): h(z, 0)) :s; h(zO, 0)0) - 8} is polynomially convex. Consequently, it is convex with respect to homogeneous polynomials (cf. 3.3, Chap. 3). It follows that there exists a homogeneous polynomial Qs(z,O)) such that I Qs(ZO, 0)°)1 > IIQsIIK' Let

Ps = 1Ii.IIK h(zO, 0)0) - 8.

Then IPsl1/s:s; h on the compact set K and from the homogeneity of the functions IPs I lis and h we obtain the inequality

IPs I lis :s; h everywhere in en+l .

8 Such compact sets are said to be pluriregular in e.

96 A. Sadullaev

On the other hand, at the point (ZO, WO), we have

IPs(ZO, wOW/s > h(zO, WO) - e.

Since e > ° is arbitrary and also the point (ZO, wo), we obtain that

h(z, w) = sup{lPs(z, w)ll/s: IPsl l/S :::;; h}.

Setting, here, w = 1, we have (7). 0

3.2. The Main Result. Let K c Cn be a pluriregular compact set and f a continuous function on K. The following connection holds between the speed at which the error emU, K) of best polynomial approximation decreases and the holomorphy of f in domains of the form

GR={zECn:V*(z,K)<lnR}, R>O.

Theorem. The function f extends holomorphically to GR if and only if

- 1 lim el/m(f K) < -

m-oo m , - R'

Proof a) If this inequality holds, then there exists a sequence of polynomials Pm(z) such that

2 Thus IlPm+l - PmllK :::;; R m and by (6)

2 IPm+l(z) - Pm(z) I :::;; Rm exp{(m + I)V(z, K)}, z E e.

00

From this it follows that the series Po + L (Pm+l - Pm) converges uniformly on m=l

compact subsets of GR , to a holomorphic function which, on K, coincides with f b) To prove the other direction of the theorem, we use (7) which, together

with Theorem 1 of 3.1, allow us, for arbitrary fixed e > 0, to select a finite set of polynomials ~, deg ~ = s,j = 1,2, ... , N, such that

supnlnl~l,j = 1,2, ... , n} > In R - e,

p. for all z E iJGR . It follows that if we set ~ = e-e; R S ' then the Wei! polyhedron

II = {I~I < 1 :j = 1,2, ... , N} is a compact subset of GR , and

KC{I~ll/s:::;;e_~R,j=I,2, ... ,N}. (8)

The function f which is holomorphic in GR can be expanded in II in a series via the polynomials ~ (cf., e.g., Shabat (1976)):

00

f(z) = L Ak&>fl ... &>tN ,

Ikl=O

where k = (k1' kz, ... , kN) are multi-indices, Ikl = k1' kz, ... , kN' and AK are polynomials of a fixed degree which depends on &>1' f1JJz, ... , f1JJN. The Ak satisfy a Cauchy type inequality: IIAkl1 n:::;; C(f), Ikl = 0, 1, ... , where C(f) is a constant independent of k.

Let q

Qj = L Ak&>fl ... &>tN,

Ikl=O be a polynomial of degree j = t + sq. Then according to (8)

where C(f, N) is a constant independent of f. From this it follows that

lim Ilf - QjllJP :::;; -1-- and, since e > ° is arbitrary, lim e;,;m(f, K) :::;; L D j~oo e R m~oo R

j=t+sq

In case the compact set K is not pi uri regular, the second part of the theorem, in general, does not hold on account of the discontinuity of V*(z, K): one can construct a compact set K and a function f, holomorphic in the domain

GR = {V*(z, K) < In R}, and continuous on K, such that lim e;,;m(f, K) > L m~oo R

However, approximating the compact set K by pluriregular compacta K~, one can show part b) of the theorem in the following weaker form: if f is holomorphic in a neighborhood of the set {V(z, K) :::;; In R}, then - 1 lim e1/m(f K) < -.

m-oo m , - R

3.3. Green Functions for Circled Sets. Projective Capacity. Let K be a circled compact set in en, i.e. for each point ZO E K the set K contains all points of the form eiCPzO, qJ E R. We show that the polynomially convex hull K of such a compact set coincides with K, the convex hull of K with respect to homogeneous polynomials. For this it is sufficient to show that if ZO E R, i.e. if IQ(zO)1 :::;; IIQIIK for all homogeneous polynomials Q, then this inequality holds also for all other polynomials.

N

Fix a number ° < (j < 1 and consider a polynomial P(z) = L Qs(z) with s=O

norm IIPllk = 1. By Cauchy's inequality on the slices {z = ),IX, A E IC} the norms IIQsllK of the homogeneous polynomials are also bounded by one and,

98 A. Sadullaev

N 1 consequently, IQs(zO)I:::;; 1. Hence IPs(UZO)1 :::;; s~o IQs(zO)lus :::;; 1 _ u· Thus,

uzo E K, since this inequality is also satisfied by the polynomials pi for arbitrary j. Letting u tend to 1, we obtain that ZO E K.

Making use of this remark, we prove the following curious result.

Theorem 1. If K is a circled compact subset of the closed unit ball B(O, 1), then

K = {z : 1 z 1 . exp V (I; I' K ) :::;; I}.

Indeed, if Qs is a homogeneous polynomial with IIQsllK = 1, then according to (6), we have

Consequently,

and, hence,

K:::> {z: Izl'exp V(I;I' K) :::;; I}.

In fact, here, instead of the inclusion :::>, we have equality, since the set on the right hand side is polynomially convex.

Corollary (compare Alexander (1981)). If a circled compact set K is contained in the unit sphere S(O, 1), then its polynomially convex hull K contains the ball

Izl:::;; exp{-sup V(e, K)}. 1~1=1

The expression on the right hand side of this inequality is related to the so called projective capacity <6'(K), introduced by Alexander (1981). A quantity close to <6'(K), namely, .-capacity, was considered by the author (1982b).

In closing this section, we introduce another application of the estimate (6), this time to questions of holomorphic continuation.

00

Proposition 1. Let L Qs(z) be a formal series of homogenous polynomials Qs' s=O

and let 2 be a family of complex lines I 3 0. If for each line I E 2 this series converges in the disc In B(O, 1), then it converges normally in the domain

G = {z E en: Izl' exp V* (1;1 ' E) < I},

where E = U In S(O, 1). IE !t'


Indeed, fix e > ° and set

According to the Cauchy inequality,

Thus, by the Bernstein-Walsh inequality

IQs(w)1 ~ (1 ~ e)S [exp V*(w, EN)]S, WE Sea, 1),

00

and, hence, the series L Qs(z) converges uniformly in the interior of the set s=O

Letting N tend to 00 and e to zero, we obtain the convergence of the series inside G.O

Proposition 2. Let

00

f = L Ps(z, z) s=O

be a formal power series such that the restriction fll represents a holomorphic function in the disc 1 n B(O, 1), for each complex line 1 3 0. Then the series converges in B(O, 1) and its sum is holomorphic there.

Indeed from the holomorphicity of the restrictions of the series to complex lines 1, it follows easily that the terms of the homogenous polynomials Ps ' which contain z are zero. The rest of the proof follows from Proposition 1. 0

From Proposition 2 we easily have the following.

Theorem 2 (Forelli, cf. Rudin (1980)). If f is infinitely differentiable at the point ° and the restriction fll is holomorphic in the disc 1 n B(O, 1) for each complex line 130, then f is holomorphic in the ball B(O, 1).

The function

Zk+1 Z f(zl,z2) = 12,

Zl Z l + Z2 Z2

which is k times continuously differentiable in (;2, is not covered by Theorem 2

(f does not determine a formal power series s~o Ps(z, z) ).

100 A. Sadullaev

The function f(zl' Z2) = IZll2 -IZ212 is identically zero on the complex lines Z2 = eiOzl> e E [0, 2n], which form a non pluripolar set. Nevertheless, f is not holomorphic at any point. This example shows that formal power series, containing Z, also are not covered by Proposition 1.

3.4. Separately Analytic Functions. Consider two domains D c Cn(z), G c

en(w), two sets E c D, F c G and set X = (D x F) u (E x G). A function fez, w), defined in X is said to be separately-analytic if it is holomorphic as a function of w in G for each fixed Z E E and holomorphic as a function of Z in D for each fixed WEF.

If E = D and F = G, we obtain a function which is holomorphic in each variable in D x G; in this case f is jointly holomorphic in D x G, according to Hartogs Theorem. There is a deep connection between the notion of separate analyticity and the well known "edge of the wedge" theorem of N.N. Bogolyubov (cf. e.g. Vladimirov (1964), Zakharyuta (1976)), which for n = m = 1 corresponds to the situation when E and F are intervals of the real axis.

Consideration of this particular case naturally gives rise to the general question of hoi om orphic continuation of f to a neighborhood of X. In this section we show the possibility of holomorphic continuation to a neighborhood which is defined in terms of Y'-measures of E and F with respect to D and G. In the sequel, for the sake of simplicity, we shall suppose that D and G are strictly pseudo convex domains. We shall also suppose that E and F are compact, although from the C-measurability property of capacity (3.4, Chap. 2), it will follow that the theorem stated below holds also for arbitrary Borel subsets F andE.

Theorem. If a function fez, w) is separately analytic on the set X = (D x F) u (E x G), then it extends holomorphically to a neighborhood of the set

x = {(z, w) E D x G: w*(z, E, D) + w*(w, F, G) + 1 < O}.

In the case of pi uri regular compact sets K and F, the set X is itself a neighborhood of X.

The theorem has content (X #- 0) if neither of the sets E and F is pluripolar, which we shall assume below.

An important part in the proof of the theorem is the construction of a special orthogonal basis for the pair E c D, however, here, we will only list the properties of this basis, refering, for further details, to the works of Zakharyuta (1974, 1976).

Let D c en be a strictly pseudoconvex domain and E c D a non pluripolar compact set. We shall associate to the domain D any Hilbert space Ho such that

(!J(15) c Ho c c(!J(15) = C(15) n (!J(D). (9)

We define also a Hilbert space Hl associated to the compact set E. It is the closure of the space (9(D) with respect to the norm

where the measure a = (ddcw*(z, E, D)t is concentrated on E (cf. Chap. 2). Let, now, {hk(Z)} be a common orthogonal basis in the spaces Ho, H1 , sat

isfying the conditions:

Such a basis exists since, by construction, Ho is densely and completely continuously embedded in the space H1 • In addition (cr., e.g. Zakharyuta (1974))

where L is a constant.

1 _kl/n < In /I < L· k 1/n L - f"'k - ,

From the continuous embedding (9), it follows that

where C is a constant.

(10)

(11)

Consider the set Ak = {z E E: Ihk(z)1 > k}. According to the Chebyshev inequality,

OCJ

U Es· Then a(E\E') = 0, and from the majorization principle (cf. 1.3, Chap. 2), k=l

we have the identity w*(z, E, D) = w*(z, E', D). Consequently, if s -+ 00

w*(z, Es ' D)! w*(z, E, D), zED. (12)

Since, by construction, I hk(z) I ::; k, Z E Es ' k ~ s, then, using (11) and with the help of the two constants theorem (2.1, Chap. 2), we obtain

(13)

where C(s) is a constant independent of k. We pass to the proof of the theorem. We fix domains D' and G' such that E c D' ~ D, Fe G' ~ G and we

denote

EN = {z E E: Ilf(z, w)IIG"::; N}

FN = {w E F: Ilf(z, w)lliY, ::;N}.

102 A.Sadullaev

OC! OC!

Then E = U EN' F = U FN and thus, after some No, the sets EN and FN N=l N=l

will be non pluripolar. Let us show that the function f(z, w) is continuous on (D' x FN) u (EN x G'), N 2:: No. Indeed, as z varies in EN the functions f(z, w) form a compact family of functions (in w) on G'. In particiar, it is equicontinuous. The same is true with respect to ZED' as w varies in FN • Moreover, since the sets EN and FN are not pluripolar, and hence are sets of uniqueness for holomorphic functions,

lim f(z, w) = f(zO, w), z, ZO E EN' W E G', z-+zO

lim f(z, w) = f(z, WO), zED', w, WO E FN • w_wO

From this follows the continuity of f on the designated set. In particular, f is continuous on the compact set EN x FN.

We construct, for EN c D' the Hilbert spaces Ho, Hl and their common orthogonal basis {hk }, satisfying conditions (10) and (13) with E = EN' D = D'; consider also the analogous basis {ek} for FN c G'.

We associate to f its formal double Fourier series

OC!

f ~ L akjhk(z)eiw), k,j

whose coefficients are defined by

with

da'z{hk) = hk(z)' (ddCw(z, EN' D'))",

d/lw(e) = eiw)' (ddCw(w, FN , G'))",

Borel measures on EN, and FN. Let us show that the series (14) converges uniformly in the open set

XN = {(z, w) E D' x G': w*(z, EN' D') + w*(w, FN, G') < -I}.

Since, for fixed WE FN , the function f(z, w) belongs to Ho, we have

Analogously, we get:

(14)

where the sequence {vJ is to the basis {ej} as {/lk} is to {hd. From this and by


(13), we obtain, for Vj ::; J1.k' the estimate

I akjhk(z)ej(w) I ::; c(s, f). k·j· J1.;1 . J1.~ +w*(z,EN ... D'). v]+w*(w,FN ... G')

::; c(s, f)· k oj' J1.~ +w*(z,EN.s,D')+w*(w,FN.s,G')

and an analogous estimate for J1.k ::; Vj' Thus, for any compact set S c XN, choosing s sufficiently large and using (12), we obtain the estimate

I akjhk(z)ej(w) I ::; c(s, f). k 'j(max {J1.k' vJ t'l,

where (j = (j(S) > O. By (10), this estimate ensures the uniform convergence in XN of the series (14).

Clearly, the sum, fN(Z, w) of this series is a holomorphic extension of the function f(z, w) to X N' Approximating the domains D and G by a non-decreasing sequence of domains D~ and G~, we construct a holomorphic extension of the function f to the set X. 0

Remarks. The results of § 1 are due to the author (1982b, 1984); the main result of this section (1.3, Theorem 2) gives a positive answer to a question posed by Gonchar. The use of convergence in measure in the definition of the class RO was needed because the question of rapid uniform approximation of functions having a pluripolar set of singularities is open for n > 1. In the particular case when f has an analytic set of singularities, the possibility of such approximation by rational functions was shown by Chirka (1974).

Theorems 1 and 2 of 2.1 are due to Chirka and Kazaryan (cf. Kazaryan (1983) where a particular case is analyzed).

An analogue of the Bernstein-Walsh Theorem in en was shown by Siciak (1962) and later, using a different method, by Zakharyuta. See also Korevaar (1986).

Theorem 2 of 3.1 is due to Zakharyuta. The proof presented here is due to Siciak. The theorem is 3.4 was proved by Zakharyuta (1976). It can be extended without difficulty to the case of several, rather than two, groups of variables; more precisely, to the case where f is a separately analytic function on the set

where

00

X = U {E I X ... X Ej - I X Dj X Ej+1 X ... x Ed, j=l

Ej c Dj c en}, nj E N, j = 1, 2, ... , k.

In Chapter 3 we presented a series of applications of complex potential theory. Of course, we were not able to cover all such applications in all fields of function theory. For example, in recent years, the basic objects of this theory (Y'-measures, Green functions, capacities, etc.) have begun to be used in Nevanlinna theory (cf. Shabat (1982)). In particular, with the help of &'-measures, it has been shown that for any holomorphic mapping f: en -+ pm, the union of its defective divisors, in the sense of Val iron, is a pluripolar set.

104 A. Sadullaev

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Engl. transl.: Math. USSR, Sb. 30, 501-514 (1978), Zb1.346.32024 Sadullaev, A. (1980): The operator (ddcu)n and condenser capacities. Dokl. Akad. Nauk SSSR 251,

No. 1,44-47. Engl. transl.: Sov. Math., Dokl. 21, 387-391 (1980), ZbI.488.31005 Sadullaev, A. (1981): Plurisubharmonic measures and capacities on complex manifolds. Usp.

Mat. Nauk 36, No.4, 53-105. Engl. transl.: Russ. Math. Surv. 36, No.4, 61-119 (1981), Zb1.475.31006

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Shabat, B.V. (1976): Introduction to Complex Analysis, Vol. 2. Moscow: Nauka. 400 pp. French transl.: Moscow: MIR 1990, ZbI.578.32001

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Slodkowski, Z. (1990): Polynomial hulls with convex fibers and complex geodesics. J. Funct. Anal. 94,156-176, ZbI.717.32009

Vladimirov, V.S. (1964): Methods of the Theory of Functions of Several Complex Variables. Moscow: Nauka. 411 pp. French transl.: Les fouctions de plusieurs variables complexes et leurs application. Paris: Dunod 1967,338 pp., Zb1.125,319

Zakharyuta, V.P. (1974): Extremal plurisubharmonic functions, Hilbert scales and isomorphisms of spaces of analytic functions of several variables, I, II. Teor. Funk. Funkts. Anal. Priloz. 19, 133-157, ZbI.336.46031; 21, 65-83, Zb1.336.46032. (Russian)

Zakharyuta, V.P. (1976): Separately analytic functions, generalized Hartogs theorem, and envelopes ofholomorphy. Mat. Sb., Nov. Ser. 101, No. 1,57-76. Engl. transl.: Math. USSR, Sb. 3D, 51-67 (1978), ZbI.357.32002

References Added (by Author) in Translation

Alexander, H., Wermer, J. (1985): Polynomial hulls with convex fibers. Math. Ann. 271, 99-109, Zb1.538.32011

Alexander, H., Wermer, J. (1989): Polynomial hulls of sets with intervals as fibers. Complex Variables, Theory Appl. II, 11-19, Zb1.673.32017

106 A. Sadullaev

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Ampere operator in C". Math. Z. 193,373-380, ZbI.624.31004 Cegrell, U. (1990): The Dirichlet problem for the Complex Monge-Ampere operator: Stability in U.

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III. Function Theory in the Ball

A.B. Aleksandrov


Contents

Preface ....................................................... 110

Chapter 1. Introduction ........................................ 111

§ 1. Preliminary Information ..................................... 111 1.1. Notation .............................................. 111 1.2. Integration on the Sphere ................................ 111 1.3. Differentiation Operators ................................ 113 1.4. Manifolds ............................................. 113

§ 2. Automorphisms of the Ball ................................... 114 2.1. Description of the Automorphisms of the Ball.. . .. . . . .. . . ... 114 2.2. The Bergman Metric .................................... 115 2.3. The Cayley Transform .................................. 116

§ 3. iff-Invariant Subspaces ...................................... 117 3.1. The Spaces H(p, q) ...................................... 117 3.2. Explicit Formulae for the Kernel Kpiz, 0 .................. 118 3.3. Generalized Functions on the Sphere S .................... 119 3.4. The Tangential Cauchy-Riemann Equations ................ 120 3.5. Multiplicative Properties of the Space H(p, q) ............... 120 3.6. Ryll-Wojtaszsczyk Polynomials ........................... 120

§4. Nonisotropic Quasimetrics on the Sphere S ..................... 121 4.1. Elementary Properties of Nonisotropic Quasimetrics ......... 121 4.2. Hausdorff Measure and Dimension ........................ 122

Chapter 2. Fundamental Integral Representations .................. 123

§ 1. Fundamental Spaces of Functions Holomorphic in the Ball ....... 123 1.1. Notation .............................................. 124 1.2. The Nevanlinna and Smirnov Classes ...................... 125 1.3. The Hardy Classes ...................................... 126

108 A.B. Aleksandrov

§ 2. Fundamental Integral Representations ......................... 127 2.1. The Cauchy Kernel ..................................... 127 2.2. The Bergman Kernel .................................... 128 2.3. The Invariant Poisson Kernel ............................ 128 2.4. The "Harmonic" Poisson Kernel .......................... 129 2.5. Which Problem does the Invariant Poisson Integral Solve? .... 129 2.6. H(p, q)-Expansion of the Cauchy and Poisson Kernels ....... 131

Chapter 3. Boundary Properties of the Cauchy Integral and the Invariant Poisson Integral ............................................ 131

§ 1. The Maximal Function ...................................... 131 1.1. Properties of the Maximal Function ....................... 131 1.2. K-Limits .............................................. 132 1.3. The Lindel6f-Chirka Theorem ............................ 133 1.4. Carleson Measures ..................................... 134

§ 2. The "Real" Hardy Class ..................................... 135 2.1. The Carleson-Duren-H6rmander Theorem ................. 135 2.2. Atomic Decomposition in Hardy Spaces ................... 136

§ 3. Dual Spaces for Hardy Spaces yt'P(S) and Spaces of Smooth Functions ................................................. 139 3.1. Dual Spaces and Spaces of Multipliers ..................... 139 3.2. The Cauchy Integral in Spaces of Smooth Functions ......... 141

§ 4. Dual Spaces of Some Spaces of Holomorphic Functions .......... 142 4.1. The Duren-Romberg-Shields Theorem ..................... 142 4.2. The Dual Space of Hl(B) ................................ 143

§ 5. The T6plitz and Hankel Operators ............................ 144 5.1. The T6plitz and Hankel Operators on the Space H2(B) ....... 144 5.2. The T6plitz and Hankel Operators on the Spaces HP(B)

(0 < p ::; +(0) .......................................... 145 5.3. T6plitz Operators and Multipliers ......................... 146 5.4. Applications of T6plitz Operators to a Problem of Gleason 147

Chapter 4. Zeros of Functions Holomorphic in the Ball .............. 149

§ 1. Characterization of Zeros of Functions in the Smirnov, Nevanlinna, and Nevanlinna-Dzhrbashyan Classes ......................... 149 1.1. One-Dimensional Results ................................ 149 1.2. The Khenkin-Skoda Theorem ............................ 150 1.3. Discussion of the Blaschke Condition ...................... 150 1.4. The Khenkin-Dautov Theorem ........................... 151

§ 2. Zeros of Functions in the Hardy Spaces HP(B) .................. 152 2.1. Uniform Blaschke Condition ............................. 152 2.2. Piecewise-Linear Analytic Sets ............................ 153 2.3. Zeros of Bounded Holomorphic Functions ................. 154

III. Function Theory in the Ball 109

Chapter 5. Interpolation, Peak Sets, A-Measures and P-Measures 156

§ 1. Representing Measures and A-Measures ....................... 156 1.1. A-Measures, Representing Measures, Totally Singular Measures

and Their Properties .................................... 156 1.2. A-Measures an Boundary Behavior of Bounded Holomorphic

Functions ............................................. 158 1.3. A-Measures and Isomorphism Classification of Banach Spaces of

Analytic Functions ..................................... 158 § 2. Null Sets and Interpolation on the Sphere S by Functions in the Class

A(B) ...................................................... 159 2.1. Z-Sets, P-Sets, I-Sets, and Null Sets ....................... 159 2.2. Examples and Properties of I-Sets ......................... 159 2.3. Boundary Uniqueness Sets ............................... 160 2.4. Maximum Modulus Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 160 2.5. Interpolation Within the Ball by Functions in the Classes A(B)

and HP(B) ............................................. 161 § 3. P-Measures ............................................... 161

3.1. Integral Representations of P-Measures .................... 161 3.2. The Khrushchev-Vinogradov Asymptotic Formula .......... 162 3.3. "Smoothness" and "Regularity" Properties of P-Measures ..... 163

§4. P-Measures and the Boundary Behavior of Holomorphic Functions ................................................. 165 4.1. P-Measures and the Hardy-Lumer Class ................... 165 4.2. P-Measures and Boundary Values of Holomorphic Functions . 167 4.3. LSC-Property .......................................... 168 4.4. Outer Functions ........................................ 169

§ 5. Peak Sets for Smooth Functions .............................. 170 5.1. Peak Sets and Local Peak Sets ............................ 170 5.2. Peak Sets and Interpolation .............................. 171 5.3. Finitely Generated Ideals in the Algebra AOO(B) .............. 171

Update on Problems from Rudin's Book (1980) Solved up to the Present Time .............................................. 171

References .................................................... 174


Preface

In the theory of functions of several complex variables, the ball occupies a crucial position: on the one hand, the ball is the simplest example of a strictly pseudoconvex domain with smooth boundary; on the other hand, it is the simplest bounded classical domain.

The theory of functions in the ball is very well and rather completely set forth in the book of Rudin (1980). Rudin's book has had a great influence both on the choice of material for the present paper as well as on its presentation. In the present paper, several results l are also exposed which were clarified in the four years following the appearance of Rudin's book. Moreover, the BM02 space and the atomic technique, which were completely absent in Rudin (1980), are herein presented. Almost all results of this paper (in which the group of automorphisms of the ball, harmonic analysis on the sphere and related aspects of the "pure" theory of functions on classical domains play no role) hold for strictly pseudoconvex domains with C2-boundary and sometimes, even with weaker conditions on the boundary.

We remark that such important topics of the theory of functions as the a-problem and related questions, analysis on the Heisenberg group, proper holomorphic mappings, and the maximal ideal space of the algebra HOO(B) will not be treated at all in this paper.

For the a-problem, we refer the interested reader to the survey of G.M. Khenkin and E.M. Chirka (1975), the book of Khenkin and Leiterer (1984), and Rudin (1980).

We remark also that the proofs of many of the results herein make use of appropriate theorems on the solution of the a-equation and variants thereof.

For proper holomorphic mappings, we refer the reader to the paper of S.I. Pinch uk (1986). Concerning the maximal ideal space of HOO(B), a certain amount of information can be gathered from Axler-Shapiro (1983), McDonald (1979), and Rudin (1983).

The corona problem, which to this day has not been solved in the multidimensional case, is discussed in Khenkin-Leiterer (1984).

The author is very grateful to G.M. Khenkin for many helpful suggestions. I also extend my sincere gratitude to S.A. Vinogradov for consultations on atomic theory and to V.V. Peller for consultations on the theory of Hankel and T6plitz operators.

1 For the convenience of the reader, at the end of this paper we append a list of those problems presented in Rudin (1980) which have since been solved. 2 BMO-Bounded Mean Oscillation.

Chapter 1 Introduction

§ 1. Preliminary Information

111

1.1. Notation. The symbols < , ) and I I will denote the scalar product and the norm in en.

def n _ def r;-:-:::: <Z, W) = L ZjWj' Izl = v <Z, Z).

j=1

Let {e1' e2, ... , en} be the standard basis in en, i.e. <z, e) = Zj for all Z E en and all j E {1, 2, ... , n}. We shall denote the open unit ball in en by Bn ~ {z E C" : I Z I < 1}. The boundary of this ball is the unit sphere Sn ~ {z E en : I Z I = 1}. We set [j) = B 1 and lr = S l' The number n will (generally) be fixed and so we will usually write Band S rather than Bn and Sn. We denote by pn-1 the complex projective space of dimension n - 1 (Le. the collection of all onedimensional linear subspaces of C"). The symbol 1t = 1tn will denote the canonical projection from en\{O} (or S) onto pn-1.

To each multi-index IY. E Z,:- ~ {IX E zn = IY.j;::: O}, we associate two numbers:

The first of these notions will not give rise to ambiguity for in any given situation it will be clear whether we are dealing with a vector in C" or a multi-index. We set

1.2. Integration on the Sphere. To every positive measure /1 we associate the space L 0(/1) of all/1-measurable /1-almost everywhere finite functons (functions equal/1-almost everywhere are identified). As usual

L 00(/1) ~ {f E L 0(/1): II I ilL", ~ ess sup III < +oo}.

The symbol (J = (In (respectively 6') will denote the unique probability Borel measure on S (respectively on pn-1), invariant with respect to all unitary transformations of en, m ~ (J l' Let v = Vn denote Lebesgue measure in en normalized by the condition v(B) = 1.

We denote by C(K) the space of all continous functions on a compact set K. The dual space of C(K) is identified in a natural way with the space M(K) of all regular Borel measures on K.

The space U(o") is identified in a natural way with the set of all absolutely continous (with respect to a) measures in M(S). In the future we will often write U(S) instead of U(a).

We present several convenient integral formulae, which will hold if either the left member or the right member has a meaning. The letter f will denote a Borel function (whose domain of definition will be defined by the formula itself).

Let Z = (u, w), where u E C\ WE cn-\ i.e.

{Ui l sj s k),

Zj = wj-k(k + 1 sj s n), Then

f f dan = (~ = ~) f (1 - Iw1 2t-1

Sn Bn-k

X (Ik f((1 _lwI2)1/2(, w) dak(o) dvn_k(w). (1)

Here and in the sequel (p) ~ ,P! . We note two important cases of (1). m m.(p - m)!

f f(Zl,Z2"",zn-ddan(Zl,Z2"",Zn) Sn

= f f(Zl, Z2, ... , zn-d dVn-1(Zl, Z2, ... , zn-d, (2) Bn - 1

f f«z, 0) dan(z) = (n - 1) f f(~)(1 _1~12t-2 dVl(~) (3) Sn ~

for all (E S. We present two more formulae.

fen f dv = 2n I'" r2n - 1 (I f(rO da(o) dr, (4)

fen f(z)lzl- 2n dvn(z) = n In (fe f((z)lzl- 2 dV1(Z)) dan(O· (5)

It is easy to see that the family {z<X} <xe z~ forms an orthogonal system in L 2(B, v) and in L 2(S, a).

The following formulae hold:

(6)

(7)

Formula (7) follows from formulae (6) and (2). In the right members of formulae (6) and (7), we have intentionally used the gamma function (instead of the factorial), in order for these formulae to make sense not only for integers !Xj , but also for complex numbers !Xj (Re !Xj > -1), if in the left members instead of Iz"12 we

n

write TI Izj I2"j. j=l

1.3. Differentiation Operators. We set

D.=~(~-i~) D.=~(~+i~) ) 2 oXj 0Yi' ) 2 oXj OYi'

where Xj = Re Zj' Yj = 1m Zj' Let Q be an open subset of en. A function J E C 1 (Q) is said to be holomorphic if Djf = 0 for all j E {1, 2, ... , n}. Every holomorphic function is infinitely differentiable and has a power series expanSiOn:

in the neighbourhood of any point a E Q, where D"f = D~' D~2 ... D:"f A function f E C2 (Q) is called pluriharmonic if DkDjJ = 0 for all k, j E {1, 2, ... , n}. A real pluriharmonic function is locally the real part of a hoi om orphic function.

n

To each holomorphic function f we associate the function f1ltof = L zjDjf j=l

If f and g are holomorphic in a neighbourhood of the ball Band f(O)g(O) = 0, then

f 2k+l-l f - ( 1 )k+l-l S f{} d(J = n(k + 1- 1)! x B (f1JttJ)(f1lt~g) log fZI Izl-2n dv(z), (8)

where k, IE Z+, and k + I > O. This formula easily reduces to the onedimensional case with the help of (5).

1.4. Manifolds. Let M be a smooth (real) manifold (with boundary). We shall denote the tangent space to the manifold M at the point p E M by Tp(M). We shall consider only manifolds which are submanifolds of en. In this case the tangent space Tp(M) can be thought of, in a natural way, as a \R-linear subspace ofe.

A smooth manifold M c e is said to be generic at a point p E M if Tp(M) + iTp(M) = e. Along with the tangent space Tp(M), we may associate to each point p EMits complex tangent space T/:(M) ~ Tp(M) n iTp(M). When M = S, we have:

Tp(S) = {z E en: Re<z, p) = O}, TpC(S) = {z E en: <z, p) = O}.

A smooth manifold M c e is said to be totally real at a point p E M if Tpc(M) = {O}. If a manifold M c e is generic (respectively totally real) at one

of its points, then dim M ~ n (respectively dim M ::::;; n). We remark also that the dimension of a manifold M c S, which is totally real at one of its points, is at most n - 1. A smooth manifold M c S is said to be integral (or complex tangential) at a point p E M if Tp(M) c TpC(S). We shall say that a manifold is generic (respectively totally real, integral) if it is such at each of its points.

1.4.1. Theorem (see Khenkin (1976), Rudin (1980)). Let M be a C1-manifold, M c S. If M is integral, then Tp(M) is orthogonal to i1;'(M) (in 1R2n ~ en) for each p E M, i.e. <~, '1 > E IR for all ~, '1 E I;,(M) and all p E M.

1.4.2. Corollary. Under the conditions of Theorem 1.4.1, the manifold M is totally real, and consequently, dim M ::::;; n - 1.

It is clear that a manifold Me S of dimension n (and even 2n - 2) can be integral at some (but not all!) of its points (n ~ 2).

§ 2. Automorphisms of the Ball

An automorphism of a domain Q c en is a biholomorphic mapping of the domain onto itself. We shall denote by Aut(Q) the group of all automorphisms of a domain Q.

2.1. Description of the Automorphisms of the Ball. The group Aut(Bn) for n = 2 was described by Poincare (see Rudin (1980)). The case of arbitrary n is completely analogous to this particular case (cf., for example, Shabat (1976) and Rudin (1980)). Each automorphism d of the ball B is a fractional linear transformation, i.e. is of the form:

n

allo + L allvzv v=l

wll=---n---

aoo + L aovzv v=l

(W = dz).

Each automorphism d is represented by a unique such matrix A = {allv }, 0 ::::;; J1 ::::;; n, 0 ::::;; v ::::;; n, up to non-zero multiple factors. In order to characterize all matrices which correspond to automorphisms of the ball, we denote by J the

matrix ( -; 1 ~). where Ik is the identity (k x k)-matrix. A matrix A is said to

be J -unitary if A * J A = J. In other words, the matrix A induces an operator in en +1 which preserves the sesquilinear form <J', . >,' also called an indefinite metric.

A matrix A determines an automorphism of the ball B if and only if A = cAo, where c E C\ {O}, and Ao is a J-unitary matrix. Thus, we may identify the group Aut(Bn) with the group of all J-unitary matrices factored by its centre, the subgroup of scalar matrices If· I n +1 •

The group Aut(Bn) is a connected Lie group of dimension n2 + 2n.

To each point a E B we may associate a unique automorphism CPa' enjoying the following properties:

I. CPa(O) = a, CPa(a) = 0; 2. CPa = cp;;l; 3. CPa has a unique fixed point.

Such an automorphism can be described by an explicit formula

Here, Pa denotes the orthogonal projection onto the one-dimensional subspace IC . a, and Qa the projection onto its orthogonal complement, i.e.

<z, a) Paz=-< )a, a,a

<z, a) Qaz = z - -< ) a. a,a

The automorphism CPb 0 CPa carries the point a E B to the point b E B. Thus the group Aut(B) is transitive on B. Any automorphism of the ball B which fixes the origin is a unitary operator. Consequently, any automorphism IjJ E Aut(B) has a unique representation of the form:

IjJ = U 0 CPa (respectively IjJ = CPa 0 U),

where U is a unitary operator. A point a E B is uniquely determined by the equation ljJ(a) = 0 (correspondingly 1jJ(0) = a).

We present two convenient formulae, in which IjJ E Aut(B) and a = ljJ-l(O),

I _ <.I,(Z) .I,(W) = (I - <a, a»)(1 - <z, w») ( B) '1','1' (I-<z,a»)(J-<a,w») Z,WE ,

(9)

, (1 - lal 2 )n+l detn;! IjJ (z) = 11 _ <z, a)12 (10)

In the ball B there exists a unique (up to a multiplicative constant) regular Borel measure which is invariant with respect to automorphisms of the ball. From formulas (9) and (10) it follows that such a measure is r,

dr(z) = K(z) dv(z),

2.2. The Bergman Metric. There exists, in the ball B, a Riemannian metric, unique up to a multiplicative constant, which is invariant with respect to the group Aut(B):

(11)


This metric is called the Bergmann metric. Formula (11) shows that this metric is a Hermitian metric. Moreover, this metric is Kahler (cf. Wells (1973)) since its associated differential form w (of bidegree (1, 1)) is exact. The exactness of this form is most easily established by noting that w = (1/2)iaa log K.

In view of its invariance, the Bergman distance between points a and b (a, b E B) is equal to the distance between the points 0 and I cpAb) I e1 • Thus,

_ f''l'a(bl ' J0+1) dt _ ~ 1 + I CPa (b) I p(a, b) - 0 1 _ t2 - v (n + 1) log 1 -ICPa(b)i'

2.3. The Cayley Transform. In order to define the Cayley transform in several variables, consider the domain D = Dn, given by

Dn d,;! {Z E en: 1m Zl > .t IZjI2}. }=2

The mapping cf>: B -+ D,

.e1 + Z cf>(z) = 1--,

1 - Zl

is called the Cayley Transform. Its inverse transform t/J can be given in the following form:

2z t/J(z)=-. ~-el'

1+ Zl

The mapping cf> induces an isomorphism between the groups Aut(B) and Aut(D). A subgroup of Aut(D) which naturally arises is the group of "nonisotropic" dilatations {br}r>o, where

brz = (t2Z1' tz 2 , .. ·, tzn )·

If t #- 1, then br fixes only two points: 0 and 00. Consequently, for t #- 1, the composition t/J 0 br 0 cf> is an automorphism of the ball B which fixes only the points e l and -e l .

Another important subgroup of the group Aut(D) is the group of "shifts" ha (a E aD), defined in the following way:

ha(w) = (WI + a1 2i .f. w}ij' W2 + a2,···, Wn + an). }=2

The set aD can be endowed with a group structure by defining an operation a # b such that ha#b = ha 0 hb. This last equation is equivalent to a # b = ha(b). The topoligical group (aD, #) is called the Heisenberg group. We remark that for a #- 0, the automorphism ha fixes only the point at infinity. Consequently, the automorphism t/J 0 ha 0 cf> (a #- 0) of the ball has the unique fixed point e1 •

Just as in the one dimensional situation, one can construct two "parallel" function theories, in B or in D (and on the sphere S or on the Heisenberg group aD, if we have boundary behavior of functuions in mind). Certain facts from

one theory automatically carryover to the other. In the present paper we shall not dwell, to any extent, on the theory of functions in the domain Q nor on the Heisenberg group aQ. For a treatment of analysis on the Heisenberg group, we refer the reader to Greiner and Stein (1977) and Rothschild and Stein (1976).

§ 3. Ol/-Invariant Subspaces

Let o/i = o/i(n) denote the group of all unitary transformations of cn. Consider the representation of the group o/i on the space L2(S) given by fU(() = f(UO (( E S), where f E L 2(S), U E O/I(n). This representation (as any representation of a compact group) can be decomposed as a direct sum of finite-dimensional irreducible representations. In the present situation, these irreducible representations turn out to be pairwise non-equivalent. In order to give this decomposition explicitely, we introduce some notations.

3.1. The Spaces H(P, q). Let C [Ul, U2 , ••• , ur ] denote the ring of all complex polynomials in the variables U 1, U2' •.• , Ur • Let H(p, q) denote the space of homogenous harmonic polynomials

of degree p in the variables Zl' Z2' .. 'Zn and of degree q in the variables Zl' Z2,

... zn. In the one-dimensional case, H(p,O) = Cz P, H(O, q) = Czq, and all remaining spaces H(p, q) (pq ¥= 0) are null. We denote the dimension of the space H(p, q) by D(p, q, n). It is not difficult to verify that

D(p, q, n) = (p +; -2)(q + ~ -2)P + ~ ~ ~ -1 (n ~ 2).

3.1.1. Theorem (cf. Rudin (1980), Sect. 12.2). All the spaces H(p, q) (p, q ~ 0) are o/i-invariant and

L2(S) = ED H(p, q). p,q?:O

Any closed o/i-invariant subspace EeL 2(S) is of the form:

E = ED H(p, q), (p,q)EA

where A c Z!. If T is a non-zero o/i-invariant operator from H(p, q) to H(r, s), then p = r, q = sand T = cI, where I is the identity operator, and c E C.

Let Kpq(z, () denote the reproducing kernel for the space H(p, q), H(p, q) c

L 2(S). This means that Kpi', z) E H(p, q) and

f(z) = <f, Kpi', z)U(S)

for all J E H(p, q) and all Z E en, i.e.

J(z) = Is J(OKpq(C, z) da(O

or (since a reproducing kernel is always symmetric: Kpiz, C) = Kpq«(, z»

J(z) = Is Kpiz, OJ(O da(O·

From the OlI-invariance of H(p, q) it follows that KpiUz, UC) = Kpq(z, 0 for all z, C E en and all U E 0lI. The integral operator with kernel Kpiz,O defines a OlI-invariant orthogonal projection from L2(S) onto H(p, q). Since the square of the Hilbert-Schmidt norm of this projection (as well as its trace) is equal to the dimension of the space H(p, q), we have

D(p, q, n) = Kpiz, z) = Is I Kpiz, 01 2 da(O

for all z E S and all p, q ~ O. We remark also that

IKpq(z, 01 s Izlp+q IClp+q· D(p, q, n),

Kpq(z, C) = KpiC, z) = Kpq(z, 0

for all z, C E en and all p, q ~ O.

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3.2. Explicit Formulae for the Kernel Kpq(z, 1;). We shall express the kernel Kpq(z, 0 via special functions. To this end we recall at first the definition of the hypergeometric Junction

F(IX, /3, y; x) ~ L (Yl (IX + j)(/3. + j») Xk.

k~O j=O (y + J) k!

The following formula is essentially contained in the proof of Proposition 12.2.6 in Rudin (1980):

( Iz121(12 ) Kpq(z, 0 = D(p, q, n)<z, OP<C, z)qF -p, -q, n - 1; 1 -I<z, 01 2 .

We may also express the kernel Kpq(z,O via Jacobi polynomials. By Y'::"P(IX, f3 > -1) we denote the polynomial of degree m, orthogonal in L2([0, 1J, x~(l - x)P dx) to all such polynomials of lesser degree, and normalized by the condition &::"P(1) = 1. The following equation holds (cf. Szeg6 (1959»:

( m IX+j) Y'::,.P(x) = (_l)m TI -/3 . F(-m, IX + /3 + 1 + m,; x).

j=l + )

It is not difficult to show that for n z 2

K ( Y) - D( )( Y»)P-qp;p-q,n-2(I(Z, (12)1 12q l YI2q pq Z, '" - p, q, n z, '" q IzI21(12 Z '" ,

if p z q.

3.3. Generalized Functions on the Sphere S. To each distribution IE g'(S) we may associate a family of harmonic polynomials {Kpqf}p,q;::o:

(KpqI)(z)~ Is Kpiz, OI(O da(O

(the integral is understood in the sense of the theory of distributions). We have:

I = L KpqI (in g'(S)). p,q;::O

Moreover, the series L K pqI converges normally in the interior of the ball B p,q;::O

and represents therein a harmonic function. Thus, the space g'(S) can be identified with the class of harmonic3 functions in the ball B such that

sup lu(z)I(1 -Izlt < +00 ZEB

for some N EN. We remark that a series L I pq , where Ipq E H(p, q), converges in g'(S) if

p,q;::O and only if II Ipq II L2(S) = O( (p + q + 1)-N), for some N E N (instead of the L 2

norm here one can substitute any L' -norm (1 ::; r ::; + CX)). The series L Ipq p,q;::O

converges in COO(S) if and only if IIIpqllu = O((p + q + 1)-N) for all N EN (1 ::; r ::; + 00 ).

aU) ~ {(p, q) E Z! : KpqI * O} (n z 2).

A distribution IE g' (S) is said to be C-invariant if I = I U for all scalar operators U E ou. In other words I can be represented in the form I = g 0 n, where g E g'([p>n-1). It is easy to see that a function I E g'(S) is C-invariant if and only if

aU) c {(p, q) E Z! : p = q}.

An analogous notion to T-invariance may also be introduced for measurable functions on the sphere S. A measurable function I: S -+ C is said to be Cinvariant if I = I U almost everywhere on S for all scalar operators U E ifIl, or (which is the same) if I can be represented in the form I = g 0 n, almost everywhere on S, where g is a measurable function on [p>n -1.

3 In some situations it is preferable to identify the space £0'(8) with the analogous class of Mharmonic functions. For the definition of M-harmonic functions cf. Chap. 2, Sect. 2.5.


3.4. The Tangential Cauchy-Riemann Equations. We shall say that a distribution f E £&'(S) (n 2:': 2) satisfies the tangential Cauchy-Riemann equations, if (z)5k - ZkDj) (f) = 0 for all k and j (the operator ZjDk - zkDj may be correctly defined in an obvious manner on £&'(S)).

3.4.1. Theorem. Let f E £&'(S) (n 2:': 2). Then u(f) c {(p, q) E Z~ : q = O} if and only if f satisfies the tangential Cauchy-Riemann equations.

For further results in this direction, see Khenkin and Chirka (1975) and Rudin (1980).

3.5. Multiplicative Properties of the Space H(p, q). Let H(p, q)H(r, s) denote N

the set of all funtions f E C(S), which can be represented in the form f = L g jhj' j=l

gj E H(p, q), hj E H(r, s). Clearly, H(p, q)H(r, s) is a OlI-invariant subspace of L2(S).

3.5.1. Theorem (see Theorem 12.4.4 in Rudin (1980)). If n 2:': 3, then /l

H(p, q)H(r, s) = L H(p + r - j, q + s - j), j=O

where J1 = min(p, s) + min(q, r).

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It is interesting to note that for n = 2 the situation is essentially more complicated (see Rudin (1980)): the left side of (14), in general, is a proper subset of the right side. Thus for n = 2 there are "more" OlI-invariant subalgebras of C(S) than for n > 2 (see Rudin (1980)).

3.6. Ryll-Wojtaszsczyk Polynomials

3.6.1. Theorem (Ryll-Wojtaszsczyk (1983)). There exists a positive number C(n) having the following property: for any p E Z+, there is a polynomial fp E

H(p, 0) such that Ifpl ~ 1 everywhere in Band Ilfp ll£2(S) 2:': C(n).

This theorem has already found several applications4 in the theory of functions holomorphic in the ball: functions with Carleman singularities (Wojtaszczyk (1982)), functions with a "large set of roots" (Alexander (1982)), and inner functions 5 (Aleksandrov (1984)). Moreover, we may find several appli-

4These applications are discussed in more detail at the end of this paper (cf. "Update on problems ... ",2). 5 Recently, with the help of Theorem 3.6.1, the author has constructed a proper holomorphic mapping of the ball B into a polydisc of sufficiently high dimension (A.B. Aleksandrov, Proper holomorphic mappings from the ball into a polydisc. Dokl. Akad. Nauk SSSR, 1985. Analogous results were obtained in a somewhat different way by Low, E.; The ball in C" is a closed complex submanifold of a polydisc. Invent. Math. 83,405-410 (1986).)

The ball here can be replaced by an arbitrary strictly pseudoconvex domain Q with C2-boundary. Low also showed the existence of a proper holomorphic mapping f from Q to a ball B of sufficiently high dimension. Moreover, in this case, one may additionally require the continuity of f up to the boundary (cf. Low, E.; Embedding and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls. Math. Z. 190,401-410 (1985).)

cations in the work of Ryll-Wojtaszsczyk (1983) itself. Most likely, Theorem 3.6.1 may turn out to be very useful in other questions of function theory in the ball also. It is unknown to the author whether one can find such polynomials fpq E H(p, q) such that Ifpql ~ 1 everywhere in Band inf Ilfpq llL2(s) > O. We

P.q:?O remark that in Ryll-Wojtaszsczyk (1983) the following inequality is essentially proven

{ lIfIIL2(S). {}} 1;: -1 sup IlfIILOO(S).fEH(P,q)\ 0 >2ynIIKpqllL1(sxs)'

It is easy to see that for p 2:: q

IIKpqllL1(sxs) = 2(n - 1)D(p, q, n) x I1 x(P-q)/2(1 - xt-2 1.?J!rq•n - 2 (x)1 dx.

One easily verifies that sup IIKpqllL1(sxs) = C(n, q) < +00 for all nand q. On the p:?O

other hand well known estimates for weighted L1-norms of Jacobi polynomials (Szego (1959» yield the following inequality:

C 1 (n)pn(3/2) ~ IIKpqllL1(s xS) ~ C2(n)pn(3/2)

for all n 2:: 2 and all p 2:: 1. Consequently,

for all n 2:: 2.

sup IIKpqllL1(sxs) = +00 P.q:?O

§ 4. N anisotropic Quasimetrics on the Sphere S

4.1. Elementary Properties of Nonisotropic Quasimetrics. For many questions in the theory of functions in the ball, different tangential directions on the sphere S turn out to have differing roles. Among these directions, a particular role is played by the complex tangential directions, i.e. those directions which are defined by vectors from the complex tangent space T{(S) (( E S).

We now introduce the nonisotropic quasimetric d on the sphere S, which in a quantitative way "pins down" this particularity of complex tangential directions. Set

d((, e) = 11 - «, 01((, e E S).

The function d 1/2 is a metric on S, and hence

for all (, e, '1 E S. For n = 1 this quasimetric d coincides with the usual Euclidean metric:

d((, '1) = Ie - '11. For n 2:: 2 we have the inequality:

tl( - el 2 ~ d((, e) ~ I( - el((, e E S).

The first inequality turns into an equality when <',0 E IR; and the second, when <" 0 E C.

Quite often (and we see this repeatedly) the quasimetric d turns out to be significantly more convenient than the usual Euclidean metric. For example, the quasimetric d is very practical in studying the Cauchy kernel C (see Chap. 2, 2). This stems from the fact that the set

g E S: Iq" ~)I > t}

is a ball for the quasimetric d. Let Q(', r) denote the open d-ball with center at the point' E S and radius r,

i.e.

dcl{ } Q(', r) = ~ E S: 11 - <" ~)I < r .

Set

B(" r) ~ g E S: I~ - " < r}.

Simple calculations show that the projection of the ball Q(', r) onto the tangent space Tc(S) is contained in the set

Q(', r) = {(w, x} E TcC(S) EB i,lR: Iwl < 2Jr, Ixl < r}

and contains the set Q (" i). Thus, the ball Q(', r), roughly speaking, represents

a "curvilinear ellipsoid" obtained by dilating the usual Euclidean ball B(" r) by a factor of approximately;:::; r- 1/2 in the complex tangential direction. From the above it follows that

(T(Q(~, r)):::=::: rn (0 < r < 2).

The symbol a:::=::: b denotes "a < c1 b and b < c2 a, for some positive C1 and c/'. We remark also that the quasimetric d admits a natural extension to a

quasimetric d1 on the ball Ii:

d1(~' 0 = 11~12 ~~~~' 01 (~,' E Ii; I~I ~ 1m

The function jd; is a metric on the ball Ii.

4.2. Hausdorff Measure and Dimension. On each quasimetric space (X, p) we may introduce, in a natural manner, the a-dimensional Hausdorff measure h/% and Hausdorff dimension. We recall the respective definitions:

h/%(A) ~ lim inf {E(diam Aj)/% : diam Aj < e, u Aj = A} (a > 0), ~--+O+

where diam A = sup{p(x, y): x, YEA}. The Hausdorff dimension ofa set A c X is the number

sup{a: h/%(A) = +oo} = inf{a: h/%(A) = O}.

To each subset E of the sphere S, we may associate its usual a-dimensional Hausdorff measure, ha(E) = ha(E, en), considering E as a subset of en. We shall denote the corresponding Hausdorff dimension by dim(E, en). However, in the theory of functions in the ball, sometimes it is natural to consider the adimensional Hausdorff measure ha(E, S) viewing E as a subset of the sphere S endowed with the quasi metric d.

Let dim E denote the topological dimension of a set E (Hurewicz and Wallman (1941)). It is well known (op. cit.) that

dim E $; dim(E, en) and dim E $; 2 dim(E, S).

The number 2 in the second inequality is explained by the fact that 2 is the smallest among all numbers a such that the function d l /a is a metric. It is easy to verify the following inequalities:

ha(E, S) $; C(a, n) {ha(E), 0 < a $; 1, h2a - 1 (E), a ~ 1.

h () ( ) { ha/2(E' S), 0 < a $; 2n - 2, aE $;Ca,n

ha+I(E, S), rx ~ 2n - 2.

From these, one obtains corresponding inequalities for dim(E, S) and dim(E, en), from which, in particular, it follows that

dim(E, en) = O<=>dim(E, S) = 0,

dim(E, en) = 2n - 1 <=>dim(E, S) = n.

4.2.1. Proposition. Let M be a compact manifold of class C l and dimension m, Me S. Then

a) if M is integral, then

o < hm/2 (M, S) < +00;

b) (f M is not integral, then

0< h(m+1)/2(M, S) < +00;

Chapter 2 Fundamental Integral Representations

§ 1. Fundamental Spaces of Functions Holomorphic in the Ball

We denote by H(B) the space of all functions holomorphic in the ball. We introduce on H(B) the topology of uniform convergence on compact subsets of

B. We define differential operators [J£o, !Yl: H(B) -+ H(B),

der n !Ylof = L zjDjf,

j=l

!Ylfd,;! f + !Ylof·

It is easy to see that if

kEZ~

is the Maclaurin series expansion of f, then

(!YlofHz) = L Ikl akz k (z E B). kEZ~

(!Ylf)(z) = LOki + 1)akzk (z E B). kEZ~

To every cr: E IR we may associate the power of order cr: of!Yl,

([J£"f)(z) = L (Ikl + l)"akzk (z E B). kEZ~

Everything which will be said below concerning the operator !Yl and its powers !Yl" will have a natural analog for the operator [J£o and its powers. However, one should keep in mind that in order to investigate the operator!Yl'O (and in particular in order to define its fractional powers), it is natural to consider the space H(B) factored by the constants, i.e. H(B)/C.

We remark that

[J£"(I;) = (!Yl"f)~ (1)

for all , E B, where I; is the slice function, i.e. I;(A) = f(AO (A E 10). This formula allows us to reduce many multidimensional results concerning the operator !Yl" to one-dimensional ones. For cr: > 0, the operator !Yl" will be called an operator of fractional differentiation order cr:, and for cr: < 0, an operator of fractional integration order ( - cr:).

1.1. Notation. We denote by Am(B) (0 ~ m ~ +00) the set of all functions f E em (B), which are holomorphic in B. Set A(B) = AO(B). We remark that Am(B) = [J£-m(A(B)) (0 ~ m < +00). This can be derived from (16) in Chap. 3.

Let AA"(B) (cr: > 0) denote the space of all functions f E H(B) such that

f der f II II A .. ' = sup I (z)1 (1 - Izl)" < +00. ZEB

The closure, in this space, of all polynomials is denoted by AA"(B). In other words, AA"(B) is the set of all f E H(B) such that

f(z) = 0«1 - Izl)-")(Izl-+ 1).

From the corresponding one-dimensional result (see Duren (1970)), and from (1) we have:

!Yl"-P(AAP(B)) = AA"(B),

!Yl"-P(AAP(B)) = AA"(B),

(2)

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where Ll, f3 E (0, +00). Formulas (2) and (3) allow us to define the spaces AA(B) and A~(B), for Ll ;::: 0, so that these formulae hold for all Ll, f3 E ~. It is well known (op. cit.) that for n = 1 and Ll E (0, 1), the class AA(B) ~ AA is the space of all holomorphic functions in the disc ID which satisfy a Holder condition of order Ll. The case Ll = 1 corresponds to the Zygmund class. In the general case

AA = AA(ID) (p < Ll ~ P + 1)

is the space of all holomorphic functions I in the disc ID such that PP) E AA-P,

In the multidimensional case, (1) shows that a function I holomorphic in B belongs to the class AA(B) if and only if {Iches is a bounded family in the space AA' We shall speak more of the degree of smoothness of the functions in AA(B) in Chapter 3. The space A~(B) is called the Bloch space. As in the onedimensional case, functions from the class A~(B) m'ay fail almost everywhere to have radial boundary values on S (see Ryll-Wojtaszsczyk (1983)). It is easy to see that n AA(B) = AOO(B). Set ~~(B) ~ U AA(B). The space ~~(B) is the set

lIeR cxeiR

of all functions IE H(B) for which lim fr, exists in the space of distributions

~'(S). Here, and in the sequel, fr(z) = I(rz), 0 < r < 1, z E B. The space ~~(B) can be identified in a natural manner with the space of all distributions in ~'(S) which satisfy the tangential Cauchy-Riemann equations (n ;::: 2).

1.2. The Nevanlinna and Smirnov Classes. The Nevanlinna class N(B) is the set of all functions I holomorphic in the ball B such that

sup r log+ Ifrl dO" < +00. O<r<l Js

It is easy to see that the class N(B) is an algebra. If IE N(B), then for almost all , E S, the slice-fuction Ic belongs to classical Nevanlinna class N ~ N(ID). Thus each function IE N(B) has radial boundary values almost everywhere on the sphere S, and we shall denote this boundary function by the same letter f If 1=1= 0 in B, then log III ELl (S). Thus the class N(B) can be identified (mod sets of measure 0) with a sub algebra of the space LO(S) of all measurable functions on the sphere S.

The Smirnov class N *(B) is the space of all functions I E N(B) such that

lim r log+ Ifrl dO" = r log+ III dO". (4) r-1- Js Js

Formula (4) is equivalent to the following condition: the family {log+ Ifrl}o<r<l has uniformly absolutely continuous integrals. For functions I E N(B) we have lEN *(B) if and only if Ic E N * for almost all , E S. Here, N * denotes the classical Smirnov class N*(ID). In the class N*(B), we may introduce the metric

p(f, g) = Is log(1 + II - gl) dO". (5)

The space N*(B) endowed with the metric p is a complete topological algebra continuously imbedded in the algebra H(B).

1.3. The Hardy Classes. The Hardy class HP(B) is the space of all functions f E H(B) such that

IlfllfIp ~ sup f 1.f..IP d(J < +00 (0 < P < +(0), O<r<l S

IlfllHoo ~ sup If(z)l· ZEB

It easy to see that HP(B) c N*(B) for all p E (0, +(0). The spaces HP(B) are Banach spaces for p E [1, +00], and are p-Banach6 spaces for p E (0, 1). The class HP(B) has a canonical imbedding as a subspace of U(S) (0 0, we associate the space J't'/(B), consisting of all functions f E H(B) such that

Ilfll~£, ~ L If(z)iP dJ1ap(z) < +00,

where

d () - F(n + s) (1 _ I 12 )S-1 d ( ) J1s z - n!r(s) z v z.

The multiplicative constant is so chosen that the measure J1s is a probability measure.

The space ~P(B) is a Banach space for p ~ 1 and a p-Banach space for p E (0, 1). Moreover, the spaces H2(B) and ~2(B} are Hilbert spaces.

To each pair of natural numbers n, m, n > m, we associate two operators: (ff':: : H(Bn) ~ H(Bm) and X-,:: : H(Bm} ~ H(Bn}.

(X-'::f)(Zl' Z2' ... , zn) = f(Zl' Z2' ... , zm}·

1.3.1. Theorem. Let m and n be natural numbers m < n, and let IX, p E (0, +(0). Then

Moreover,

ff'::HP(Bn) = £(~-m)/p(Bm)'

ff'::(~P(Bn)) = ~P+(n-m)/p(Bm)'

Ilff'::FII.J!'p :::;; IIFIIHP' (n-m)/p

Ilff'::FIIHP = IlfIIJl"l:'_mIlP'

Ilff'::FIIJI":'+<n_mJ!p:::;; IIFII.J!'£,'

II X-'::f IIJI"£, = II f IIJI"!+<n_mJ!p' This theorem follows easily from (1) in Chapter 1 and Fubini's Theorem.

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6 A p-Banach space differs from a Banach space in that instead of the triangle inequality, one requires the weaker inequality Ilx + yilP ::; IlxilP + IlyliP (0 < P < 1).

The following assertion follows from Theorem 1.3.1 for m = 1 and from the corresponding one-dimensional result.

1.3.2. Theorem. Let IX, p E (0, +(0). Then

HP(B) c A.:4n/p(B),

Yf!(B) c A.:4(n/p)-«(B).

We introduce several more results concerning the spaces HP(B) and Yf!(B).

n n 1.3.3. Theorem. Let IX, p, q E (0, +(0), where - - - = IX. Then

p q

~-«(HP(B)) c Hq(B),

HP(B) c Yf,.q(B).

It is not difficult to give a simple proof of this theorem (see Aleksandrov (1981) p. 25) using Theorem 1.3.2 and a theorem on the maximal function:

sup 1.f..1 E U(S), (8) O<r<l

when J E HP(B). The inclusion (8) easily reduces to the corresponding onedimensional result.

1.3.4. Theorem. Let IX, p E (0, +(0). Then

~-«(Yf,.q(B)) c HP(B) (p =:; 2),

~«(HP(B)) c Yf!(B) (p ~ 2).

This theorem reduces to the one-dimensional case with the help of (5) in Chapter 1.

§ 2. Fundamental Integral Representations

Various integral representations playa very important role in the theory of holomorphic functions (see Aizenberg-Yuzhakov (1979) and Khenkin-Leiterer (1984)). In this section we briefly pause to consider only a few of these.

2.1. The Cauchy Kernel. The Cauchy (or Cauchy-Szego) kernel is the reproducing kernel C(z, 0 for the space H 2 (B). This means that

J(z) = (f, C(., z) >H2 for all J E H2(B) and all z E B, i.e.

J(z) = Is J(Oc«(, z) du(O

or (on account of the symmetry property C(z, 0 = C«(, z))

J(z) = Is C(z, 0 du(O (9)

for all J E H2(B) and all z E B.

The following formula holds

C(z, 0 = (1 - <z, 0 )-n(z, , E B).

Formula (9) automatically carries over to functions of Hardy class H1 (B). Moreover, it is also valid for all f E ~~(B), provided the integral therein is interpreted in the sense of distributions. The operator

(Cf)(z) = Is C(z, Of(O d(J(O (10)

is the orthogonal projection from L 2(8) onto H2(B) and is called the Riesz projection. We remark that the right member of (10) is meaningful for f E L1(8), and even for f E ~'(S), provided the integral is understood in the sense of distributions. In the latter case (i.e. when f E f0' (S)), we shall also use the notation Cf. In particular, if J.l E M(S), then

(CJ.l)(z) = Is C(z, 0 dJ.l(O·

The operator C plays a very important role in the theory of functons in the ball and we shall speak of C again many times (in particular, in Chap. 3).

2.2. The Bergman Kernel. The Bergman kernel is the reproducing kernel K(z, 0 for the space £172(B). It has the following form:

K(z,O = (1 - <z, 0 r n - 1 (z" E B).

Consequently, the orthogonal projection T from L 2(B) onto £172(B) can be given in the following form:

(Tf)(z) = L K(z, Of(O dv(O·

In particular,

f(z) = L K(z, Of(O dv(,).

for all z E B and all f E £02 (B), and therefore, for all f E £/(B). The reproducing kernel for the space ~72(B) has the following form:

Ka(z,O = (1 - <z, 0 )-n-a (z" E B).

The corresponding orthogonal projections are studied in Forelli-Rudin (1974) and Rudin (1980). These authors also investigated integral operators with kernel K a , for complex IX.

2.3. The Invariant Poisson Kernel. The invariant Poisson kernel is the kernel

P( r)~C(z, OC(', z) = ( 1 -lzl2 )n ( B r S) z, .. C(z, z) 11 _ <z, 012 Z E , .. E .


1 C(', Z) 1 . Let f E H (B). Then g = f· C(z, z) E H (B). Applymg (9) to the functon g we

obtain

f(z) = I P(z, ()f(O da(O (14)

for all z E B. Formula (14) holds also for all f E H1(B) since P(z, 0 ::2: O. The measure P(z,·) da is invariant with respect to the group of all

automorphisms of the ball which fix the point z E B.

2.4. The "Harmonic" Poisson Kernel. In the ball B, there exists one more important kernel, the usual Poisson kernel Ph(z, 0, which solves the Dirichlet problem

1 -lzl2 Ph(z,O = " _ zl2n'

Since every function f E H1(B) is harmonic in B, we have

f(z) = I Ph(z, ()f(O da(O·

However, for n ::2: 2, the kernel Ph does not reflect the "complexness" of the ball B (the analog of this kernel can be considered also in [R2n +1) and has no specific connection with the Cauchy kernel C nor with the group of all automorphisms of the ball B. Since the equality Ph = P holds only for n = 1, the invariant Poisson kernel does not solve the Dirichlet problem for n ::2: 2.

2.5. Which Problem does the Invariant Poisson Integral Solve? By the invariant Laplacian 3, we mean the Laplace-Beltrami operator (see de Rham (1955)) on the ball B, associated with the Bergman metric, i.e.

where bjk • is the Kronecker symbol. This operator is invariant with respect to all automorphisms of the ball. From the obvious equality (3f)(0) = (Af)(O), it thus follows that

(3f)(a) = (A(f 0 qJa))(O) (15)

for all a E B. We introduce two more formulae for the invariant Laplacian:

where fa: I~I [D -+ C is the slice function fa(t) = f(ta);

(3f)(a) = (1 - laI 2 )((Af)(a) - (Afa)(1)),

- 4nf (Af)(a) = lim 2: (f(qJa(rO - f(a)) da(o. r-O+ r s

A function f E C2 (B) is called M-harmonic if Af == O. A function f E C(B) is M-harmonic if and only if

f(ljJ(O)) = Is (f(ljJ(rO) da(()

for allljJ E Aut(B) and all r E (0, 1). Many properties of M-harmonic functions are analogous to properties

of harmonic functions. For example, the maximum principle holds for Mharmonic functions. Moreover, each M-harmonic function is real analytic.

The following theorem follows from the fact that the invariant Poisson kernel is an approximate identity.

2.5.1. Theorem (see Rudin (1980)). Let u be a function M -harmonic in the ball B.

1. If 1 < p :::; +co, then

u(z) = Is P(z, ()f(O da(O (16)

for some function f E U(S) if and only if sup Ilurlb(s) < +co. Moreover O<r<l

lim Ur = f in the space U(S) (for p = +co one should use the weak topology r-+1-

a(V'"', U). 2. Formula (16) holds for some function f E Ll(S) if and only if lim Ur exists

in the space L 1 (S), and then f = lim ur. r-+1-

3. Formula (16) holds for some function f E C(S) if and only if the function u is uniformly continuous in B, and f = lim Ur (in C(S)).

r-+1-

4. A function u can be represented in the form

u(z) = Is P(z, 0 dJl(O, (17)

where Jl E M(S) if and only if sup IlurllL' < +co and in the weak topology O<r<l

a(M(S), C(S)) on the space M(S) Jl = lim ur. r-+1-

5. Formula (17) holds for some positive measure Jl if and only if u ~ O.

Every pluriharmonic function u is M-harmonic since u is annihilated by all operators Di5k. However, the class of M-harmonic functions is far from being exhausted by the pi uri harmonic functions.

We remark that the operator A "interacts" very poorly with dilatations. The following assertion speaks to this.

2.5.2. Proposition (Rudin (1980)). If the function u is M-harmonic in the ball B and if r E (0, 1), then the M-harmonicity of the functions Ur is equivalent to the pluriharmonicity of the function u.

We remark further that a function U E C(B) is pluriharmonic if and only if u is both M-harmonic and harmonic (see Rudin (1980)).

2.6. H(p, q)-Expansion of the Cauchy and Poisson Kernels. The following expansions hold:

1 (p + n - 1) 1 _ ( on = L Kpo(z,O = L (z,OP «z,O E II:D),

Z, p"2:0 p"2:0 P

1 -lzl2 ,,_ 12n = L Kpq(z,O (z E B, (E S),

Z p,q"2:0

( l-lzl2 2)n = L Kpiz,O.F(p,q,p+q+n;~zI2). 11 - (z, 01 p,q"2:0 F(p, q, p + q + n, 1)

Here, F(rx, p, y; X) denotes the hypergeometric function (see Chap. 1,3.2).

Chapter 3 Boundary Properties of the Cauchy Integral and

the Invariant Poisson Integral

§ 1. The Maximal Function

1.1. Properties of the Maximal Function. To each measure {l E M(S) we may associate the maximal function M {l : S --+ [0, + 00],

(M )(Y) deC l{ll (Q«(, r)) {l <, = SUp .

r>O a(Q«(, r))

For such a maximal function (associated with the nonisotropic quasimetric d), the usual properties of the "isotropic" maximal function hold. We present some of these.

1.1.1. Theorem. The operator M is of weak type (1, 1), i.e.

for all t > O.

a{M{l> t} :::;; C(n) 11{l11 t

Moreover, it is clear that M(L OO(S)) c L 00 (S). Invoking, now, the Marcinkiewicz Interpolation Theorem (see Rudin (1980)), we obtain the following theorem.

1.1.2. Theorem. For each p E (1, +00], there is a constant A = A(p, n) such that

Each measure Jl has a Lebesgue decomposition Jl = fa + Jls. The function f can be found by the formula

f(O = lim Jl(Q(', r)), r-+O+ a(Q(', r))

valid for almost all , E S. We remark also that the singular part Jls of the measure Jl is concentrated on the set {MJl = +oo}. Almost every point' E S is a Lebesgue point for the summable function f, i.e.

for almost all , E S.·

1.2. K-Limits. To each point' E S and IX> 1, we associate a domain Da(O,

Da(Od,;! {Z E B: !<z -" 01 < ~(1 -IZI2)}.

We remark that for each unitary operator U E Olt(n), we have

Hence, in order to try to better imagine the "structure" of the domain Da(O, it is sufficient to restrict one's attention to the case' = e1 • The intersection Da(e 1 ) n e . e 1 is the usual angular domain

However, the intersection of this domain with the (2n - 1 )-dimensional space

1R2n - 1 = {z E en: 1m Zl = O}

is the ball

{Z E 1R2n - 1 : (Zl - ~y + J2 IZj l2 < (1 - ~y}, which is tangent to the sphere S at the point e l'

Thus, the boundary of the domain DiO is tangent to the sphere in the complex tangential direction and the tangency is of order 1 (i.e. as for a circle or a line).

To each function F, given on the ball B, and to each IX > 1, we may associate the maximal function MaF : S -+ [0, + 00],

deC { } (MaF)(O = SUp IF(OI: Z E Da(O .

The function MaF is measurable since it is lower semicontinuous.

1.2.1. Theorem (Koninyi, see Rudin (1980)). For each IX> 1 there is a constant A = A(IX, n), such that

Ma(PJi.) :=; AMJi..

1.2.2. Definition. We shall say that a function F: B -+ C has K-limit A E C at (E S if

A = lim Flv.(c) c

for each IX > 1. In this case, we write A = K-lim F. c

From Theorem 1.2.1, the following result of Koranyi (see Rudin (1980)) easily follows.

1.2.3. Theorem. Let Ji. E M (S) have Lebesgue decomposition; Ji. = f· a + Ji.s. Then

for almost all ( E S.

f(O = K-lim PJi. c

We present also the multidimensional analog of Plessner's Theorem (see Garnett (1981)).

1.2.4. Theorem (see Rudin (1980)). Let f E H(B). Then the sphere S can be written as the union of three measurable sets Ek , Ee, and EN' such that

1) a(EN) = 0; 2) the function f has a finite K-limit at each point of Ek ;

3) the image f(Da(m is everywhere dense in C, for all ( E Ee and all IX> 1.

1.2.5. Theorem (Koranyi-Stein, see Rudin (1980)). Every function f in Nevanlinna class N(B) has finite K-limits almost everywhere on S.

1.3. The Lindelof-Chirka Theorem. We pause now briefly to consider a multidimensional generalization of a theorem of Lindelof discovered by E.M. Chirka. A mapping r: [0, 1) -+ B will be called a (-curve (( E S), iflim r(t) = (. By a special (-curve we mean a (-curve r, such that t-+1

I· Ir(t) - y(tW - 0 1m 2 - ,

t-+1 1 - ly(t)1

where y is the orthogonal projection of the curve rinto the complex line C(, i.e.

y = <r, 0(.

A special (-curve will be called a (B,O-curve, if there are numbers IX> 1 and to < 1 such that r(t) E Da(O for all t E (to, 1). Finally, we shall say that A. E IC is the B-limit of a function f at a point ( E S, if

A. = lim f(r(t)) t-+1

for each (B, O-curve r. In this case, we shall write A. = B-lim f ~

It is clear that the existence of a K-limit follows from that of a B-limit. The converse is not true. For example, the function

z~ f(z1' zz) = -1--z - Z1

bounded and holomorphic in the ball B Z has B-limit at the point (1, 0) (equal to zero), but does not have a K-limit at (1, 0) since

f(t, cj1=ti) = cZ•

The function f is also interesting in that its power series

f(z1' zz) = L ZikZ~ k;:"O

converges absolutely in the closed ball BZ, although the function f in not continuous on the closed ball (In the one-dimensional case, this is impossible.).

1.3.1. Theorem (E.M. Chirka (1973)). Let f E HOO(B), ( E S, and suppose r is a special (-curve such that

Then B-lim f = L. ~

lim f(r(t)) = L. t-1-

Z

The example of the function f(zl, zz) = (1 _z;i)1+e (not having a B-limit at

the point (1, 0)) shows that the condition f E HOO(B) is essential. Further results and examples relating to multidimensional generalizations of

the Lindel6f Theorem can be found in the paper of Chirka (1973), the survey of Khenkin and Chirka (1975), and Rudin's book (1980).

1.4. Carleson Measures. Stein (1970, Chap. 7,4.4) remarked that the angular maximal function is convenient not only for studying the boundary behavior of functions, but also for obtaining imbedding theorems in the spirit of Carles on (see Garnett (1981)). We shall need an analog of Stein's result in the ball; for this we require the following definition.

1.4.1. Definition. A positive Borel measure J1. on the ball B is called a Carleson measure if

dd { } C(J1.) = sup rnJ1.(B((, r)): ( E S, r > 0 < +00, (1)

where

B(" r) = {z E B: I<z - " 01 < r}.

S.A. Vinogradov (see Nikol'skij (1985)) suggested another convenient form of condition (1): i I C(z, 01 2 dfl(')

sup r P(z, () dp«() ~ sup [ < +00. ZEBJB zEB Ic(z, 01 2 da(')

s

1.4.2. Theorem. Let f be a continuous function in the ball B and let fl be a Carleson measure on B. Then

L IflP dfl ~ A(n, ex)$(fl) L IMJIP da (p > 0, ex > 2).

Theorem 1.4.2 was implicitely proved by Power (1985).

1.4.3. Definition. A positive Borel measure fl on the ball B is called a Carleson measure of order t, if

$( fl, t) ~ sup {r-n(fl(B(" r)))l/t: , E S, r > O} < +00,

1.4.4. Theorem Let f be a continuous function on the ball B; 0 2).

In order to deduce Theorem 1.4.4 from Theorem 1.4.2, it is sufficient to invoke the following equality

(L Iflq dfl Ylq = sup {L IflP Igl dfl: 9 E U/(q-P)(fl), f Iglq/(q-P) dfl ~ I}.

§ 2. The "Real" Hardy Class

2.1. The Carleson-Duren-Hormander Theorem. Let .Yl'P(S) (0 1. Every function u E .Yl'P(S) is the invariant Poisson integral of a distribution f E .@'(S); we shall identify functions in the class .Yl'P(S) with (generalized) functions in .@'(S) (u 1-+ f), and we shall not introduce a special notation for such functions. It is easy to see that .Yl'P(S) c U(S), for p ;?: 1. In fact, .Yl'P(S) = U(S), if 1 < p < +00.

From Theorem 1.4.4 we have the following:

2.1.1. Theorem (Carleson-Duren-H6rmander, (see H6rmander (1967), Duren

(1970))). Suppose 0 < p ~ q < +00 and J1. is a Carleson measure of order 'i. Then p

.1l'P(S) c U(J1.}.

The space .1l'P(B} is a subspace of .1l'P(S). For n ;?: 2, .1l'P(B) consists of all functions f E .1l'P(S} C ~'(S) which satisfy the tangential Cauchy-Riemann equations.

Each function f E .1l'P(S} (0 < p < +oo) has finite K-limits almost everywhere on S. However, for p < 1, a function f E .1l'P(S} is no longer uniquely determined by these boundary values. Indeed, it easy to see that M(S} c .1l'P(S}, for all p < 1. However, the invariant Poisson integral of a singular measure has zero boundary values almost everywhere on S.

2.2. Atomic Decomposition in Hardy Spaces

2.2.1. Theorem. The Riesz projection C projects the space .1l'P(S} onto the space HP(B} (0 < p < +oo).

For p E (1, +oo) this result was obtained by Koninyi and Vagi (1971). In this case, Theorem 2.2.1 is equivalent to the continuity in U(S} of the singular operator determined by the Cauchy kernel C(z, 0 = (1 - <z, 0 }-n (z, ( E S). As in the one-dimensional case, this operator is of weak type (1, I), i.e.

for all J1. E M(S}.

A(n} u{ICJ1.1 > t} ~ -11J1.11

t

The case p E (0, 1] was considered by Garnett and Latter (1978). They used their own atomic decomposition of the space .1l'P(S), for 0 < p ~ 1. We now pass to this decomposition.

2.2.2. Definition. A function bE LOO(S} is said to be p-atomic (0 0 such that simultaneously i) supp b c Q(C r}; ii) IlbllLoo ~ (u(Q«(, r)))-l/p;

iii) Is bP du = 0 for all polynomials P E C[Zl' Z2' ... , Zn' Zl' Z2"'" zn] of

degree no greater than 2n (t - 1).

In particular, if p > 2n2: l' then condition iii} means that Is b du = O.

From the following two assertions, it follows that IIbllxP(s) ~ C(p, n) for any p-atom b (0 < p ~ 1).

2.2.3. Proposition. Let C E S, r > 0, and 0 < p < 1. Suppose a measure Il E

M(S) satisfies the following properties: 1) supp Il c Q(C, r);

2) Is P dll = o for all polynomials P E lC[z1' Z2' Zn' ... , Z1' Z2' ... , znJ of degree

no greater than 2n G -1). Then Il E J('P(S) and 111l1I.Jf'p ::;; C(p, n) 111l1iM' r(n/p)-n.

2.2.4. Proposition. Let C E S, and r > O. Suppose that f E L log L (i.e. f E

U (S) and Is Ifllog(1 + If I) du < +00) and satisfies the following conditions:

1) supp f c Q(C, r);

2) Is f du = O. Then f E J('1(S) and Ilfll.Jf" ::;; C(n) IlfllLlog L'

H ere, II II Liog L' denotes the norm in the Orlicz space L log L, i.e.

def. { f If I ( If I) } IlfllLlogL=mf A>O: -:4 log 1+-:4 ::;;1.

It is not possible to weaken the condition f E L log L in Proposition 2.2.4, for if f E J('1(S) and f;::: 0, then f E L log L (see Rudin (1980)).

From Propositions 2.2.3 and 2.2.4 it follows that

(2)

for any sequence {bdk;>:l' of p-atoms provided that

f = L Ill(klP < +00 (Il(k E IC). k;>:l

Garnett and Latter (1978) showed that the functions of type (2) exhaust the space J('P(S) (0 < p ::;; 1).

2.2.5. Theorem (op. cit.). Let 0 2n (~ - 1)). We may also require

that all the atoms bk are orthogonal to all polynomials of degree at most N (of course, in this case the constants A and B in (3) will depend also on N).

2n - 1 We remark that, for p::;~, Garnett and Latter considered somewhat

different atoms, requiring in iii) that P E C[el' e2' ... , e2n-l], where el' e2, ... , e2n -l are specially chosen local coordinates in Q((, r) c S. Their approach allows one also to diminish the degree of the polynomials P with respect to the coordinates which are "orthogonal" to the complex tangent space (at the point O. However, this approach has its own drawback: the orthogonality condition depends on the ball Q((, r) (more precisely, on its center). For n = 1, Theorem 2.2.5 was obtained by Coifman (1974).

Theorem 2.2.1 for p E (0, 1] follows easily from Theorem 2.2.5.

2.2.7. Definition. A function f, which can be represented in the form f = Cb, where b is a p-atom, is called a holomorphic p-atom.

The following theorem on holomorphic p-atoms follows from Theorems 2.2.1 and 2.2.5.

2.2.8. Theorem (Garnett-Latter (1978)). Each function f E HP(B) (0 < p ::; 1) can be represented in the form f = L (i.kbk' where the bk are holomorphic p-atoms

k:<:l and (i.k E C; moreover

For p = 1, this theorem was proved by Coifman, Rochberg, and Weiss (1976) by-passing Theorems 2.2.1 and 2.2.5.

Theorems on atomic decompositions are very strong tools in the theory of Hardy spaces (see Coifman-Weiss (1977)).

The following theorem is deduced in Coifman-Rochberg-Weiss (1976) and Garnett-Latter (1978) from Theorem 2.2.8.

2.2.9. Theorem. Let f E HP(B) (0 < p ::; 1). Then there are two sequences of functions {Fdk:<:l and {Gdk:<:l from H 2P(B) such that

1)

2) f = L FkGk (the series converges in HP(B) on account of (4)). k:<:l

(4)

In the one-dimensional case, thanks to the availability of the inner-outer factorization (see Hoifman (1962)), a stronger assertion holds:

HP = HP' H' G = ~ + ~). Horowitz (1977) proved an analogous assertion for the one-dimensional

spaces Yl;"P:

Gowda (1983) showed that in the multidimensional case the set Hq(B)' W(B) (respectively the set Yt'ajq(B)' JIt';jr(B) is a set of first category in HP(B) (respectively in Yt'aIfp(B)).

At the present time, it is unknown whether in Theorem 2.2.9 one may require that the sequences {Fdk;;'l and {Gkh;;'l be finite.

Theorems on atomic decompositions (with different atoms) were obtained by Coifman and Rochberg (1980) for the spaces Yt'aP(B) (0 < p, ct < +00). These results yield the following theorem of Lindenstrauss and Pelczinski (1971), even for p :::;; 1.

2.2.10. Theorem. Let ct, p E (0, +00). Then the space Yt'aP(B) is isomorphic to [P.

Recall that

We remark that the space HP(B) is isomorphic to [P only for p = 2. For further results concerning isomorphisms and non-isomorphisms of

spaces of holomorphic functions, see Chap. 5, 1.3.

§ 3. Dual Spaces for Hardy Spaces ~P(S) and Spaces of Smooth Functions

3.1. Dual Spaces and Spaces of Multipliers

3.1.1. Definition. A topological vector space X will be called a space of gen-eralized functions on the sphere S, if the following two conditions are satisfied:

1) COO(S) c X c ~'(S) (here, c means continuously imbedded); 2) the set COO(S) is dense in X. To each such space X, we may associate its dual space X' as well as the space

of multipliers .AX. The space X' consists of all distributions qJ E ~' (S) for which the functional

f 1-+ (f, qJ),

defined at first on the space COO(S), can be extended to a continuous linear functional on the entire space X. The space .A X is the set of all distributions qJ E ~'(S) such that the operator

at first defined on COO(S), can be extended as a continuous operator from X into X.

It is clear that (JIt'P(S))' = Jlt'q(S), for 1 < p < +00, where q is the conjugate

f . 1 1 s-o p, I.e. - + - = 1. Moreover, .A(JIt'P(S)) = LOO(S), lor 1 < p < +00.

P q

In order to describe the spaces (Jf"P(S»)' and vII(Jf"P(S» for p :::;; 1, we require several definitions.

We define, first of all, the degree of smoothness (with respect to the quasimetric d) of functions in Aa(s).

3.1.2. Definition. A function IE C(S) is said to belong to the space Aa(s) if for each, E S and each r > 0, there is a polynomial P E C[Zl' Zz, ... , Zn' Zl' zz, ... , znJ such that deg P :::;; 2et and

(5)

everywhere on Q(', r). We set IIIIIA" ~ IIIllc + CJ where CJ, is the smallest quantity Cf ' for which

(5) holds for all , E S and all r > 0.

3.1.3. Definition. We denote by BMO(S) the space of all functions I EL I (S) such that

def • 1 f. IIfIlBMO = IIIllv + sup mf (Q('» II - cl dO" < +00. ,eS,O<r eeC 0" ,r Q("r)

3.1.4. Definition. We denote by BMO]og(S) the space of all functions IE U(S) such that

10 log-

I def I . r f. I II IIBMO = II Ilv + sup mf (Q(' » I - cl dO" < +00. ,eS,O<r<Z ee CO", r Q("r)

None of the spaces Aa(s), BMO(S), and BMO]og(S) are separable. The closure of Coo(S) in these spaces are denoted respectively by A a(s), VMO(S), and VM010g(S). It is not difficult to verify that

A a(s) = {I E Aa(s): lim C f(8) = o}, £-+0+

where Cf (8) is the smallest quantity Cf ' for which (5) holds for all , E S and all r E (0, 8]. Moreover (see Coifman-Rochberg-Weiss (1976»

VMO(S) = {I E BMO(S): lim sup inf ( ;, » f. II - cl dO" = a}. ,--+0+ 'eS,O<r';;, Ce cO" Q ,r Q("r)

We shall not require an analogous description of the space VM01og(S).

3.1.5. Theorem (Garnett-Latter (1978». Let 0< p < 1. Then

vII(Jf"P(S» = (Jf"P(S»)' = A(n/p)-n(s).

3.1.6. Theorem (Coifman-Rochberg-Weiss (1976) and Janson (1976».

(Jf"l(S»)' = BMO(S),

vii Jf"l(S) = L oo(S) n BM01og(S).

(6)

(7)

3.1.7. Corollary.

.Jt(BMO(S)) = L <X.l(S) ('\ BM01og(S).

Theorems 3.1.5 and 3.1.6 can be deduced from Theorem 2.2.5. Formula (7) in the one-dimensional case was obtained by Janson (1976).

From the inclusions £1(S) ::J U(S) = £P(S) for all p> 1, we obtain the inclusions BMO(S) c U(S) for all p < +00. Moreover, the following result of John and Nirenberg (see Garnett (1981)) holds.

3.1.8. Theorem. There exist positive constants A(n) and B(n) such that

Is e lfl du ::;; B(n)

for all f E BMO(S) such that IIfIIBMO::;; A(n).

3.2. The Cauchy Integral in Spaces of Smooth Functions. Let us denote by {.iJa(S)}a>o the classes of smooth functions on S with respect to the usual Euclidean metric on S. The space ;Ja(S) is the set of all functions f E C(S) such that for each point' E S and each r > 0, there is a polynomial P E IC[ZI' Z2' ... ,

Zn' ZI' Z2' ... , zn] such that deg P ::;; I/. and

If(z) - P(z)1 ::;; Cf ' r a

for all z E S such that Iz - 'I ::;; r. It is easily seen that /ia(s) ::J Aa(s) for all I/. > O. <-

Analogously, one can define the space BMO(S) (the BMO-space associated with the Euclidean metric on S).

One can show that AA(B) = Aa(s) ('\ A(B). Let us set BMOA(B) = HI (B) ('\ BMO(S), and VMOA(B) = Hl(B) ('\ VMO(S).

The following theorem follows from Theorem 2.2.1 and duality.

3.2.1. Theorem. The operator C projects the space Aa(s) onto AA(B) (I/. > 0) and the space BMO(S) onto BMOA(B).

We remark also that BMOA(B) = C(LOO(S)) and VMOA(B) = C(C(S)).

3.2.2. Theorem (Ahern-Schneider (1980)). Let f E Ll(S). Suppose that for almost every' E S, the slice function ~ belongs to Aa(lr) (I/. > 0), and

ess sup II~IIA" < +00. 'ES

Then Cf E AA(B).

3.2.3. Corollary. The opertor C projects /ia(s) onto AA(B).

From Theorem 3.2.2 (and no less from Corollary 3.2.3), the "doubling of smoothness" phenomenon, first detected by Stein (1973), appears.

3.2.4. Corollary. A(B) ('\ /ia(s) = AA(B).

142 A.1i. Aleksandrov

This corollary shows that a function of class A(B) n A"(S) is "twice smoother" in the complex tangential direction. We remark also that an analogous phenomenon holds also for functions satisfying a Holder condition with respect to the LP-metric.

We now present several results of Krantz (1980) on the inter-relation between r--...J

the spaces BMO(S) and BMO(S). r--...;

An analog to Corollary 3.2.3 for BMO(S) holds only partially: r--...;

C(BMO(S)) = BMOA(B), (8)

however, r--...;

BMOA(B) ¢ BMO(S). r--...J

Thus, the operator C does not act on the space BMO(S). From (8) it follows r--...J

that BMO(S) n Hl(B) c BMOA(B). However, the last inclusion is strict, i.e. r--...J

there is no analog of Corollary 3.2.4 for BMO(S).

§ 4. Dual Spaces of Some Spaces of Holomorphic Functions

4.1. The Duren-Romberg-Shields Theorem. A topological vector space X will be called a space of hoi om orphic functions if

1) A"'(B) c X c ~~(B) (continuous imbeddings); 2) A "'(B) is dense in X. We define the dual space as the set of all functions <p E ~~(B) such that the

functional f 1--+ (f, <p), defined at first on A"'(B), extends as a continuous linear functional on all of X. Just as in §4 we may define the multiplier space JltX.

It is easily seen that JIt HP(B) = HOJ(B) for all p > O. Moreover, (HP(B))* = 1 1

Hq(B), for 1 < p < +00, - + - = J. p q

4.1.1. Theorem. If p E (0, 1), then

(HP(B))* = A~/p)-n(B).

This theorem follows easily from the theorem on atomic decompositions in HP(B) (see Garnett-Latter (1978)). We remark that in the one-dimensional case, Theorem 4.1.1 was proved by Duren, Romberg, and Shields (1969) five years before the appearance of the theorem of Coifman (1974) on the atomic characterization of HP. A proof of Theorem 4.1.1 analogous to the proof of the "onedimensional" theorem of Duren-Romberg-Shields can be given without using the rather non-trivial Theorme 2.2.5.

4.1.2 Theorem. If 0 < p ::;; 1, then

(~P(B))* = A~+(n/p)-n(B),

(A.~(B))* = ~1(B).

(9)

(10)

One essentially derives (9) for p = 1 and (10) in the process of proving Theorem 4.1.1. The case of arbitrary p < 1 reduces to Theorem 4.1.1 and the case p = 1 with the help of the inclusion

n n where - = - + IX.

q P

Hq(B) c .1{'l(B) c ~;'(n/p)-n(B),

One can show that, for p > 1 and IX > 0, one has

(~P(B))* = 9l-ap(~~/q(B));

where q is the conjugate exponent. The spaces 9l-P(~q(B)) (f3 > IX) are Besov spaces and have a more direct description. At this point, we shall not dwell further on these spaces.

4.2. The Dual Space of Hl(B)

4.2.1. Theorem (Coifman-Rochberg-Weiss (1976)).

(H1(B))* = BMOA(B),

(VMOA(B))* = H1(B).

4.2.2. Theorem. Let f be a function holomorphic in B. Then the following assertions are equivalent:

1) f E BMOA(B);

2) The measure I jt1

ZjDjf 12 (1 - Izl) dv(z) is a Carleson measure;

3)

IlfIIG ~ ~~~ II (1 ~ ~;,(:~ r 1In2 (l - lal 2r/2 < +00. (11)

The equivalence of 1) and 2) was proved in Coifman-Rochberg-Weiss (1976). The functional IlfliG is a seminorm. The equivalence of 1) and 3) in the one-dimensional case was proved by Garsia (see Garnett (1981)). For the multidimensional case, see Axler-Shapiro (1983). An important property of the seminorm of Garcia is its invariance with respect to automorphisms of the ball, I.e.

Ilf 0 cpliG = IlfliG

for all cP E Aut(B). From this it follows that if f E BMOA(B), then

sup inf Ilf 0 cp - cllBMo < +00. cpeAut(B) Ce C

The following theorem follows easily from the Hahn-Banach Theorem.

4.2.3. Theorem.

(C(U(S))* = HOO(B),

(A(B))* = C(M(S)).

§ 5. The Toplitz and Hankel Operators

5.1. The Toplitz and Hankel Operators on the Space H 2 (B). The theory of Hankel and Toplitz operators is very rich, already in the one-dimensional case (see, for example, Nikol'skij (1985)). We remark also that this theory is very useful in many questions in function theory, in operator theory, and in probability theory (see Nikol'skij (1985) and Khrushchev-Peller (1982)). It will not be possible for us to dwell on this theory and its applications in detail. We shall restrict ourselves merely to some definitions and isolated results.

To each distribution qJ E ~'(S), we may associate the Toplitz operator TqJ and the two Hankel operators HqJ and VqJ (defined at first on the set Aoo(B)):

TqJf <i,;[ C( qJf),

f dee HqJ = C_(qJf),

dee VqJf = qJf - C( qJf),

where

f h(O f (C_h)(z) = s (1 _ <" z»" du(O - s h(O du(O (z E B).

The operator C_ projects the space U onto the space 7 {J: f E H2(B) e C}. It is clear that HqJ = VqJ in the one-dimensional case.

We remark also that

TqJ! = TqJ2 <=>qJl = qJ2'

HqJ! = HqJ2 <=> C-qJl = C-qJ2'

VqJ! = VqJ2 <=>qJl - CqJl = qJ2 - CqJ2·

A nonzero operator TqJ cannot act compactly on any space HP(B) and a nonzero operator VqJ cannot have a finite-dimensional image (n ~ 2).

5.1.1. Theorem (Davie-Jewell (1977)). The operator TqJ acts continuously on H2(B) if and only if qJ E Loo(S); moreover, II TqJll = IIqJIILoo.

Let us denote by uLoo(qJ) the essential image of a function qJ, i.e. the spectrum of qJ, with respect to the algebra L 00(S). We denote by conv(K) the convex hull of a set K.

5.1.2. Theorem (Davie-Jewell (1977)). Let qJ E L oo(S). Then

uLoo(qJ) c u(TqJ) c conv uLoo(qJ),

where u(TqJ) is the spectrum of the operator TqJ .

7 The symbol L e M denotes the orthogonal complement of a subspace M of a Hilbert space L.

We remark that in the one-dimensional case the spectrum of a T6plitz operator is connected (Theorem of Widom (1964». In the multidimensional case, however, the spectrum of the operator T", need not be connected even for q> E

C(S) (see Davie-Jewell (1977». An example can be constructed (op. cit.) of the form q> = h(lzli). In general, it should be remarked that if the function q> is lr-invariant, then T",(H(n,O» c: H(n, 0) (n ~ 0), i.e. the space H2(B) can be decomposed as a sum of finite-dimensional T",-invariant subspaces.

T6plitz operators can be characterized by the following identity (DavieJewell (1977»:

n

L Tz.T",~. = T",. j=l J J

We set HI!..(B) = {f E U(S):] E HP(B), ](0) = O}.

5.1.3. Theorem (Coifman-Rochberg-Weiss (1976». The operator H", acts continuously from H2(B) to H:'(B) if and only if C_q> E BMO(S) or (which is the same) there exists a function t/J E L OO(S) such that C_q> = C-t/J. The compactness of the operator H", is equivalent to the condition C_q> E VMO(S), i.e. C_q> = C-t/J for some function t/J E C(S).

We remark further that

(12)

A theorem of Nehari (see Nikol'skij (1985» asserts that in the onedimensional case, we may take 1 for C(1) and both inequalities in (12) become equalities. It is not difficult to show that this is not the case in the multidimensional situation:

C(n) ~ II zIllH2 > 1. IlzIllHl

From Theorem 5.1.3 and the equality H", = C_ 0 V", we obtain the following

5.1.4. Proposition. If the operator V", acts continuously from H2(B) into L2(S) e H2(B), then C_q> E BMO(S).

In addition the following proposition is obvious:

5.1.5. Proposition. If q> - Cq> = t/J - Ct/J for some function t/J E L 00 (S), then the operator V", acts from H2(B) into L 2(S) e H 2(B).

Necessary and sufficient conditions for the continuity of an operator V", are unknown.

5.2. The Toplitz and Hankel Operators on the Spaces HP(B) (0 E L 00 (S), moreover, if p # 2, equality of norms (as in Theorem 5.1.1) fails. The inclusion aLoo(q» c: a(T",) holds in any space HP(B) (0 < p ~ +(0) regardless of the T6plitz operator T",. However, the inclusion a(T",) c: conv aLoo(q» for p # 2 is false already in the

one-dimensional case (see Gokhberg-Krupnik (1973». If 1 < p < +00, then the inclusion H'P(HP(B» c H'!..(B) is equivalent to the condition C_cp E BMO(S).

5.2.1. Theorem. Let 0 < p < 1; cP E A(n/p)-n(s). Then

T'P(HP(B» c HP(B), V'P(HP(B» c yt'P(S), H'P(HP(B» c H'!..(B).

5.2.2. Theorem. Let cP E LOO(S) n (BM01og(S). Then

T'P(H 1(B» c H 1(B), V'P(H 1(B» c yt'1(S), H'P(H 1(B» c H:(B).

Theorems 5.2.1 and 5.2.2 follow easily from Theorems 3.1.5 and 3.1.6 respectively.

In all theorems of this section (on the continuity of the operators H", and V'P)' H'P and V'P turn out to be compact provided we assume that the function cP belongs to the closure of COO(S) in the corresponding space.

5.2.3. Theorem (Rudin (1980». If the modulus of continuity W'P of the function cP satisfies the Dini condition, i.e.

f2 w'P(t) dt < +00 o t '

then

Moreover, the operator V'P : HOO(B) - C(S) is compact.

5.3. Toplitz Operators and Multipliers. Let X be a space of hoi om orphic functions (see §4). We define

deC { } ff(X) = cP E f!)' (S) : T'P E .P(X) ,

where .P(X) is the set of continuous operators on the space X. It is clear that ~(X) n f!)~(B) = JlX.

We further define

JI*X~ {cp E ff(X): (jj E f!)~(B)}.

It is not difficult to show that JlX = JI*X* and JlX* = JI*X. As an application, we have the following theorem.

5.3.1. Theorem. Let cP E HOO(B). Then

Tq;(A~(B» c A~(B), Tq;(A~(B» c A~(B) (a> 0);

Tq;(BMOA(B» c BMOA(B), Tq;(VMOA(B» c VMOA(B);

Tq;(C(U(S))) c C(L1(S».

We remark also that Tq;(C(M(S))) c C(M(S», if cP E A(B).

5.4. Applications of Toplitz Operators to a Problem of Gleason. Let f E X C

H(B), a E B. Are there functions gl' g2' ... gn E X such that n

f - f(a) = L (Zj - aj)gj? j=l

This question was posed by Gleason for X = A(B), a = O. An affirmative answer was obtained by Lejbenzon (see Khenkin (1971)). At the present time there are several ways to solve this problem (see Rudin (1980)). We shall consider the method of Ahern and Schneider (see Rudin (1980)).

To each point a E B we associate an operator &ta : H(B) --+ H(B),

deC n &taf = L (Zj - a)DJ

j=l Set

deC 11 gj(a, z) = 0 (Djf)(a + t(z - a)) dt.

It is easy to verify the following equality

Set

n

f(z) - f(a) = L (Zj - aj)gj(a, z). j=l

deC 11 dt (&t;;lf)(Z) = (f(a + t(z - a)) - f(a))-. o t

(13)

Then gj(a, .) = Djg£;;lf, and (13) signifies that &tag£;;lf = f - f(a). We remark that the power series expansion of the function 9 j(a, .) in the vicinity of the point a is easily expressed via the corresponding expansion of the function f:

(14)

for Iz - al < 1 - Ia!-We shall say that the Gleason problem is canonically solvable in the space X

at the point a E B if (DjG£;;l)X C X for allj E {1, 2, ... , n} A direct calculation shows that if f E '@~(B), then

gia, z) = Is C(z<~~-a~g' 0 f(O(j da(O

(the integral is to be understood in the sense of distributions). The identity

C(z,O - C(a,O = C(z, 0 nf (1 - <z, O~l <z - a, 0 j=O (1 - <a, 0 Y

(15)

shows that


where ({Jk,a are functions antiholomorphic in some neighborhood of the closed ball E. Consequently, the following theorem holds.

5.4.1. Theorem. Let X be a space of holomorphic functions. Suppose 1) ZjX c X for allj E {t, 2, ... , n}; 2) Tq;X c X for all functions ({J holomorphic in some neighbourhood of the

closed ball E. Then the Gleason problem is canonically solvable for the space X and for all a E B.

The results of the previous section allow us to apply Theorem 5.4.1 to the following spaces X:

A (B), HP(B) (p > 0), A~(B) (a> 0), A,4(B) (a> 0),

BMOA(B), VMOA(B), C(U(S)), C(M(S)).

The canonical solvability of the Gleason problem is also known in the spaces .Yt;.P(B) (a, p > 0) and Am(B) (m ~ +00).

We remark that

Using precisely this equality, Lejbenzon obtained the first solution to Gleason's problem for the space A(B) and a = O.

If the space Xc H(B) allows multiplication by coordinate functions (i.e. ZjX c X for all j E {I, 2, ... , n}), then the canonical solvability in X of the Gleason problem for a = 0 is clearly equivalent to the following property of the space X:

for allj E {I, 2, ... , n},

for f E H(B). Making use of (6) from Chap. 1, it is easy to verify that

aj (DafHO) a zkTz.fHz) = L I I 1 ,Z .

J aez': a + n - a. a#O

(16)

This equality together with (14) shows that canonical solvability of the Gleason problem for a space X at the point a = 0 is "almost equivalent" to the following assertion:

for allj E {I, 2, ... , n}. (17)

In particular from (15) for a = 0, it is easy to see that (17) implies the canonical solvability of the Gleason problem for a = 0 if the space X sustains multiplication by the coordinate functions.

Chapter 4 Zeros of Functions Holomorphic in the Ball

§ 1. Characterization of Zeros of Functions in the Smirnov, Nevanlinna, and Nevanlinna-Dzhrbashyan Classes

149

1.1. One-Dimensional Results. We recall, first of all, the well known characterizations of zeros offunctions in the Nevanlinna, Smirnov, and Hardy spaces.

Let {ank"l be a sequence in the unit disc satisfying the Blaschke condition

L (1 - lanl) < +00. (1) n;o,l

Then the Blaschke product

b(z) = IT h(an , z), (2) n;o,l

where

{ a-z a

h(a, z) = 1 - az . raT' . z,

a =f. 0,

a =0,

is a bounded holomorphic function in the unit disc !D, having {an}n;o,l as its precise set of zeros (counting multiplicities).

Conversely, the zero set (counted according to multiplicities) of a non-zero function of Nevanlinna class satisfies the Blaschke condition (1). Thus, the Blaschke condition gives a complete characterization ofzero sets offunctions in all classes X such that H OO eX c N.

We turn now to zero sets of functions in the Nevanlinna-Dzhrbashyan class. Let us denote by Na the set of all functions f holomorphic in the disc []l such that

f[) (1 -lzl)a-1log+ If(z)1 dv1(z) < +00 (IX> 0).

A sequence {an}n;o,l of points in the unit disc is the set of zeros (counting multiplicities) of a function in the class Na if and only if

L (1 - lanl)l+a < +00 (3) n;o,l

(see, for example, Dautov-Khenkin (1978)). Korenblum (1975) characterized zero sets of functions in the class 92~([]l).

However, this characterization is rather complicated. We remark that the Blaschke condition is no longer necessary here, however, it is still necessary if all of the zeros lie on a line. Moreover, from what we have said above, it follows that condition (3) is necessary for all IX > 0.

1.2. The Khenkin-Skoda Theorem. Throughtout the rest of this chapter, r will denote (2n - 2)-dimensional Hausdorff measure h2n - 2 in tC". Let M be an analytic subset ofthe ball B of dimension (n - 1) at each of its points (see Shabat (1976)). In other words, M is the zero set of some non-zero holomorphic function in B. Let us denote by M the set of all regular points of the set M, i.e. the set of all points of M in the neighborhood of which M is a complex manifold (of dimension n - 1). The set M\M of all critical points of M is an analytic set of dimension at most n - 2. Consequently, r(M\M) = O. By a divisor on M we shall mean a locally constant function k: M --+ N.

To each function f holomorphic in the ball B and each point a E B, we may associate the degree kf(a) of the zero of f at the point a (if f(a) =F 0, then kf(a) '!;f 0) defined as follows:

kf(a) '!;f inf{ loci: (D"f)(a) =F O}

Thus, the function kf maps B into Z+ u {(f)}. We shall say that a divisor k: M --+ N is the divisor of the function f E H(B)

if f- 1(0) = M and kflM = k. Of course, each divisor k : M --+ N has a natural extension k to all of M such

that kflM = ~ implies kflM = k. However for our purpose this is not essential since r(M\M) = O.

In the sequel, we shall say for brevity that M is an analytic set of dimension n - 1, omitting the words "at each of its points".

1.2.1. Theorem (G.M. Khenkin (1977a, b), Skoda (1976)). Let M be an (n - I)-dimensional analytic subset of the ball Band k a divisor on M. Then the following assertions are equivalent:

1) k is the divisor of some function f E N*(B); 2) k is the divisor of some function f E N(B); 3) the divisor k satisfies the following Blaschke condition

fM k(z)(1 -Izl) dr(z) < +00.

The necessity of the Blaschke condition is proved analogously to the one dimensional case. The essential diffulty in the multi-dimensional theorem is the proof of sufficiency, i.e. the construction of a holomorphic function having the given divisor.

1.3. Discussion of the Blaschke Condition. Let k be a divisor on an (n - 1)dimensional analytic set M in B. To each point ( E S we associate the quantity

@J(k,O = I k(z()(1 - Izl). ZED

zCEM

We consider also the function v,.: [0, 1) --+ IR,

Vk(r) = f k(z) dr(z). M''>rB

1.3.1. Proposition. The Blaschke condition is equivalent to each of the follow-ing two conditions:

1) JsBl(k, () du(() < +00; 2) J6 "k(t) dt < +00.

We mention also two consequences of the Khenkin-Skoda Theorem.

1.3.2. Theorem. A divisor of a function f E H(B) is the divisor of some function g E N*(B) if and only if

sup r log Ifrl du < +00. 0<r<1 Js

1.3.3. Theorem. Let k 1 , and k2 be divisors on (n - 1)-dimensional analytic subsets M 1 and M 2 of the ball B. Suppose that M 1 C M 2 and kl :::::; k2 Lv l' If the divisor k2 is the divisor of some function in N*(B), then the same is true of the divisor k 1 .

We remark that the analog of Theorem 1.3.3 for HOO(B) (n ~ 2) fails (see 7.3.6 in Rudin (1980) and Amar (1982)).

1.4. The Khenkin-Dautov Theorem. Let us denote by N,,(B) the space of all holomorphic funcions f in the ball such that

L (log+ Ifl)(1 - Izl),,-1 dv(z) < +00,

where IX> O.

1.4.1. Theorem (Dautov-Khenkin (1978)). Let M be an (n - 1)-dimensional analytic subset of the ball B and IX > O. Then a divisor k on M is the divisor of some function f E N,,(B) if and only if

fM k(z)(1 -lzl)"+1 dr(z) < +00.

We remark that if IX is a positive integer, this theorem reduces to Theorem 1.2.1; it is sufficient to notice that if f E N(Bn+"), then

f(z l' Z2, ... , Zn, 0, ... ,0) E N,,(Bn).

1.4.2. Proposition. Let k be a divisor on an (n - 1)-dimensional analytic subset M of B. The following assertions are equivalent

1) JM k(z)(1 - Izl),,+l dr(z) < +00; 2) Jb "k(t)(1 - t)" dt < +00; 3) JG Bl,,(k, 0 du(O < +00,

where

Bl,,(k, 0 = L (1 -lzly+lk(zO· ZED

ZI;EM

Moreover, there are analogs to Theorems 1.3.2 and 1.3.3. We present here only the analog of Theorem 1.3.2.

1.4.3. Proposition. A divisor oj a Junction J E H(B) is the divisor oj some Junction g E NIT.(B) iJ and only iJ

Il (1 - r)IT.-l (Is log 1/,1 da) dr < +00.

We remark also that from the easy part (i.e. the necessity) of Theorem 1.4.1, we have the following

1.4.4. Proposition. IJ k: Nt --+ N is the divisor oj a Junction J E !0~(B), then

fM (1 -lzl)d1k(z) d"C(z) < +00,

Jor all r:t. > O. The converse is Jalse, even in the one-dimensional case (see 1.1).

§ 2. Zeros of Functions in the Hardy Spaces HP(B)

The problem of characterizing the zero sets of functions in the Hardy classes HP(B) (0 < p < +(0) appears at the present to be very difficult. In any case, in view of Theorem 1.3.1 of Chap. 2, such a characterization would follow from a (at present unknown) characterization of zeros of functions in the onedimensional classes ~P for an integer r:t.. From this remark it follows that the zero sets of functions in the HP(B) classes are different for different p.

Also unknown is a characterization of the zeros of functions in the class U HP(B). We present here a sufficient condition obtained by Varopoulos

p>O

(1980).

2.1. Uniform Blaschke Condition. Let k be a divisor on an (n - l)-dimensional analytic subset M of the ball B. We shall say that k satisfies the uniJorm Blaschke condition if

sup f (k 0 cp)(z)(l - Izl) d"C(z)) < +00. ",eAut(B) ",-'(M)

In the one-dimensional case the uniform Blaschke condition is equivalent to the discrete measures (1 - Izl)k(z) d"C(z) being a Carleson measure, which in turn is equivalent to the corresponding Blaschke product

(a - z li )k(a) n -'-

aeM\{O} 1 - liz lal

8 Here and hereafter it is convenient to consider that k has been extended as the zero function on B\M.

being a finite product of interpolating Blaschke products (see Garnett (1981)). In the multidimensional case the uniform Blaschke condition is equivalent to

k(Z)((1 - Izl) + (1 - I<z, l:i~)12)) dr(z), (4)

being a Carleson measure, where n(z) is the unit vector orthogonal to the manifold M at the point z E M. The measure (4) first appeared in the work of Malliavin (1974). From the results of Malliavin (1974) and the Khenkin-Skoda Theorem, it follows that the finiteness of the measure (4) is equivalent to the Blaschke condition, i.e. the finiteness of the measure

k(z)(1 - Izl) dr(z). (5)

However, the condition that the measure (4) be a Carleson measure is, in general, not equivalent to the condition that the measure (5) be a Carleson measure (see Varopoulos (1980)). By the same token in the multidimensional situation, the uniform Blaschke condition may fail even if the measure (5) is a Carleson measure.

2.1.1. Theorem (Varopoulos (1980)). If a divisor k satisfies the uniform Blaschke condition, then it is the divisor of some function in the family

{J E N*(B): log If I E BMO(S)} c U HP(B). p>O

It is worth mentioning that for each p > 0 there exists an analytic set of dimension n - 1 which is a set of uniqueness for HP(B) (i.e. f E HP(B) and flM == 0= f == 0) and which nevertheless satisfies the uniform Blaschke condition.9

2.2. Piecewise-Linear Analytic Sets. We now direct our attention in more detail to the situation when M is a piecewise-linear analytic set of dimension n - 1, i.e.

M = U (lj!1 B), jeJ

where the lj are complex hyperplanes in e" and diam (lj!1 B) --+ O. We shall assume that 0 rt M. Then for each hyperplane lj' there is a unique point aj E B such that

lj = {z E e": <z, aj ) = lajI2}.

Here, as in the one-dimensional case, instead of considering a divisor on M, we shall consider that the sequence {lj}jEJ (or what amounts to the same, {aj}jeJ) allows repetitions.

The Blaschke condition for M becomes:

L (1 - lajl)" < +00. jeJ

9 More precisely, the divisor which is identically one on M satisfies the uniform Blaschke condition.

The Dautov-Khenkin condition for M becomes

L (1 - lajW+<X < +00. jEJ

The uniform Blaschke condition for M becomes the requirement that the measure 10

be a Carleson measure.

2.2.1. Theorem (Varopoulos (1980)). Let A c B. If

L (1 - lajltc5a aE A

is a Carleson measurell , then there exists a piecewise-linear (n - 1)-dimensional analytic set containing A and satisfying the uniform Blaschke condition.

We remark that under the conditions of Theorem 2.2.1, the set

U {z E en : <z, a) = Ian ae A\{O}

may not satisfy the uniform Blaschke condition. For example, such is the case if the set

{a E A: «, a) = lal 2 }

is infinite for some ( E S (see Varopoulos (1980)). From Theorems 2.1.1 and 2.2.1 we have the following.

2.2.2. Corollary (Varopoulos (1980)). Under the conditions of Theorem 2.1.1 there exists a function

fE U HP(B) p>O

not identically zero such that f- 1(0) ::::l A.

One can find other examples of applications of Theorem 2.2.1 in the works of Varopoulos (1980) and Amar (1982).

2.3. Zeros of Bounded Holomorphic Functions. If k is a divisor of a bounded holomorphic function not identically zero in the ball B, then, in addition to the Blaschke condition, the divisor k satisfies the following additional condition (absent in general for divisors of HP-functions (p < +00)).

If I is a complex affine subspace of en, then either In B c M or the divisor kl1nM satisfies the Blaschke condition (in the ball I n B).

10 Here 1 E signifies the signifies the characteristic function of the set E. 11 The symbol /ja denotes the /j-measure at the point a.

However, as the following theorem shows, this additional necessary condition for divisors of HOO-functions does not superimpose (as compared to the Blaschke condition on the ball B) any additional restrictions on the "massiveness" of the set of zeros of bounded holomorphic functions.

2.3.1. Theorem (Hakim-Sibony (1982a)). Let h: (0, 1] ~ (0, +(0) be a function infinitely large at zero. Then there exists a function f E HOO(B) whose divisor has the following property

f k(z)·(1 -lzl)h(1 -Izl) dr(z) = +00.

We remark that we may additionally require that the zero set of the function f be piecewise-linear and that f E A(B). Amar (1982) and Alexander (1982)

showed independently that for h(t) = ~ we may also require that f E An(B) t

(n < + (0). In the case of a piecewise-linear analytic set

M = U {z E en: <z, aj ) = laj l2} jeJ

the inequality

L (1 - lajl} < +00. (6) jeJ

is clearly a sufficient cbndition to be the divisor of an HOO-function. As the required function f E HOO(B), we may take the analog

f(z) = n <aj - z, aj )

jeJ 1 - <z, aj )

of the Blaschke product. It is useful to remark that it is not possible to weaken condition (6) on the

terms lajl since it is a necessary condition when all the points lie on a complex line.

The following beautiful result is unfortunately so far proven only for n ~ 3.

2.3.2. Theorem (Berndtsson (1980)). Let k be a divisor on an analytic subset M of the ball B. Set

V(r) = f k(z) dr(z) (0 < r < 1). MnrB

Suppose the (n - 2)-nd derivative of the function V is bounded. Then if n = 2 or n = 3, the function k is the divisor of some function in HOO(B).

We remark that the boundedness of the (n - 2)-nd derivative of the function V for a piecewise-linear set

M = U {z E en: <z, a) = lal} jeJ

is equivalent to condition (6).

The methods of Berndtsson (1980) yield some sufficient (but not so elegant) conditions for divisors of Hoo-functions also for n 2 4.

Chapter 5 Interpolation, Peak Sets, A-Measures and P-Measures

§ 1. Representing Measures and A-Measures

1.1. A-Measures, Representing Measures, Totally Singular Measures and Their Properties

1.1.1. Definition. A measure J1. E M(S) is called an A-measure (analytic mea

sure, L-measure, or Khenkin measure), if lim f In dJ1. = 0 for any bounded se-n-+oo S

quence Un}n?:l in A(B) which tends to zero everywhere in B. We denote the set of all A-measures by HM(S).

In the one-dimensional case (as is well known, see for example Hoffman (1962)), any closed subset Fe 1" of zero Lebesgue measure is a peak set for some function IE A(D). This means that IIF == 1 and III < 1 everywhere on ID\F. If

J1. E HM(S), then J1.(F) = lim If" dJ1. = o. Consequently, in the one-dimensional n-oo

case, HM(1") c U(1"). In fact, HM(lr) = U(1"), for the reverse inclusion L l(S) c HM(S) holds for all n 2 1 (see Rudin (1980)).

1.1.2. Definition. A positive measure J1. E M(S) is called a representing mea

sure for the point Z E B if I(z) = Is I dJ1. for all IE A(B). We denote by Mz the

set of all representing measures for the point z. For n = 1 each point z E ID has a unique representing measure, the Poisson

kernel P(z, .). m. In the multidimensional case each point z E B has many representing measures, for example, the Poisson kernel Ph(z, .). (J, and the invariant Poisson kernel P(z, . )(J. There are also singular representing measures: if , E S then

1(0) = L I(~O dm(~), for all I E A(B).

Clearly, any representing measure is an A-measure.

1.1.3. Definition. A measure J1. E M(S) us said to be totally singular if it is singular with respect to each representing measure (or, equivalently, if it is singular with respect to each measure in Mo). We shall denote by TM(S) the set


of all totally singular measures in M(S). A Borel set E c S is said to be a null set if m(E) = 0 for all fl E Mo.

In the one-dimensional case, total singularity is equivalent to singularity and a null set is just a set of Lebesgue measure zero on lr.

From the general Theorem of Glicksberg, Konig, and Seever (see Theorem 9.4.4 in Rudin (1980)) we deduce the following.

1.1.4. Theorem. Any measure fl E M(S) can be uniquely represented in the form fl = fla + fl .. where the measure fla is absolutely continuous with respect to some measure in M 0 and the measure fls is concentrated on a null set of type Fe!"

We denote by A(B)-L the set of all measures fl E M(S) such that f f dfl = 0 for all f E A(B).

1.1.5. Theorem (Val'skij (1971)).

HM(S) = A(B)-L + U(S).

In other words each measure fl E HM(S) determines a functional on A(B) which extends to a functional on H OO which is continuous with respect to the weak topology (J(L 00, U) on Hoo(B).

1.1.6. Corollary. Let fl E M(S). If m is an A-measure, then

lim Y(Jg E S: J(Cfl)(OJ > y} = o. y .... + co

The converse is false for n ~ 2. In order to show this, we start with a singular measure fl on the circle lr such that

lr (ld~(g)2 = 0 C ~ JZJ) (JzJ-d -).

It is known that such a measure fl exists (see Piranian (1966)). Now in order to obtain our required counterexample, we have only to "transfer" the measure fl to the sphere S with the help of the imbedding Z f--+ (z, 0, ... , 0) (z E lr).

1.1.7. Theorem. In order that a measure fl E M(S) be an A-measure, it is necessary and sufficient that it be absolutely continuous with respect to some representing measure (or measure in Mo).

This theorem was proved by Khenkin and then reproved by several authors (see Chirka-Khenkin (1975)).

1.1.8. Corollary. If fl E A(B)-L, fl is absolutely continuous with respect to some measure in Mo.

For n = 1 this corollary is well known from the F. and M. Riesz Theorem (Hoffman (1962)).

Theorem 1.1.7 allows us to reformulate Theorem 1.1.4 in the following form.

1.1.9. Theorem. M(S) = HM(S) EB TM(S).

1.2. A-Measures and Boundary Behavior of Bounded Holomorphic Functions

1.2.1. Theorem (see 11.3 in Rudin (1980)). Let Ji E M(S). Then the following assertions are equivalent:

1) for any function f E Hoo(B), the limit lim Ir exists in L oo(IJiI) in the r--+l-

(J(L 00, L I)-topology;

2) for any function f E Hoo(B), the limit lim fir dJi exists; r-+l-

3) JiEHM(S).

It is no known whether lim Ir exists Ji-almost everywhere for all f E Hoo(B) r-+l-

and all Ji E H M(S). In other words, is the set of all, E S such that lim f(rO, fails r-+l-

to exist necessarily a null set for all functions f E Hoo(B)?

1.2.2. Theorem (Rudin (1980), Nagel-Wainger (1981)). Let cp: [0,1] --+ S be an absolutely continuous curve. Suppose that 1m < cp', cp> > 0 almost everywhere on [0, 1]. Then any function f E Hoo(B) has a B-limit at cp(t) for almost all t E [0, 1].

In Nagel-Wainger (1981) there is also an analogous assertion proved under weaker assumptions on the curve cpo

In view of Theorems 1.2.1 and 1.2.2 we can construct examples of A-measures.

1.3. A-Measures and Isomorphism Classification of Banach Spaces of Analytic Functions. The notion of A-measure was introduced by Khenkin (1968) in order to show the non-isomorphism of the spaces A(B) and A([j)k) (k ~ 2). Here, A([j)k} denotes the space of all holomorphic functions in the polydisc [j)k which are continuous up to the boundary. Subsequently, a stronger result was obtained by Pelcziitski (1977).

We now present several more results on isomorphism or non-isomorphism of Banach spaces of holomorphic functions.

The space HP(B} is isomorphic to the space U(S) for p E (1, +oo) (see Boas (1955), Aleksandrov (1982), and Wojtaszczyk (1983)). Analogous assertions are also true for Hardy spaces HP([j)k) on the polydisc [j)k (1 < p < +00).

Wojtaszczyk (1983) showed that HI(Bn) is isomorphic to HI for all n. It is interesting that for the polydisc, the situation is completely different: Bourgain (1982, 1983a) showed that if Hl([j)k) is isomorphic to HI ([j)i), then k =j. Mityagin and Pelcziitski (see Pelcziitski (1977)) showed that A(Bn) is not isomorphic to A(D) for n ~ 2. Bourgain (1983b) showed that if A([j)k) is isomorphic to A([j)i) then k = j. He also proves substantially more general assertions.

So far as the (non Banach) spaces H(B) and H([j)k) of all holomorphic functions on the ball and polydisc are concerned, we have the following assertions: H(Bn) is isomorphic to H([j)n) for all n; H(Bn} is not isomorphic to H(Bm) if n =1= m (see Khenkin-Mityagin (1971)).

§ 2. Null Sets and Interpolation on the Sphere S by Functions in the Class A(B)

159

2.1. Z-Sets, P-Sets, I-Sets, and Null Sets. From Theorem 1.1.7 it follows that a Borel set E c S is a null set if and only if fl(E) = 0 (or Ifll(E) = 0) for every A-measure fl. From this and from Theorem 1.5 there follows another necessary and sufficient condition in order that E be a null set: Ifll (E) = 0 for all fl E A(B).l.

2.1.1. Theorem. Let K be a compact subset of the sphere S. Then the following assertions are equivalent:

1) K is the set of zeros (Z-set) of some function in A(B), i.e. there exists a function f E A(B) such that flK == 0 and f(z) =F 0 for all z E B\K;

2) K is a peak set (P-set) for A(B), i.e. there is a function f E A(B) such that flK == 1 and If I < 1 for all B\K;

3) K is an interpolation set (I-set), i.e. for each function hE C(K), there is a function f E A(B) such that flK == h;

4) K is a null set.

This theorem has a rich history already in the one-dimensional case. Various parts of the one dimensional case where obtained in turn by Fatou, the brothers F. and M. Riesz, Rudin, and Carleson. Abtract versions of certain parts of Theorem 2.1.1 were obtained by Bishop and Varopoulos.

In Rudin (1980) one can find the proof of Theorem 2.1.1 as well as more detailed historical remarks.

2.2. Examples and Properties of I-Sets 1. The real sphere defined as S R = S 11 ~n is the Z-set of the function

n

L zJ - 1. j=l

2. The torus Su = {z E S: Ii Zj = n-n/2} is the Z-set of the function n J=l

TI Zj - n-n/2. j=l

We remark that SR and Su are smooth manifolds of dimension n - 1. It is convenient to bear in mind that for n = 2 there exists a unitary transformation of en which transforms SI!? to SUo

3. Davie and Oksendal (1972) (see also Rudin (1980)) showed that if hl (K, S) = 0 (see Chap. 1,4), where K is a compact subset of the sphere, then K is an I-set. In the one-dimensional case the converse is also true.

4. (See Rudin (1980).) The trace of a C1-curve y : [0, 1J --+ S is an I-set if and only if <y'(t), y(t) = 0 for all t E [0,1]. The necessity of this condition follows from Theorem 1.2.2, and the sufficiency from 3.

5. (Tumanov-Khenkin-Burns-Stout-Rudin (see Khenkin-Leiterer (1984) and Rudin (1980))) Let M be a C1-smooth manifold. Then every compact subset K of M is an I-set if and only if M is integral. Notice that the necessity follows from 4.

6. A countable union of I -sets is an I -set if it is compact. 7. Stout (1982) showed that the topological dimension of any I-set is at most

n - 1. 8. Henrikson (1982) constructed an example of an I-set K such that

dim(K, en) = 2n - 1 and consequently dim(K, S) = n (see Chap. 1, 4). The first non-trivial result in this direction had been obtained by Tumanov (1977).

2.3. Boundary Uniqueness Sets. A subset K of the sphere S is called a uniqueness set (for A(B)) if any function f E A(B) which vanishes on K vanishes identically.

Here are some examples of sets of uniqueness and sets of non-uniqueness. 1. Any Z-set is not a set of uniqueness. 2. The sphere {z E S : Zn = o} of dimension 2n - 3 is neither a Z-set nor a set

of uniqueness (n ~ 2). 3. Suppose y: [0, 1] -+ S is a C l integral curve, i.e. <y', y) == 0. Then, because

of Example 4 of 2.2, y([O, 1]) is a set of non-uniqueness. 4. Consider the curve

This curve is not integral. The set y([O, 1]) is a set of uniqueness if and only if {a l , a2' ... , an} is a Q-linearly independent system of numbers (see 10.6.2 in Rudin (1980)).

5. If O"(K) > ° then K is a set of uniqueness. In the one-dimensional case, for compact sets the converse is also true.

6. (Pinchuk (1974)). Let M be a C2 submanifold of S. If M is generic at some point, then M is a set of uniqueness. In particular, if dim M = 2n - 2, then M is a set of uniqueness. In the survey by Chirka and Khenkin (1975), it is shown that an analogous assertion is true also for C l submanifolds.

Sadullaev (1976a) showed that if M is generic at each of its points and O"M(K) > ° for some Borel set K c M, then K is a set of uniqueness. (Here O"M signifies Lebesgue measure on M.)

Joricke (1982) obtained a more precise quantitative version of Pinch uk's (1974) result in the spirit of the Two Constants Theorem.

All of the assertions of 6 can be proved by using a modification of the method of "gluing analytic discs" which originated with Bishop (1965).

2.4. Maximum Modulus Sets. There is also a definite interest in studying maximum modulus sets (M-sets) for A(B). A subset K of the sphere S is called a M -set if there exists a non-constant function f E A (B) such that

We list some examples and properties of M-sets. 1. Any Z-set is an M-set.


n

2. The sets lr· S ~ and lr· STare M -sets respectively for the functions L zJ n j=l

and f1 Zj. j=l 3. (Duchamp-Stout (1981)). If f E A(B), If I ::; 1, g E S: If(OI = I} ::::> lr· ST

then f = b (nn/2 )] Zj) for some finite Blaschke product b. An analogous asser-

tion is true also for the set lr· S R; (in this case, f = b (jtl zJ) ). From this it follows that the union of two (even disjoint) M-sets need not be

an M-set. 4. (Duchamp-Stout (1981)). The topological dimension of an M-set is at

most n. Example 2 shows that this estimate is sharp. 5. (Aleksandrov (1983, 1984)). There exists an M-set of positive Lebesgue

measure on S. 6. If K is the M-set of a function f E A~(B) and (X > 1/2, then a(K) = 0 (11.4,

Rudin (1980)). With stronger smoothness assumptions, this was shown by Tumanov and Sibony (see Chap. 3, Khenkin-Leiterer (1984)). Therein, assertion 6 is also announced for (X = 1/2.

2.5. Interpolation Within the Ball by Functions in the Classes A (B) and HP(B)

2.5.1. Theorem (Khenkin-Leiterer (1984». Let M be a closed complex submanifold of the ball r B (r > 1). Let f: M n B -+ If:- be a function holomorphic in M n B and bounded (respectively uniformly continuous). In this case we write f E HOO(M n B) (respectively f E A(M n B)). Then there exists a function f E HOO(B) (respectively F E A(B» such that FIMnB == f There exists a continuous linear operator R : HOO(M n B) -+ H'°(B) which generates the function F from the function f

If, moreover, the manifold intersects the sphere transversally, then we may require additionally that R(A(M n B)) c A(B) (Khenkin (1972)).

The case of holomorphic functions smooth up to the boundary is considered by lacobczak (1983).

Analogs of Theorem 2.5.1 for the spaces HP(B) and Y{'/(B) were obtained by Cumenge (1983) (in the case of transversal intersection of M with S) and Amar (1983b) (in the general case). The results of Cumenge and Amar can be viewed as far-reaching generalizations of Theorem 1.3.1 of Chap. 2.

§ 3. P-Measures

3.1. Integral Representations of P-Measures

3.1.1. Definition. We shall call a measures J.l E M(S) a P-measure if its (invariant) Poisson integral is a plurisubharmonic function in the ball B.

We remark (see Chap. 2, 2.5) that a measure J.l E M(S) is a P-measure if its Poisson integral is identically equal to its invariant Poisson integral. Moreover,


a measure 11 E M(S) is a P-measure if and only if Isf dl1 = 0 for all real functions f E C(S) such that f· a E A(B).L. We denote the set of all P-measures by PM(S).

To every point' E [pn-1 there corresponds a circle lr{ = {~ E S: n(~) = nand a disc []){ = g E B: n(~) = n u {a}. It is easy to see that if u is the Poisson integral of a measure 11 E PM(S), then for almost all , E [pn-1 the function ul [b is , the Poisson integral of a measure 11{ E M(lr{) and we have

11 = Ln-, 11{ d8(O (1)

in the following weak sense: the function' f---+ I TJ dl1{ is summable in [pn -1 and

Is f dl1 = Ln-, (f..., f dl1{) dc1(O

for all f E C(S). We remark that in the same weak sense, we have the following identities:

1111 = f 111{1 dc1(O, (2)

l1 a = f I1Z dc1(O, (3)

I1 s = f I1t dc1(O· (4)

Here, l1 a is the absolutely continuous (with respect to a) part of the measure 11, I1 s is the singular part, I1Z is the absolutely continuous (with respect to Lebesgue measure on lr~) part of the measure 11{, and I1t is the singular part.

We remark that if the measure 11 E PM(S) is positive, then the measure 11{ is defined for all , E [pn-1 and the mapping' f---+ 11{ is continuous from [pn-1 to M(S) (we identify M(lr{) with the set of all measures 11 E M(S) for which supp 11 c lr{) endowed with the weak-* topology a(M(S), C(S)).

We introduce several results which follow easily from the integral representations (1 )-(4).

3.1.2. Theorem. Let E be a Borel subset of the sphere Sand h2n - 2 (E) = O. Then 1111 (lr . E) = 0 for each measure 11 E PM (S).

A somewhat weaker result was obtained by Forelli (1974). One can find other results of the same kind in the works of Forelli (1974, 1975).

3.2. The Khrushchev-Vinogradov Asymptotic Formula. Let 11 E PM(S). Then

lim yag E S: I (CI1)(O I > y} = ~ I1l1 s ll. y-+co 7r

(5)

In the one-dimensional case this result was obtained by Khrushchev and Vinogradov (1981). The multidimensional case reduces to the one-dimensional case with the help of formulas (1) to (4).

Since in the multidimensional case there exist singular A-measures, Corollary 1.1.6 shows that for n ~ 2 formula (5) does not always hold for measures Jl E

M(S). We remark further (see Rudin (1980)) that if Jl is a <5-measure on the sphere S, then

lim yug E S: I( C")(OI > y) ~ ((' ))' > ~, y .... +oo 4 r ~ + 1 rc

2

for n ~ 2. One can show that

lim sup Y.{( E S: I(C")(OI > y}"" ((' ))' 11"'11 y .... +oo 4 r ~ + 1

2

for each measure Jl E M(S). From (5) and Corollary 1.1.6 we have the following.

3.2.1. Proposition. HM(S)nPM(S) c Ll(S).

We can amplify somewhat on the results presented earlier for positive Pmeasures.

3.2.2. Boole's Formula (see Khrushchev-Vinogradov (1981)). If Jl is a positive singular measure in PM(S), then

erg E S: 12(CJl)(O - Jl(S) I > y} = ~ arctg IIJlII. rc y

Several of the "one-dimensional" proofs of this formula set forth in Khrushchev-Vinogradov (1981) go through in the multidimensional case.

3.3. "Smoothness" and "Regularity" Properties of P-Measures. Forelli (1974) has shown that if e is a Borel subset of projective space IP'n-l, then

Jl(rc- 1(e)) = a-(e)· Jl(S)

for any measure Jl E PM(S). From this it follows that

Jl(B(" r)) ::;; Cn· r2n -2 Jl(S),

(6)

(7)

for each positive measure Jl E PM(S), each r E (0, 2), and each, E S. Thus, we also have the following:

(8)

for each positive measure Jl E PM(S), each r E (0, 2), and each, E S. Along with formula (6), we mention also (see Forelli (1974)) one more for

mula for P-measures:

Jl(P- 1 (E) n S) = f u(PO der(O, p-l(E)nS

(9)


where u is the (invariant) Poisson integral of the measure p. and P is the orthogonal projection of C" onto Ck (k ~ n - 1).

From (7) and (8) we may deduce the following.

3.3.1. Proposition. If M is a C1-submanifold of S with dim M ~ 2n - 2, then p.(M) = 0 for each positive measure p. E PM(S).

If dim M ~ 2n - 3, then in view of Theorem 3.1.2 a stronger assertion holds: ip.i (M) = 0 for each p. E PM(S). It is unknown to the author whether this is true when dim M = 2n - 2.

The following lower estimates hold:

p.(B((, r)) ~ Cn·r2n +1p.(S),

p.(Q((, r)) ~ Cnrn+1p.(S),

for each positive measure p. E PM(S), each r E (0, 2), and each (E S (n ~ 2).

(10)

(11 )

The estimates (7), (8), (10), and (11) are best possible. Indeed, if the Poisson

. I f h . I 1 + z 1 (. (1 + z 1) l" mtegra 0 t e measure p. IS equa to Re -- I.e. p. = Re -- (J lor

) 1-Z1 1-Z1

n ~ 2 ,then

p.(B(e1, r)):::=:::: r 2n - 2 (r --+ 0),

p.(Q(e1, r)):::=:::: r n- 1 (r --+ 0),

p.(B(-e1,r)):::=::::r2n +1 (r--+O) (n~2),

p.(Q(-e1,r)):::=::::rn+1 (r--+O) (n~2).

All of the estimates of this section easily reduce to a few estimates for harmonic functions of two real variables with the help of (9) and (1) of Chapter 1.

We remark that for a signed measure p. E PM(S), the functions r ~ ip.i(Q((, r)) and r ~ ip.i (B(C r)) can tend to zero rather quickly but not arbitrarily quickly (n ~ 2).

3.3.2. Theorem (Joricke (1971)). If p. E PM(S) is not the zero measure and n ~ 2, then

for all (E S.

L log(ip.i (B((, r))) dr > -00,

L log(ip.i (Q((, r))) dr > -00,

Joricke (1971) shows also that this result is in some sense best possible.

3.3.3. Corollary (Forelli (1974)). If J1 E P(S) is not the zero measure and n ~ 2, then supp J1 = S.

We remark further that, for n ~ 2, any positive measure J1 in PM(S) has the following homogeniety property (with respect to the quasimetric d):

for all ( E S and all r > O. In other words, for n ~ 2, the space (S, d, J1) is a space of homogeneous type (see Coifman-Weiss (1977)). For the usual Euclidean metric on the sphere S, the situation is in general different: if the Poisson integral of the

. 1 + Zl measure J1lS Re -1--' then

- Zl

J1(B«(,2r)) sup = +00.

~ES.r>O J1(B«(, r))

Let u be the Poisson integral of a positive measure J1 E PM(S). Consider the set

XJl~{(ES: lim u(rO= +oo}. r-l-

It is well known (see 5.2.7, 5.4.11, and 5.4.12 in Rudin (1980)) that the singular part J1s of the measure J1 is concentrated on the set Xw It is easy to see that p(XJl) = 0 for each p E Mo. Consequently the measure J1s is singular with respect to each measure in Mo. It is unknown to the author whether the analogous assertion is true for all measures in PM(S).

§4. P-Measures and the Boundary Behavior of Holomorphic Functions

4.1. P-Measures and the Hardy-Lumer Class. Let us denote by LHP(B) the set of all functions f holomorphic in the ball B for which there exists a pluriharmonic function u such that IflP ::; u everywhere in B. We set

IlIflllp ~ inf U(O)l/P,

where the infimum is taken over all positive pluriharmonic functions u. The classes LHP(B) were introduced by Lumer (1971). For n = 1, LHP(B) = HP(B). The spaces LHP(B) are Banach spaces for p ~ 1 and Polish spaces for p < 1. It is not difficult to see (see 7.4.2 in Rudin (1980)) that the zeros of functions in the class LHP(B) have the same structure as the zeros of bounded holomorphic functions in the ball. However, linear-topological properties of the class LHP(B), for n ~ 2, are rather "bad": the spaces LHP(B) are not separable and the space LH2(B) is not isomorphic to a Hilbert space (see 7.4.6 in Rudin (1980)).

We remark further that if Ref> 0 (f E H(B)), then f E LHP(B) for all p < 1. Let us denote by CM 0 the space of all functions f E C(S) such that f· a E Mo.

4.1.1. Theorem. Let f be a function holomorphic in the ball B and let 0 < p < +00. Then f E LHP(B) if and only if fEN * (B) and

A(lfIP) ~ sup d {Is IflPcp da: cP E CMo} < +00;

moreover A(lfIP) = Illflll~.

A similar characterization of functions in the class LHP(B) was obtained by Lumer (1971). However, in Theorem 4.1.1 functions in the class LHP(B) are characterized by their boundary values, whereas Lumer's characterization is in terms of their values inside the ball B.

Theorem 4.1.1 is implicitely proved in 9.7 of Rudin (1980). It follows from the following assertion which is also implicitely proved in 9.7 of Rudin (1980).

4.1.2. Theorem. Let v be a positive measure in M(S). Then

sup {f cP dv : cP E CMo} = inf{Il(S): Il E PM(S), Il ~ v}. (12)

In particular, the existence of a measure Il E PM(S) such that Il ~ v is equivalent to the inequality

sup{f CPdV:CPECMo}< +00.

It is easily seen that the infimum on the right side of (12) is attained if both sides of (12) are finite. For n ~ 2, the measure for which this infimum is attained may not be unique. For example, if

( . ( 1 + Zl 1 - Zl)) V = mm Re--, Re-- a, 1-Z1 1+Z1

then

sup{f cP dv: cP E CMo} = 1,

( 1+Z1 1-Z1) . and for all rx E [0, 1] the measure Ila = rx· Re -- + (l - rx) Re -1 -- . a IS 1 - Zl + Zl

a P-measure, Ila ~ v, and lla(S) = 1 (n ~ 2). Theorem 4.1.2 is an easy corollary of (2) in 9.7.4 of Rudin (1980). We remark also that from Theorem 4.1.2 and from the Minimax Theorem

(see, e.g., 9.4.2 in Rudin (1980)), we deduce the following.

4.1.3. Theorem. Let K be a compact subset of the sphere S. Let W denote the set of all probability measures in PM(S). Then

sup {1l(K) : Il E W} = inf{-. 1_: cP E CMo}. mmKCP

The extreme points of the compact set Ware studied in the works of Forelli (1977, 1979) (see also 19.2 in Rudin (1980)).

4.2. P-Measures and Boundary Values of Holomorphic Functions. Throughout the present section q> will denote a positive lower semicontinuous function defined on the ball B (B = BuS).

4.2.1. Theorem (Aleksandrov (1982,1984)). If q>ls E L l(S) then there is a positive singular measure j1 E M(S) such that

j1(S) = Is q> da, q>a - j1 E PM(S)

and the Poisson integrazt 2 of the measure q>a - j1 is dominated by q> everywhere in the ball B.

4.2.2. Corollary. There exists a positive singular P-measure on the sphere S.

4.2.3. Corollary (Aleksandrov (1982, 1984), LOW (1982, 1984)). If log als E

L 1 (S) then there exists a function fEN *(B) such that If I ::; q> everywhere in Band If I = q> almost everywhere on S.

4.2.4. Corollary (Aleksandrov (1982), Low (1982)). There exists a nonconstant inner function f : B -+ C.

We recall that a function f is an inner function if

f E HOO(B) and If I = 1 almost everywhere on S.

The analog of Theorem 4.2.1 for the polydisc was proved by Rudin (1969).

4.2.5. Theorem (Aleksandrov (1982, 1984)). If log q> Is EL l (S), then the closure of the set {f E N*(B): If I ::; q> in Band If I = q> almost everyhwere on S} in the topology of uniform convergence on compact subsets of the ball B contains the set {f E A(B) : If I ::; q> everywhere on B} .

. This theorem can be seen as a multidimensional variant of Schur's Theorem on approximation by inner functions.

Let us denote by A(B) the set of all functions f E HOO(B) such that the limit lim f( 0 exists for almost all , E S. ~ .... , ~EB

4.2.6. Theorem (Aleksandrov (1982, 1984)). Let f, g E A(B). Suppose there exists a function h E A(B) different from zero such that If I + Ihl ::; q> everywhere in B and almost everywhere on S. Then if log q>ls E U(S) there exists a function FE N*(B) such that IFI < q> everywhere in B, IFI = q> almost everywhere on S, and (F - f)g-l E N*(B).

12Since cpa - J.I. E PM(S), it makes no difference which Poisson kernel we consider (invariant or "harmonic").

This theorem can be viewed as a multidimensional variant of the PickNevanlinna Interpolation Theorem. In the one-dimensional case stronger assertions hold (see Garnett (1981)). For example, in Theorem 4.2.6, instead of the space A(B), one can consider the space HCO(B). In the multidimensional situation, such is not the case, even for qJ = 1 and f = O. We may construct the appropriate counterexample using Theorem 4.3.1 below; see also Rudin (1983).

4.2.7. Theorem (Aleksandrov (1983, 1984)). Let e > O. Then there is a function f E A(B) such that If I :s; qJ everywhere in Band u{' E S : If(OI =f. qJ(O} < e.

4.2.8. Corollary (Aleksandrov (1983)). There exists a non-constant function f E A(B) such that If I :s; 1 everywhere in Band u{ If I = I} > O.

4.2.9. Corollary (Aleksandrov (1983)). There exists a function f E A(B) such 1

that f(z) =f. 0 for all z E Band 7 ¢ N*(B).

It is unknown to the author whether this can occur if:7 E U (S) (n ~ 2).

4.2.10. Theorem (Hakim-Sibony (1982b) and Hakim (1982/83)). Let e > 0 and qJ E C(B). Then there exists a compact set K c S and a function f holomorphic in some neighborhood of the set B\K such that u(K) = 0, If I :s; qJ everywhere in B, and If I ~ qJ - e everywhere on S\K.

Further results 13 in this direction (in particular, assertions 4.2.1 through 4.2.10 for strictly pseudoconvex classical domains and several others) can be found in Aleksandrov (1982, 1984), Low (1982, 1984), Hakim-Sibony (1982b, 1983), Rudin (1983), Tomaszewski (1984), and Hakim (1982/83).

4.3. LSC-Property14. Let rp be a real continuous (not necessarily linear) functional on the space of all functions holomorphic in some neighborhood of the disc [D. To each function f holomorphic in the ball B, we associate the function

rpf: S --+ IR u { +oo}, rpf(') ~ sup rp(f.5), O<r<l

where f,.{(z) = f(r(z) (z E [D). It is easy to see that the function rpf is lower semicontinuous.

This simple assertion has a whole series of interesting consequences (see Rudin (1983)). As rp, it is particularly useful to take functionals which enjoy the following monotonicity property

rp(f,.) :s; rp(f) (0 < r < 1),

13 See also the references to applications in the footnotes to Sect. 3.6. 14This notion was introduced by Rudin (1983): LSC = Lower Semicontinuous.

for example,

LIOg(I+lfl)dm, IIfllHP (0 < p::s; +00), LIReflPdm (p~I). Thus, if we take as rp the first of these functionals and invoke Corollary 4.2.3,

then we obtain the following assertion.

4.3.1. Theorem. Let IjJ be a measurable lr -invariant function on the sphere S. Then the following assertions are equivalent:

1) there exists a function fEN *(B) distinct from zero such that If I = IjJ almost everywhere on S;

2) there exists a homogeneous polynomial FE C[Zl' Z2' ... , zn] distinct from zero such that IFI = IjJ almost everywhere on S, log IjJ E U(S), and the function IjJ agrees almost everywhere on S with a lower semicontinuous function.

This theorem shows that it is not possible to drop the condition of lower semicontiniuity in Theorems 4.2.1, 4.2.5, and 4.2.6. For other results on the necessity of lower semi continuity in the multidimensional case see HakimSibony (1983).

Various modifications of the LSC-property are possible (for example, by considering analytic discs parallel to the vector en, we may analogously consider a functional rpf: Bn - 1 - (-00, +00]; moreover, defining rpf somewhat differently, one can guarantee that this function belongs to the first Baire class (see Jewell (1980)).

Using the LSC-property, Rudin (1983) showed several results on the impossibility of approximation in the L CO-norm. See also Jewell (1980) where one of the above described modifactions of the LSC-property is exploited.

4.4. Outer Functions. Let f be a non-zero function in HCO(B). Consider the following for the function f:

a) for each function g E HCO(B), from the inequality Igl ::s; If I almost everywhere on S, it follows that Igl ::s; If I everywhere in B;

b) log I f(O) I = flOg If I du; 1 s

c) 7 E N*(B);

d) f· A(B) is dense in N*(B); e) f· A(B) is dense in H 2 (B). In the one-dimensional case it is not hard to verify that e) => d) <==> c) <==>

b) => a). The example f(z) = exp Zl + 1 (see 4.4.8 in Rudin (1969)) shows that Zl - 1

property d) does not imply property e) (n ~ 2). The function f(z) = Zl shows that b) does not follow from a).

It is not difficult to prove that any function in class A(B) satisfies property a). We remark also that if II f II H'(B) = 1 and f satisfies property a), then f is an extreme point of the unit ball in the space H1(B).

Let us denote by (J(f) the spectrum of a function f E L "'(S), i.e.

(J(f) = pEe: (A - f)-I ~ L"'(S)}.

From the Hartogs Theorem on the removability of compact singularities, it follows that

f(B) = (J(f), (13)

for all f E A(B) (n 2 2). From the existence of non-constant inner functions in the ball, it follow that (13) does not in general hold for functions f E H"'(B). If property a) holds for all functions of the form f + c (where CEq, then (13) clearly holds. Tamm (1982) proved (13) for all functions f E H"'(B) such that

II f - f..IIHP(B) = 0((1 - r)I/2P)(r ~ 1-) (p > 0).

In Tamm (1982) the sharpness of this result is asserted (at least for p = 2). Improving on a result of Sadullaev (1976b), Rudin (1983) showed that non

constant inner functions have extremely pathological boundary behavior.

§ 5. Peak Sets for Smooth Functions

5.1. Peak Sets and Local Peak Sets. A set K c S is called a peak set for Am(B) if there exists a function f E Am(B) such that flk == 1 and If I < 1 everywhere on B\K. A set K c S is called a local peak set for Am(B) if each point of K has a closed neighborhood V such that V n K is a peak set for Am(B).

It is easy to see that the real sphere S Rand S"U (see Sect. 2) are peak sets for A"'(B); as functions f E A"'(B) we may take

~ Ctl zJ + 1) and ~ (nn/2 Jj Zj + 1).

5.1.1. Theorem. Let K be a compact subset of the sphere S. The following assertions are equivalent:

1) K is a peak set for A"'(B); 2) K is a local peak set for A"'(B); 3) each point of K has a neighborhood V such that K n V is contained in some

totally-real C"'-manifold of dimension n - 1 and having the following property:

Tp(M) c TpC(S)

forallpEKn V; 4) each point of K has a neighborhood V such that K n V is contained in an

integral C'" -manifold.

The implicaton 1) = 2) is trivial. Hakim and Sibony (1978) proved the implication 4) = 2) = 3). Chaumat and Chollet (1979) proved the implication 3) = 4). Fornaess and Henriksen (1982) proved the implication 2) = 1).

It is easy to see that, for n = 2, the existence of an integral C<Xl-manifold M such that K c M follows from 4). Fornaess and Henriksen (1982) showed the analogous assertion for n = 3. Chaumat and Chollet (1982) showed that for n ;;::: 4 this is generally not the case, if instead of the ball one considers a strictly pseudoconvex domain with smooth boundary.

5.2. Peak Sets and Interpolation. A compact subset K of the sphere S is called an interpolation set (for A<Xl(B)) of order r (r E N u {oo}), if for each function f E C<Xl(Cn ) such that Dilaf = 0 everywhere on K for each multi-index 0: E

z~n, with 10:1 < r, there exists a function FE A<Xl(B) for which DIlF = Dilf everywhere on K, for all multi-indeces 0: E z~n with 10:1 :s; r. For n ;;::: 2, the sphere S is an interpolation set of order r for all r E N u {oo}.

5.2.1. Theorem (Chaumat-Chollet (1980)). Let K be a peak set for A<Xl(B). Then K is an interpolation set of order r for all r E N u { 00 }.

Let rbe a C<Xl-smooth curve on the sphere S. Suppose that r;,(T) ¢ TpC(S), for all PEr. In the paper of Chaumat-Chollet (1983) the following two problems are solved:

1. Which compact subsets of the curve r coincide with the set f- 1 (0) for some function f E A<Xl(B)?

2. Which compact subsets of the curve r are interpolation sets of infinite order for A <Xl (B)?

The answers to both of these questions (see Chaumat-Chollet (1983)) are respectively analogous to the one-dimensional results (see Garnett (1981)).

5.3. Finitely Generated Ideals in the Algebra A<Xl(B). Let M be a closed C<Xl_ submanifold of some neighborhood of the ball B;

B n M = {z E B: gl(Z) = g2(Z) = ... = gk(Z) = O},

where gj E A<Xl(B). When does the ideal in A<Xl(B) generated (in the algebraic sense) by the functions gl' g2'" gk' coincide with the ideal {f E AC()(B) : fllinM == O}? By way of an answer to this question we refer the reader to BartolomeisTomassini (1981) and Amar (1983a).

Update on Problems from Rudin's Book (1980) Solved up to the Present Time

1°. 7.1.7 and 19.1.9Y. Let n ;;::: 2. Does there exist a constant C = C(n) such that I Igl d(J :s; C I IRe gl d(J

for all functions g E A(B) whose value at the origin is real?

15 Here and in the sequel a.b.c. denotes the corresponding section in Rudin (1980).

A negative answer to this question follows easily from the existence of a nonconstant inner function on the ball (see Chap. 5, 4). Indeed, if I is an inner function and 1(0) = 0, then

i I 1 + 1'1 i 1 + I, Re -1 _ drr = Re -----=- drr = 1 s I, s 1 I,

for all r E (0,1). However, i 11 + I rf. B1(B) since any function in B1(B) which is -I

real almost everywhere on S is constant. Consequently,

lim r I~I drr = +00. ,~1- Js 1-1,

We remark that a negative answer to this question can be obtained without using inner functions (see A.B. Aleksandrov, BP Hardy classes for p < 1 and semi-inner functions in the ball. Dokl. Akad. Nauk SSSR 262, No 5, 1033-1036 (1982)).

2°. 7.3.6 and 19.3.2. Does there exist, for n 2 3, a function IE A(B) not identically zero and such that the set 1-1(0) has infinite (2n - 2)-dimensional Hausdorff measure?

A corresponding example for n = 2 is constructed in 7.3.6 of Rudin (1980) (the one-dimensional case is trivial). A positive answer to Problem 2 was obtained independently by Alexander and Amar (see Chap. 4, 2.3). Further results in this direction were obtained by Hakim and Sibony (Chap. 4, Theorem 2.3.1) and also by the author (A.B. Aleksandrov, The Blaschke condition and zeros of bounded holomorphic functions, in "Multidimensional Complex Analysis", Inst. Fiz. SO Akad. Nauk SSSR, Krasnoyarsk, 1985,23-26). Amar used the theorem of Varopoulos (Chap. 4, Theorem 4.1.1). The construction of Amar gives a function having a piecewise-linear zero set. Alexander used the Ryll-Wojtaszsczyk polynomials (Chap. 1, 3). The Ryll-Wojtaszsczyk polynomials also allow one (see Ryll-Wojtaszsczyk (1983)) to strengthen and extend, to arbitrary dimensions n, practically all results (see Chap. 7 in Rudin (1980)) based on 7.2.8 in Rudin (1980). We now enumerate several of these:

7.2.9. There exists a function IE B2(B) (for n = 2,3) for which almost no slice function has a Taylor expansion absolutely convergent in the closed unit disc. This result was essentially duplicated by Wojtaszczyk (1982) (see also Chap. 1, 3.6). He showed that with no restrictions on the dimension n, one can also require the following properties:

1) IE A(B); ~. . ~

2) ~ 1.t;;(kW = +00 for each p < 2 and for almost every' E S; where .t;;(k) = k;,:O

.t;;<k)(O) -------rr- .

7.2.10. There exists a function I E Yt';,72 (B) (for n = 2, 3) such that

sup I/(rOI = + 00 O<r<l

for almost all , E S.

173

(1)

This assertion also can be strengthened for all n. In order to convince oneself of this, we consider the function

1= L _1 12k ,

k~l Jk where Ip (p > 0) are Ryll-W ojtaszsczyk polynomials (see Chap. 1, Theorem 3.6.1). It is not difficult to verify that the functon I satisfies (1) and I E yt'/(B) for all a, p E (0, +(0). This same function allows one also to strengthen the results of7.2.11 in Rudin (1980).

3°. 10.5.7. In this section the conjecture was made that the topological dimension of any I-set is no greater than n - 1. This conjecture was proved by Stout (see Chap. 5, 2).

4°. 10.7.4. For A <Xl (B), is every local Peak set global? A positive answer to this question was obtained by Fornaess and Henriksen (see Chap. 5, 5).

5°. 11.4.1 and 19.3.15. Let I be a non-constant function in A(B), n ~ 2. Is it true that

(J {, E S: 1/(01 = max I/(Z)I} = O? ZES

A negative answer to this question was obtained by the author (see Chap. 5,4).

6°. 13.4.6 and 19.3.7. Let X be a closed subalgebra of the algebra C(B) of all continuous functions on the ball B endowed with the topology of uniform convergence on compact subsets of the ball B. Suppose that X is ./It -invariant, i.e. 1 0 cP E X for all I E X and all cP E Aut(B). Is it then true that X coincides with one of the following five algebras

{O}, C. H(B), H(B), C(B)?

This conjecture was proved by Rudin (W. Rudin, Moebius-invariant algebras in balls. Ann. Inst. Fourier, 33, No.2, 19-41 (1983)). Rudin's result is new even in the one-dimensional case.

7°. 19.1.1. Do there exist non-constant inner functions in the ball B (n ~ 2)? A positive answer to this question was obtained independently by the author and by Low (see Chap. 5, 4). The example of a non-constant inner function refutes essentially all conjectures in 19.1 of Rudin (1980) and also gives an example of a non-surjective isometric operator on the space HP(B) (19.3.8 in Rudin (1980)) and an example of a nontrivial inner mapping (19.3.9 in Rudin (1980)).


8°. 19.3.1. Let n ~ 2. Does there exist a function in f E HI(B) which cannot be written as the product of two functions in H2(B)? Gowda obtained a positive answer to this question. He showed that for n ~ 2 the set

{gh: g, h E H2(B)}

is a set of first category in HI (B) (see Chap. 3, 2).

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IV. Complex Analysis in the Future Tube

A.G. Sergeev, V.S. Vladimirov


Contents

Introduction ................................................... 182

Chapter 1. Geometry of the Future Tube .......................... 184

§ 1. The Future Tube ........................................... 184 1.1. Definition, Description of the Boundary ................... 184 1.2. Tangent Bundle, Levi Form ............................. 185 1.3. Group Structure, Automorphisms ........................ 186

§2. The Future Tube as a Classical Domain ....................... 187 2.1. A Realization of the Future Tube as the Generalized

Unit Disc ............................................. 187 2.2. Geometry of the Generalized Unit Disc ................... 188 2.3. A Realization of the Future Tube as the Lie Ball ............ 190

§ 3. Penrose Representation and Some Physical Applications ........ 192 3.1. Penrose Representation and Twistor Transform ............ 192 3.2. Conformal Compactification of the Minkowski Space ....... 194

§4. Holomorphic Non-straightening ............................. 196 4.1. Holomorphic Non-straightening ......................... 196 4.2. Approximation by Strictly Pseudoconvex Polyhedra ........ 198

§ 5. Generalizations ............................................ 199 5.1. Tube Cones ........................................... 199 5.2. Tuboids .............................................. 200

Bibliographical Notes ........................................... 201

Chapter 2. Boundary Properties of Holomorphic Functions .......... 202

§ 1. Boundary Values in U and Yfs ............................... 202 1.1. The Spaces HP(TC ) .•••••••••••••••••••••••••••••••••••• 202 1.2. The Spaces H(S)( C) ...................................... 203

180 A.G. Sergeev, V.S. Vladimirov

§ 2. Boundary Values in Spaces of Distributions and Hyperfunctions 203 2.1. The Space H(C) ......................................... 203 2.2. Boundary Values in the Sense of Hyperfunctions ............ 204 2.3. Distributional and Hyperfunctional Boundary Values in

Tuboids ............................................... 205 § 3. Boundary Values of Bounded Holomorphic Functions ........... 206

3.1. Auxiliary Results ........................................ 206 3.2. Fatou and Lindelof Theorems ............................ 207 3.3. Uniqueness Theorems ................................... 209

§4. Inner Functions and Holomorphic Mappings ................... 210 4.1. Rational Inner Functions .................... . . . . . . . . . . .. 210 4.2. General Inner Functions ................................. 211 4.3. Holomorphic Mappings ...... . . . . . . . . . . . . . . . . . . . . . . . . . .. 211

§ 5. Interpolation Sets ........................................... 212 5.1. Properties ofInterpolation Sets ........................... 212 5.2. Interpolation Manifolds ................................. 213


Chapter 3. "Edge-of-the-Wedge" Theorem and Related Problems ..... 215

§ 1. "Edge-of-the-Wedge" Theorem ............................... 215 1.1. Theorem of Bogolubov .................................. 215 1.2. Theorem of Martineau ................................... 216

§ 2. "C-convex Hull" Theorem .................................... 217 2.1. "C-convex Hull" Theorem ................................ 217 2.2. Holomorphic Hulls and Dyson Domains ................... 218

§ 3. Analytic Representations ..................................... 219 3.1. Decomposition of Hyperfunctions in Tuboids. Extensions of

the "Edge-of-the-Wedge" Theorem ........................ 219 3.2. Generalized Fourier and Radon Transforms ................ 222 3.3. Factorization of Hyperfunctions .......................... 223


Chapter 4. Integral Representations 224

§ 1. Cauchy-Bochner Integral Representation ....................... 224 1.1. Cauchy-Bochner Integral in Tube Cones ................... 224 1.2. Cauchy-Bochner Integral for Classical Domains ............. 226 1.3. Hilbert Transform .................. . . . . . . . . . . . . . . . . . . .. 226 1.4. Estimates of the Cauchy-Bochner Integral .................. 227 1.5. Schwartz Representation ................................. 228

§ 2. Poisson Integral Representation ....... . . . . . . . . . . . . . . . . . . . . . .. 229 2.1. Poisson Integral in Tube Cones ........................... 229 2.2. Poisson Integral in Classical Domains ..................... 230 2.3. Boundary Properties of the Poisson Integral ................ 231

IV. Complex Analysis in the Future Tube 181

2.4. Pluriharmonic Functions ................................ 233 2.5. Functions given by Poisson Integrals ...................... 234

§ 3. Other Integral Representations ................ . . . . . . . . . . . . . .. 235 3.1. Bergman Representation ................................. 235 3.2. Cauchy-Fantappie Type Representations ................... 236 3.3. The lost-Lehmann-Dyson Representation .................. 238 3.4. Representations for Solutions of the a-equation ............. 240

§4. Functions with Nonnegative Imaginary Part .................... 241 4.1. Properties of Functions with Nonnegative Imaginary Part in

Tube Cones ............................................ 241 4.2. Integral Representation .................................. 242 4.3. Tauberian Theorems .................................... 244 4.4. Linear Passive Systems .................................. 245


References ..................................................... 246

Introduction

The future tube in en+! is the unbounded domain

r+ = {z = (zo, ... , zn) E en+! : (1m ZO)2 > (1m ZI)2 + ... + (1m zn)2, 1m Zo > O}.

In other words, r+ is a tube domain over the future cone V+ = {y E IRIn+! : Y6 > yf + ... + y~, Yo > O}. The domain r+ is biholomorphically equivalent to a classical Cartan domain of the IVth type, hence to a bounded symmetric domain in en+ l . The future tube r+ in e4 (n = 3) is important in mathematical physics, especially in axiomatic quantum field theory, being the natural domain of definition of holomorphic relativistic fields. These specific features of the future tube motivated its investigation by mathematicians and physicists. Beginning with Elie Cartan's classification of bounded symmetric domains, these domains were examined in many papers where the complex structure of their boundaries, integral representations, boundary values of hoi om orphic functions and so on were considered. The proof of the "edge-of-the-wedge" theorem by N.N. Bogolubov generated the rapid development of applications of the theory of several complex variables to axiomatic quantum field theory. During this period the "C-convex hull" and "finite covariance" theorems were proved, the lost-lehmann-Dyson representation was found et cetera. Recently R. Penrose has proposed a transformation connecting holomorphic solutions of the basic equations of field theory with analytic sheaf cohomologies of domains in e[p>3.

These two directions developed, to a large extent, independently from each other, and some important results obtained in axiomatic quantum field theory still remain unknown to specialists in several complex variables and differential geometry. One of the goals of this paper is to give a unified presentation of advances in complex analysis in the future tube and related domains achieved in both of these directions.

Among the main classes of bounded domains of holomorphy, the following classes are usually considered: strictly pseudoconvex domains, smooth weakly pseudoconvex domains, and analytic polyhedra. As basic examples of these classes we can consider respectively the unit ball {z E en: I IZjl2 < 1}, a domain {z E en: I Izjl2Pj < 1, Pj ~ 1 and not all Pj equal to 1}, and the unit poly disc {z E en: IZjl < 1}. These domains are distinguished by the complex geometry of their boundaries: the Levi form of the ball is non-degenerate (the complex tangent hyperplane has 2nd order contact with the boundary at each of its points), the boundary of the second domain has only finite type points (the complex tangent hyperplane has finite order of contact with the boundary at each of its points), the Levi form of the polydisc is identically zero on the smooth part of the boundary (the complex tangent hyperplane "sticks" to the boundary). All these domains belong to the general class of pseudo-convex polyhedra. The Levi form of the future tube r+ degenerates at all points of the smooth part of the boundary because any point of or+ is contained in a complex halfplane (complex light ray) lying entirely on the boundary. So from this point of view, the


future tube is similar to analytic polyhedra. However for n ~ 2 there is a principal difference - the boundary of r+ cannot be "straightened" along complex light rays by a biholomorphic transformation. In other words, for n ~ 2, r+ is not (even locally) a pseudo convex polyhedron.

There are two ways of generalizing the future tube: one way is to consider bounded symmetric domains in en, the other is to consider arbitrary tube cones and tuboids. Some results of this survey are true for these generalizations and are formulated in their maximal generality though we are mainly interested in the future tube. We have included in our list ofreferences related papers published after 1970 and several earlier papers (the reader can find references before 1970 in the books Vladimirov (1964,1979), Rudin (1969), Stein-Weiss (1971), and the articles Vladimirov (1969c), 1983a, 1971, 1982), Zharinov (1983), Koranyi (1972), Morimoto (1973), Stein (1971). This list is, of course, not complete and, to some extent, reflects the authors' interests. We provide all chapters with bibliographical notes where we have collected some easy-to-find books and papers, review articles and also some additional references.

Some notations: the Euclidean (complex linear) inner product of vectors z, ( E en and Hermitian norm of Z are denoted respectively by (z, 0 = Z 1 (1 + ... + zn(" and IzI2 = Iz 112 + ... + IZnI2, the Lorentzian inner product of z, ( E en+1

and Lorentz norm of z are denoted by z·( = zo·(o - Zl(l - ... - zn(n and Z2 = z6 - zi - ... - z;. A proper convex cone in IRn is denoted by C, V+ is the future cone in IRn+1, sn-l is the unit sphere in IRn. For an open Q in IRn, we call a complex neighborhood of Q any domain of holomorphy Q in en such that Q (\ IRn = Q. The space of distributions in Q is denoted by ~'(Q), the space of tempered distributions in IRn by [/', the space of bounded functions with compact supports in IRn by L;"(lRn). The space of holomorphic functions in a domain D in en is denoted by (!)(D), the sheaf of holomorphic functions in en by (!).

The authors are grateful to E.M. Chirka, Ju. N. Drozhzhinov, G.M. Khenkin, B.I. Zavialov and V.V. Zharinov for their remarks which helped the authors to improve the original text of the paper.


Chapter 1 Geometry of the Future Tube

§ 1. The Future Tube

1.1. Definition, Description of the Boundary. The future tube .+ is a domain in C4 of the form

.+ = {z E C4 : y2 = y~ - yi - y~ - y~ > 0, Yo> a},

where z = (zo, Zl' Z2' Z3), Zj = Xj + iYj. Using the Lorentz inner product this definition can be rewritten in the form:.+ = {z: y2 > 0, Yo> a}. So, ,+ is a tube domain over the future cone

v+ = {y E jR4: y2 > 0, Yo> a}.

The section ,+ n {z : Re Z = x} of ,+ for arbitrary fixed x E [R4 coincides with the cone V+, the section ,+ n {z: 1m Z = y} of ,+ for arbitrary fixed y E V+ coincides with the whole of [R4.

The boundary 0,+ of.+ consists of the smooth part

S = {' = ~ + il] E C4 : 1]2 = 0, 1]0 > o}

and the distinguished boundary

M = g E C4 : I] = o} = [R4.

M is the set where the boundary 0'+ degenerates. Through any point' E S there passes a generator I, of the cone

r+ = oV+ = {I] E [R4: 1]2 = 0, 1]0 ~ a},

called a (real) light ray. The complexification A, of the ray I, which coincides with a complex halfplane A, = {~ + (XI] : (X E C, 1m (X > o} is called a complex light ray (Fig. 1). The complex light ray A, goes through the point' and lies entirely on S; the section A, n {z: Re z = x} for any fixed x E~, n [R4 coincides

URI,

11

Fig. 1

with the generator l~ moved to the point x. At any point ~ of the distinguished boundary we have a collection of real and complex light rays parametrized by points of the 2-dimensional sphere

g + i'1 : '1~ = '1I + '1~ + '1~ = I}.

The future tube r+ = r+(n) in cn+1, n ~ 0, is defined by analogy with the future tube in C4 as a tube domain over the future cone

r+(n) = [Rn+1 + iV+ = {z E cn+1 : y2 = y~ - YI - ... - y; > 0, Yo> OJ,

V+ = V+(n) = {y E [Rn+1 : y2 > 0, Yo > OJ.

Its boundary has the same structure as for n = 3, namely, through any point of the smooth part of the boundary or+ there passes a complex light ray lying entirely on the boundary and in any point of the distinguished boundary we have a collection of complex light rays parametrized by points of the (n - 1)dimensional sphere. We keep for r+(n) the notations introduced above for n = 3 (henceforth we omit usually the index "n" in the notation r+(n)).

For n = ° the domain r+(O) coincides with the upper halfplane (z E C : 1m z > OJ; for n = 1 the domain r+(1) can be transformed by a linear transformation onto the domain {z E C2 : Yo> 0, Y1 > O} which is the unbounded realization of the bidisc. These cases are degenerate in the sense that for n ~ 2 the future tube r+(n) is not equivalent biholomorphically to the polydisc. We suppose in the sequel that n ~ 2.

1.2. Tangent Bundle, Levi Form. Denote by r the real function

r(z) = - y2, Z E cn+1.

This is a local defining function of r+ in the sense that

r+ = {z: r(z) < O} n {Yo> OJ.

Differentials of this function having the form

n or(z) n or(z) dr(z) = L: - dZj + L: -_- dZj = -2y-dy

j=O OZj j=O OZj

= -2(yo dyo - Y1 dY1 - ... - Yn dYn),

n or(z) or(z) = L: -",- dZj = iy- dz = i(yo dzo - Y1 dZ 1 - ... - Yn dzn),

j=O vZj

are non-degenerate at all points Z E r+\M. Let us consider in more detail the structure of the boundary or+ at points

( E S. Denote by Tc,S the tangent space of S at ( and by Tc,c S the complex tangent space of S at (. The latter space is defined as the linear space of tangent vectors Z = L:Zj OZj E Tc,t:n+l satisfying the condition

(or(o, Z) = i('1oZo - '11Z1 _ ... - '1nzn) = 0, (= ~ + i'1 E S.


Consider the following vectors at a point' E S

n . a Zk = L Xk~' k = 1, ... , n - 1,

j=1 UZj

where the vectors X k = (Xl, ... , X~), k = 1, ... , n - 1, form an orthonormal n

base in the hyperplane L I'/jXj = 0. The vectors Zo, ZI"'" Zn-l generate the j=1

space 7;;c S and the vectors

Xo = 2 Re Zo, ... , X n- 1 = 2 Re Zn-l' T = 2 Re Zn,

Yo = -21m Zo, ... , ¥,,-1 = -21m Zn-l

form an orthonormal base of the space 7;;S. The vector Zo points in the direction of the complex light ray As'

The Levi form 2, at a point' is defined by

- - - 1 - - -Y'r(Z, W) = oor(O(Z, W) = 2( - ZO W O + ZI Wi + ... + zn W")

on vectors n

Z = L Zjo/OZj' j=O

n

W = L Wj%zj . j=O

The matrix of the restriction of the Levi form 2,(Z, Z) to the complex tangent space 7;;c S in the base {Zo, ZI' ... , Zn-l} is diagonal and has the form diag (0, 1/2, ... , 1/2), i.e. the restriction of 2, to 7;;c S has one zero eigenvalue and a positive eigenvalue of multiplicity n - 1.

1.3. Group Structure, Automorphisms. The Lorentz group L consists of all linear transformations of [Rn+1 preserving the quadratic form y2 = Y6 -yi - ... - Y;, Y E [Rn+l, and fixing the origin. Denote by Lt the subgroup of L consisting of transformations preserving the cone V+ (i.e. preserving the orientation of "time" Yo). Linear automorphisms of the future tube r+ are given by transformations of the form Z -> Az + b where A is a linear transformation of [Rn+l preserving the cone V+ and fixing the origin (in other words, A is a composition of transformation of Lt and dilatations), and b is an arbitrary vector of [Rn+l. Transformations of this type exhaust all analytic automorphisms of r+

continuous in the closure i+. Conformal transformations of the space M with the metric y2 are generated by Poincare transformations x -> Ax + b where A E L t, b E [R1I+1, dilatations, and inversions (inversion with respect to the origin is given by x -> X/X2). An arbitrary analytic automorphism of the future tube r+ is a composition of transformations of this type (cf. Vladimirov (1964) and also Sect. 2.2, 2.3 and Chap. 2, Sect. 4.3).

§ 2. The Future Tube as a Classical Domain

2.1. A Realization of the Future Tube as the Generalized Unit Disc. We construct here a biholomorphic map of the future tube r+ = r+ (3) onto a bounded homogeneous domain - the generalized unit disc. This map is a composition of two mappings. The first of them is a realization of r+ as the generalized upper halfplane. It is given by the formula

(1)

where eTa is the unit 2 x 2-matrix, eT; for i = 1, 2, 3 are the Pauli matrices

(0 1) (0 -i) (1 0) eT l = 1 0' eT2 = i ° ' eT3 = ° -1 .

The mapping (1) biholomorphically maps the future tube r+ onto the generalized upper ha({plane H consisting of complex 2 x 2-matrices Z with positive

definite imaginary part 1m Z = ~(Z - z*). The mapping inverse to (1) is given by

the formula

( 1 1 1 1 ) z -> z = .2 Tr Z, .2 Tr(zeTd, .2 Tr(zeT2), .2 Tr(zeT3) .

The mapping (1) has the following properties

det Z = Z2, det(lm Z) = y2.

Its extension to the distinguished boundary M maps M bijectively onto the space of Hermitian 2 x 2-matrices.

The second mapping is a realization of the generalized upper halfplane as the generalized unit disc. It is given by the Cayley transform

(2)

mapping the generalized upper halfplane H biholomorphically onto the generalized unit disc

B = {Z E e[2 x 2J : ZZ* < I}.

In other words, B consists of complex 2 x 2-matrices Z such that the matrix / - Z* Z is positive definite. The inverse Cayley transform has the form

Z -> Z = i(l - Z)(/ + Z)-I.

The composed mapping z -> Z -> Z maps the future tube r+ biholomorphically onto the generalized unit disc B and is given by the formula

(3)

where LI(z) = det(l- if) = 1 - Z2 - 2izo = -(z + i)2, i = (i, 0, 0, 0). We have

16y2 16y2 det(I ZZ*) - - ,..,---------:c..,...,..,.

- -ILI(zW -I(z + i)21 2'

The extension of the mapping (3) to the distinguished boundary M maps M injectively into the distinguished boundary U = {Z: ZZ* = I} of the generalized unit disc B which coincides with the group U(2) of unitary 2 x 2-matrices. The image of the mapping (3) coincides with the set U\ Uo where

Uo = {X E U: det(l + X) = O}.

2.2. Geometry of the Generalized Unit Disc. The generalized unit disc B is a convex domain with the boundary given by

8B = {Z E e[2 x 2]: det(l- Z*Z) = 0, ZZ* :::; I}.

Note that the set {det(I - Z*Z) = O} has two parts - the bounded part consists of Z subject to the condition ZZ* :::; I, and the unbounded one given by the inequality Z*Z ~ I. These parts intersect in the distinguished boundary U.

In terms of the polar representation of matrices Z E B

Z=XA,

where X E U(2), A is a Hermitian operator (A = A*) with 0:::; A :::; I. We can rewrite the boundary 8B in the form

aB = {Z = XA: det(l- A) = 0,0:::; A:::; I}.

Let us consider the structure of the boundary at points z E aB\ U. After diagonalization of the matrix A we represent the matrix Z in the form

Z = XV(l 0) V* o it ' (4)

where V E U(2), 0 :::; it < 1. The matrix V in this representation is defined up to multiplication from the right by a diagonal unitary matrix. The matrix X parametrizes points ofthe distinguished boundary U and the set U(2)jdiag U(2) which parametrizes classes of matrices V is a 2-dimensional sphere S2. At any point Z E aB\ U given by (4) we have a complex disc consisting of points

(1 0) * XV 0 a V, lal < 1, a E e,

lying entirely on aB\ U. This disc is an analogue of a complex light ray in the future tube. Denote by p the real function

p(Z) = p(Z, Z*) = -det(l - Z*Z), Z E e[2 x 2].

This is a local defining function of B at points of aB\ U which means that any point Zo E aB\ U has a neighborhood Q such that B (\ Q = {Z: p(Z) < O} and

dp(Z) =1= 0 for Z E Q. The last inequality follows from the explicit formula for the differentials of p:

ap(Z) = -a(det Z) det Z* + Tr(dZ' Z*),

dp(Z) = - 8(det Z*) det Z - a(det Z) det Z* + Tr(dZ' Z*) + Tr(Z' dZ*).

These expressions are derived using the following identity for p

p(Z) = - 1 - det(ZZ*) + Tr(ZZ*).

In particular, for Z = Zo = (~ ~) we obtain that

a(det Z) = A. dZll + dZ22 , 8(det Z*) = A. dZll + dz22 ,

dp(Z) = (1 - A.2)(dzll + dzll ) =1= 0 for A. < 1.

It follows using the homogeneity of aB\ U that dp(Z) =1= 0 for any Z E aB\ U. The Levi form of p in points Z E aB\ U is computed as follows

ff'p = a8p(Z) = - a(det Z) 1\ 8(det Z*) + Tr(dZ 1\ dZ*).

So, in particular, at the points Zo = (~ ~). 0 S A. < 1 it is equal to

ff'p = (1 - ,1.2) dZ ll 1\ dZll + dZ 12 1\ dZ12 + dZ 21 1\ dZ21

- A.(dz ll 1\ dZ22 + dZ22 1\ dzll ).

The complex tangent space at a point Z E aB\ U is given by the equation

G~(Z)' w - Z) = 0 where (A, C) = Tr(AC') is a complex linear inner product

in the space of matrices. At the point Zo the complex tangent space is given by W 11 = 1. The restriction of the Levi form to this space has the form

- - 2 2 ff'p(W - Zo, W - Zo) = Iwul + IW211 ,

so it has one positive eigenvalue 1 of multiplicity 2, and one zero eigenvalue. Because of the homogeneity, the same assertion is true at any point of aB\ U.

Analytic automorphisms of the generalized unit disc are given by the mappings (cf. Siegel (1949), Hua (1958), Piatetski-Shapiro (1961»:

Z -+ (AZ + B)(CZ + Dfl, Z -+ rz,

where the block 4 x 4-matrix M = (~ ~) belongs to the unitary group

U(2, 2), i.e.

M*JM = J where J = (~I ~). The generalized unit disc Bm is defined analogously as

Bm = {Z E C[rn x rn]: ZZ* < I}.

For any m, Bm is biholomorphic to a tube domain over a cone in Cm2 (cf. Sect. 5.l), however for m> 2 the domains Bm and ,+(m2 - 1) are not biholomorphically equivalent.

2.3. A Realization of the Future Tube as the Lie Ball. The existence of the biholomorphic equivalence between the future tube ,+(3) and the generalized unit disc (a classical Cart an domain of the 1st type, cf. Sect. 5.1) is, as was noted above, a low-dimensional effect. Analogously, for n = 2 the future tube ,+(2) can be realized as a classical domain of the IIlrd type given by the set of symmetric matrices belonging to the generalized unit disc B2 , i.e. {Z E B2 : tz = Z}, which is also a low-dimensional phenomenon. In this Section we shall construct a bounded realization of the future tube ,+(n) for any n as a classical domain of the IVth type called the Lie ball. The biholomorphic mapping of ,+(n) onto the Lie ball is a composition of two mappings.

The first mapping is a realization of ,+ = ,+(n) as a domain on a complex quadric in CII])n+2. Let us introduce the new variables

Sl Sn Sn+1 Zl=-""'Zn=-' Zo=-·

So So So

In these variables the domain, = {z E e+1 : y2 > o} will transform to the domain

~' = {s E cn+3: -lsol2 - '" -lsnl2 + ISn+112 + 2 Re(sosn+2) > 0,

- s6 - ... - s; + S;+l + 2s0sn+2 = o}

(note that So i= ° for S E ~' so we can divide out so). Changing the variables So, S l' ... , Sn+2 to the variables to = So - Sn+2, t 1 = Sl' ... , tn+2 = Sn+2 we can write ~' in the form

~ = {t E e+3 : -ltol2 - ... -ltnl2 + Itn+112 + Itn+212 > 0,

- t6 - ... - t; + t;+1 + t;+2 = a}.

The domain ~ is a section of the domain flj = {t E e+3 : Itol2 + ... + Itnl2 < Itn+112 + Itn+212} by the complex quadric {t6 + ... + t; = t;+l + t;+2}' The domains ~ and flj are given by homogeneous relations so it's more natural to consider them as domains in ClPn+2. Note that the Levi form of the domain flj being restricted to the complex tangent space of aflj at a point t with tn+2 i= ° has one negative and n + 1 positive eigenvalues. The domain ~ has two compo-

nents distinguished by the sign of 1m tn+1 . The future tube ,+ is biholomorphic tn+2

to the domain ~+ on the quadric in ClPn+2 given in homogeneous coordinates as follows

This representation of r+ as a domain on the quadric in ClPn+2 is closely related to the Penrose representation considered in the following Section.

The second mapping is a realization of.@+ as the Lie ball and is given by the formula

to Wo = . , ... ,

t n+1 + Itn+2

Under this mapping the domain .@+ transforms biholomorphically onto the domain

BL = {w E e+1 : IW5 + ... + w;1 2 + 1 > 21wol2 + ... + 21wn12,

I w5 + ... + w; 12 < 1}

called the classical domain of the IVth type or the Lie ball. (In Hua's (1958) book this domain is called the Lie sphere. We prefer to call it the Lie ball reserving the name "Lie sphere" for the distinguished boundary of Bd.

The composed mapping of r+ onto BL is given by the formula

. 1 + Z2 . 2z 1 . 2zn Wo = 1 (z + i)2' WI = 1 (z + i)2' ... , Wn = 1 (z + i)2

where i = (i, 0, ... ,0). In particular, points of the form z = (iyo, 0, ... ,0) trans-

" . (.1 -Yo ° ) lorm to pomts w = -1--, , ... , ° . 1 + Yo

The distinguished boundary of the future tube transforms into the set SL = {lwol2 + ... + IWnl2 = 1, IW5 + ... + w;1 = 1}. Let us consider this set in more detail. Set w = u + iv, (z, w) = zowo + ZI WI + ... + ZnWn' Then the intersection of SL with the complex sphere ..[1 = {w : (w, w) = I} is given by the equations lul2 = Ivl2 + 1, (u, v) = 0, lul2 + Ivl2 = 1. It follows that v = 0; hence ..[1 intersects SL in the n-dimensional real sphere {u E [Rn+1: lul2 = I}. So the set SL can be written as

SL = {w = e i8 u: lul 2 = 1}.

This set is called the distinguished boundary of BL or the Lie sphere. Consider now the smooth part of the boundary of BL

aBL\SL = {I(w, w)12 + 1 = 21w1 2, I(w, w)1 < I}.

The complex sphere..[;. = {w: (w, w) = A}, 1..1.1 < 1, intersects aBL \SL in the set

{ . 11 + AI 11 - AI 1m A} w = u + IV: lui = -2-' Ivl = -2-'(u, v) = -2-

which coincides with the product of spheres sn X sn-l. This defines a fibration of aBL \SL by (2n - I)-dimensional real submanifolds parametrized by points of the disc {A. E C: 1..1.1 < I}.

The local defining function of BL at points of aBL \SL is given by

PL(W) = 21wl2 -I(w, wW - 1.


Its differentials have the form

apL(W) = 2(w, dw) = _2W2(W, dw),

apL(W) = 2(w, dw) - 2W2(W, dw)

whence the Levi form is computed as follows

2L = aapL(W) = 2 dw /\ dw - 4(w, dw) /\ (w, dw).

The restriction of the Levi form to the complex tangent space T~(aBL \SL) has the following properties: it is positively defined on vectors belonging to T~(aBL n I).) where I). is a complex sphere through the point w, and equals to zero in the transversal direction (defined by the projection of the vector field a/a), on T~(aBd).

Analytic automorphisms of the Lie ball BL are given by the following transformations (cf. Hua (1958))

w ~ {[ (w, w1 + 1, /w, w1- I)A + wC JC) r1

-[ (w, w1 + 1, /w, w1- I)B + wD 1 where A, B, C, D are respectively real 2 x 2,2 x (n + 1), (n + 1) x 2, (n + 1) x (n + 1) matrices subject to the condition

tMJM=J,

where M, J are the block (n + 3) x (n + 3) matrices

M = (A B) J = (12 0) CD' 0 1.+1 .

§ 3. Penrose Representation and Some Physical Applications

3.1. Penrose Representation and Twistor Transform. Denote by J an Hermitian 4 x 4-matrix having the eigenvalues (+ 1, + 1, -1, -1) and consider the set QJ of block 4 x 2-matrices

P=(~J of the form

Q J = {P : P* J P > O}.

We introduce an equivalence relation in QJ by setting two matrices P and P' of the above type equivalent if there exists a non-degenerate matrix R such that z~ = ZlR, Z; = Z2 R. The quotient of QJ with respect to this equivalence relation is denoted by f2;J.

The domain f0 J can be identified with the Grassmann manifold of 2-dimensional (complex) subspaces in C4 which are positive with respect to J, i.e. zJ z* > 0 for any non-zero vector z of the considered 2-subspace (in other words, the restriction of the Hermitian form corresponding to J to the 2-subspace is positive definite). To prove this assertion, consider a vector z E C4 as a pair of two vectors z = (w, n) where w, n E C2 and assign to a matrix P E DJ a 2-subspace p in C4 by the equations

nZ1 = wZ2·

It is clear that a matrix P' E DJ equivalent to P defines the same 2-subspace. Thus, we have assigned to an arbitrary element of f0J a 2-subspace in C4 . We show that this subspace is positive with respect to J. The notion of positivity is invariant with respect to unitary transformations of C4 so we can assume that the matrix J has the diagonal form

J = (~f ~). Then the condition P*JP > 0 is reduced to Z!Z2 > ZiZ1 so the matrix Z2 is non-degenerate. Hence we can identify the domain f0J with the set of matrices

P = (~) such that Z*Z < f, i.e. with the generalized unit disc B. The positivity

condition for the corresponding subspace

p = {(w, n): nZ = w}

can be written as nn* > ww* for 0 1= (w, n) E p which is equivalent to the inequality n(l - ZZ*)n* > O. We have thus defined a correspondence between f0J

and the Grassmann manifold of positive 2-subspaces in C4 • It is easy to show that it is a one-to-one correspondence.

The space C4 with the Hermitian form cP(z) = IZl12 + IZ212 -lz312 -lz412, Z = (Zl' Z2' Z3' Z4) E C4, given by the matrix J, is called the twistor space and denoted by If. A twistor z E If is called positive (respectively, negative, null) if cP(z) = zJz* > 0 (respectively, cP(z) < 0, cP(z) = 0). The corresponding subspaces of If are denoted by If+, If-, N respectively. We have shown above that the domain f0J which can be identified with the generalized unit disc B coincides with the Grassmann manifold G2 (lf+) of 2~subspaces in If+. So, B is identified with G2 (If+).

If we take another matrix representation of cP (or J), namely

( 0 if) J = -if 0 '

we obtain a realization of f0J as the generalized upper halfplane H, so G2 (lf+) is identified also with H. The representation of the future tube ,+ = ,+(3) which is biholomorphic to H, as the Grassmann manifold G2(lf+) will be called the Penrose representation. The correspondence ,+ ~ G2 (lf+) is extended to the distinguished boundary M of ,+ and to the whole space C4. If we identify M

with the complexified Minkowski space CM, we obtain the embeddings M ~ G2 (1\J), CM ~ G2ClI} The space G2 (N) is the twistor model of the Minkowski space; G2 Clf) is the twistor model of the complexified Minkowski space.

Using these embeddings we can transform relativistic (conformally invariant) fields on the Minkowski space to the twistor space lr. This transformation is called the twistor (or Penrose) transform. Under this transform conform ally invariant objects on M correspond to complex analytic objects on lr such as holomorphic bundles, cohomologies with coefficients in these bundles and so on (cf. Twistors and Gauge Fields (1983) and references therein).

We note in conclusion that the constructed Penrose representation r+ ~ G2 (lr+) is closely related to the realization of r+ as the domain fi)+ on the complex quadric in CIP 5 defined in this Section. To see this it is sufficient to represent G2 (lr) as a complex quadric in CIP 5 (cf. e.g. Chern (1956)).

3.2. Conformal Compactification of the Minkowski Space. The twistor model G2 (N) of the Minkowski space constructed in the last Section is a compact space so it defines through the embedding M ~ G2 (N), a natural compactification M of the Minkowski space M. Using the correspondence between G2 (lr+) and the generalized unit disc B (cf. last Section) which can be extended to a homeomorphism of the distinguished boundaries V ~ G2 (N), we can identify M with V and study the compactification M through the embedding M --+ V constructed in Sect. 2.1. The compactification M coincides with the conformal compactification of Minkowski space known in quantum field theory (cf. Penrose (1980, 1967), Uhlmann (1963)). It has the following properties. The "points at infinity" of M correspond to the points ofthe set Vo = {X E V: det(/ + X) = O} (cf. Sect. 2.1). We may represent elements of V in the form V 3 X = eiq>/2 u, where o :::;; <p :::;; 2n, U E SU(2), so

u = (.:: 7J !), lexl 2 + IPI 2 = 1, ex, P E C2.

(This representation will be uniquely defined if we identify the pairs (<p = 0, u) and (<p = 2n, -u) for any u E SU(2)). The set Vo in this parametrization is equal to {(<p, u) E V: Re ex + cos <p/2 = O}. Thus, topologically Vo is the torus S2 x S\ with one of the equators (corresponding to <p = 0 and <p = 2n) shrunken to a point.

Let us consider the topology of M in a neighborhood of the points at infinity using formula (3) from Sect. 2.1. Denote the points of M by x = (xo, x) = (xo, Xl' X 2 , x 3 ) and consider the limits of various straight lines in M. It follows from (3) that the limits in M of all "time" lines XO = xg + t, x = XO (where XO

is a fixed point of M) and all "space" lines Xo = xg, x = XO + ext (where ex = (ex l , ex2 , ex 3 ) is a fixed point of the sphere lexl 2 = 1) for t --+ ±oo coincide with each other and are equal to the unique point at infinity of M corresponding via (3) to the matrix X = - I. This point is denoted by 1o and called the spacetime infinity. From the other side, the limits of the "light" line x = ex(xo - r) (where lexl 2 = 1, r is a fixed real number) for X o --+ ±oo coincide and are equal to the

IV. Complex Analysis in the Future Tube

point at infinity of M corresponding via (3) to the matrix

[

r + iex 3.

-r-I x=

iex 1 - ex 2

-r-i

iex 1 + ex~ I -r-I

r - iex3

-r-i

195

(5)

The set of points of M corresponding to matrices of the form (5) is called the light irifinity and denoted by ,3. Generalizing the last assertion we can prove that the limits in M of a "light" line x = Xo + exxo with (XO, ex) + r = 0 (where lexl 2 = 1, r E IR, ex and r are fixed) for X o ~ ±oo coincide and are equal to the point (5) at infinity. The limits in M of all non-light lines are equal to the spacetime infinity 10' So the set of points at infinity of M is parametrized, according to (5), by pairs (r, ex) where ex E S2, -00 ::::; r ::::; 00 and all points of

the form (±oo, ex) are identified. ( This parametrization is related to the para-

metrization of Vo defined above through the change of variables ei<p = r - ~). r+1

One can imagine the set of points at infinity of M as a "spinning top" with the equator shrunken to a point and identified with the vertices, and the upper and bottom cones identified along opposite generators (cf. Fig. 2). This interpretation was proposed by Penrose.

It is also possible to describe neighborhoods of points at infinity. Consider first the spacetime infinity 10' Introduce the sets

+ -V: = V; u V;, r E ~,

where + V: = {x: (xo - r)2 > Ix1 2, Xo > r},

U: = {x: (xo + r)2 > Ix1 2, Xo < -r}.

Fig. 2

------~~----r---~r-----~X1

U;

Fig. 3

+ In other words, U: is the interior of the future light cone with vertex at the point (r, 0, 0, 0), U: is the interior of the past light cone with vertex at point

+ -( - r, 0, 0, 0) (cf. Fig. 3). Denote by U: the complement of the set U!"r U U!"r. The set U: can be obtained by rotation of the cone {(XO, Xl' 0, 0) : (Xl - r? > X6, Xl > r} around the axis (xo) in M. Finally, put Ur = U: u U: and denote by v,. the complement of Ur in M. Then the completions Or of sets Ur in the topology of M (i.e. Or is the union of u,. and limit points at infinity of Vr in M) form, for r --+ +00, a fundamental system of neighborhoods of the spacetime infinity 10 ,

Neighborhoods of a point of the light infinity 3 with parameters r = 0, a = aO can be described as follows. Consider the subset v,.e(aO) of the set v,. filled out by light lines x = a(xo - s) with la - aOI < e, lsi < r and denote by VA,r(aO) the intersection of v,.e(aO) with the exterior of the ball: {ixi :$; R}. Then the comple-

r------.J

tions VA,r(ao) of sets VA,r(aO) in M form for R --+ +00, r --+ + 0, e --+ + ° a funda-mental system of neighborhoods of the point (0, aO) E 3. Neighborhoods of the other points of 3 can be described in an analogous way.

§ 4. Holomorphic Non-straightening

4.1. Holomorphic Non-straightening. In a neighborhood of any point ( E S, the future tube r+ = r+(n) looks locally like the product of a strictly pseudoconvex domain in cn and a complex line. More precisely, we can find a neighborhood U of ( and a diffeomorphism cp of this neighborhood onto an open subset V in e+l mapping r+ (\ V onto (C l x ~') (\ V where~' is a strictly pseudoconvex domain in e. Indeed, this diffeomorhism is given by the formula W = cp(O where Wo = (0' WI = (tiI10, ... , Wn = (nI110' The domain ~' has the form

~'= {w' = (WI' ... , wn):(Im WI? + ... + (1m Wn )2 < 1}


which is a convex and strictly pseudoconvex domain (note that !!))' is not strictly convex because the tangent space at any point of o!!))' sticks to o!!))' along an n-dimensional real plane).

The constructed local diffeomorphism qJ "straightens" the hypersurface S along complex light rays lying on S. However, there is no biholomorphism with the same property. Namely, we have the following.

Theorem 1 (Sergeev (1983, 1986), Sergeev-Vladimirov (1986)). The hypersurface S cannot be biholomorphically straightened along complex light rays in a neighborhood of any of its points.

This theorem is proved by checking the necessary condition for biholomorphic straightening found by Freeman (1970, 1977). In fact, the assertion of the theorem remains true if we weaken the definition of the straightening biholomorphism qJ assuming only that qJ is defined and holomorphic in a one-sided neighborhood ,+ n U and smooth up to Un 0,+ (Sh. Tsyganov) or even that qJ

is a CR-diffeomorphism in a neighborhood of , in S (S. Pinch uk (1990), Sh. Tsyganov).

In Khenkin-Sergeev (1980) a notion of strictly pseudo convex polyhedra was introduced unifying the notions of strictly pseudoconvex domains and that of analytic polyhedra. A domain Q in em is called a strictly pseudo convex polyhedron if there exist a domain Q' ::::J D, holomorphic mappings Xa, a = 1, ... , N, of Q' onto domains Q~ c em, with ma :s; m and smooth strictly pseudo convex domains Qa, Da c Q~, such that Q has the form

Thus, Q is the intersection of the preimages of domains Q a with respect to the mappings Xa. The boundary of Q consists of smooth pieces Sa = {' ED: Xa(,) E aQa}, a = 1, ... , N and each of these pieces is fibered by complex submanifolds of the form (X a)-l(W), WE aQa. It is evident that the map Xa defines a biholomorphic straightening of the hypersurface Sa along these complex submanifolds in a neighborhood of any point on Sa. Moreover, if a polyhedron Q is nondegenerate (cf. Khenkin-Sergeev, op. cit.), i.e. some conditions of general position type are satisfied on edges

then also these edges SA can be biholomorphically straightened in a neighborhood of any of their points along complex submanifolds of SA- Conversely, any pseudo convex domain (with piecewise smooth boundary with general position conditions satisfied on edges) which can be locally biholomorphically straightened in the above sense is locally a strictly pseudoconvex polyhedron. Hence Theorem 1 asserts that the future tube ,+ gives an example of a pseudo convex domain which is not (even locally) a strictly pseudo convex polyhedron. However, it can be approximated up to the 2nd order by strictly pseudoconvex polyhedra as is shown in the next section.

\01 \V! \ / )i

/ \ I~ /'c.j\

Fig. 4

4.2. Approximation by Strictly Pseudoconvex Polyhedra. Fix a point K E

1R"+l and consider the domain

£4K = {z E en+l: Izo - Kol2 > Iz' - K'12} where z' = (Zl' ... , zn). The domain £4K has the following properties. For any x = Re Z belonging to the cone with vertex at K: {x E IRn+l : (z - K)2 = O}, the section £4K n {z : Re z = x} of £4K with the fixed x is the interior of the light cone {y E IRn+l : y2 > O}. For other x the section of £4K with fixed x coincides with the interior of the hyperboloid {z: (x - K)2 + y2 > O} which has one cavity for (x - K)2 > 0 and two cavities for (x - K)2 < 0 (cf. Fig. 4). Note that £4K is invariant under the subgroup of the Poincare group in M fixing the point K and sections of £4K with fixed x are invariant under the action of the Lorentz group on these sections.

The holomorphic mapping

z· - K· Z --+ XK(z) = (X~(z), ... , X:(z)), Xr(z) =} } ,

Zo - Ko

transforms £4K onto the ball {lxW + ... + Ix:12 < 1}. SO £4K is the preimage of the ball under the map XK, however this map degenerates on the boundary of £4K

at the point K (o£4K also degenerates at this point). Let us extend the definition of a strictly pseudoconvex polyhedron Q given above by allowing the maps Xa

to degenerate on oQ. In this case we shall say that Q is a strictly pseudoconvex polyhedron with singularities. Thus, £4K is a strictly pseudoconvex polyhedron with singularities.

As was noted above, for x = K the section of £4K with fixed x coincides with the section of ,+ for Yo > o. We can assert more than that. Namely, denote by SK the smooth hypersurface o£4K n {Yo> O}. Then Sx coincides with S to the 1st order at any point z = x + iy E S, i.e.

~Sx = ~S, ~cSx = ~cS.

The Levi forms of £4K and ,+ also coincide at these points (the Levi form of £4K

is computed using the defining function rK(z) = -!Izo - Kol2 + !Iz' - K'12).

§ 5. Generalizations

5.1. Tube Cones. A tube cone or a Siegel domain of the 1 st kind is a domain of the form

T C = {z = x + iy E em: y E C} = IRm + iC

where C is an open cone in IRm with vertex at the origin. According to Bochner's Tube Theorem (cf. Vladimirov (1964)), any function holomorphic in T C can be holomorphically extended to the tube cone ychC where ch C is the convex hull of C. Hence, it is natural to suppose that the cone C is convex. We shall also assume that the cone C is proper, i.e. its closure C does not contain a whole line (cf. the motivation of this condition in the note to Theorem 3 from Chap. 2, Sect. l.2).

Besides the future cone, we have the following examples of convex proper cones:

1) The octant IR~ = {y E IRm: Y1 > 0, ... , Ym > O}. The tube cone T+ = TIR':.' is biholoplOrphic to the polydisc {z E em: IZ11 < 1, ... , IZml < 1}.

2) The cone .Yt, c IRm with m = F consisting of all complex positive definite Hermitian I x I-matrices. For I = 2 the tube cone T.Yf2 coincides with the generalized upper halfplane H (cf. Sect. 2.1). For any I the tube cone T.Yti is biholomorphic to the generalized unit disc Bl (cf. Sect. 2.2) which is a particular case of a classical Cartan domain of the 1st type (Cartan (1935), Siegel (1949), Piatetski-Shapiro (196l)) consisting of complex p x q-matrices Z, p ~ q ~ 1, subject to the condition ZZ* < I. This domain is biholomorphic to a tube cone only for p = q.

3) The cone f!i>l C IRm with m = l(l ; l) consisting of all real positive definite

symmetric I x I-matrices. For I = 2 the tube cone T"'2 is biholomorphic to the future tube ,+(2). For any I the tube cone T"" is biholomorphic to the classical Cartan domain of the IIIrd type consisting of complex I x I-matrices Z such that ZZ* < I and tz = Z.

4) The cone Ql C IRm with m = 2F - I consisting of all quaternion positive definite quaternion-Hermitian I x I-matrices. The tube cone TQ, is biholomorphic to the classical Cart an domain of the IInd type consisting of complex p x pmatrices Z such that ZZ* < I, tz = - Z with p = 21.

To characterize the common properties of these cones including the light cone let us give the following definitions. For a cone C we call the cone C* =

{17 E IRm : (17, y) ~ 0, Vy E C} the dual cone. A cone C is self-dual if C* = C. A cone C is called homogeneous if the group of linear automorphisms of C (i.e. linear non-degenerate transformations of IRm mapping C into itself) acts transitively on C, i.e. for any y, y' E C there exists an automorphism of C mapping y to i. All the cones listed above are self-dual and homogeneous; such cones are also called domains of positivity (Koecher (1957), Rothaus (1960)). It turns out that almost the only examples of self-dual homogeneous cones are the ones listed

above. More precisely, any self-dual homogeneous cone C (which is convex and proper) can be represented as the direct sum of light cones V+ (n), cones of type 2)-4) and an exceptional cone in 27-dimensional space which can be realized in the space of matrices over the Cayley numbers (cf. Vinberg (1963)). Tube cones T C over domains of positivity can be realized as the direct sums of classical Cart an domains of the types I-IV (domains of the IVth type were introduced in Sect. 2.3) and an exceptional domain in 27-dimensional space. So they form a subclass of bounded symmetric domains in em (cf. Helgason (1978)) which can be realized as tube cones and for this reason they are called bounded symmetric domains of tube type (arbitrary bounded symmetric domains in em can be realized as Siegel domains of the Ilrid kind, cf. below).

Arbitrary tube cones have the following general properties. Any tube cone is biholomorphic to a bounded domain because it can be mapped by a nondegenerate linear transformation into the tube cone T+ biholomorphic to the polydisc. Analytic automorphisms of a tube cone T C continuous in the closure of T C have the form z -+ Az + b where A is an affine transformation of the cone C onto itself, b E IRm.

A further generalization of tube cones is connected with the notion of Siegel domains of the IInd kind. Recall (cf. Piatetski-Shapiro (1961) that a Siegel domain of the lInd kind is a domain in ek+m of the type

{(z, w) E ek x em: 1m z - F(w, w) E C}

where F: em x em -+ ek is a sesquilinear non-degenerate form with values in ek which is C-Hermitian in the sense that F(w, w) E C for any WE em and F(w, w) = 0 only when w = O. Tube cones (Siegel domains of the 1st kind) correspond to the case m = 0, F = O. The other extreme case is k = 1, C = IR+. In this case the Siegel domain coincides with the unbounded realization of the ball in em+!. We have restricted ourselves here to the case of tube cones.

Another generalization of tube cones is considered in the next section.

5.2. Tuboids. Let us call a profile A = A (Q) over an open set Q in IRm a domain in em of the form

A = {z = x + iy E em: x E Q, y E AJ where the fiber Ax for any x E Q is an open proper cone in IRm. We call the fiber convex hull ch A of a profile A the profile having the fibers (ch A)x equal to the convex hull of Ax for any x E Q. A profile !\ is compact in a profile A, !\(Q) <£

A (Q), if !\x <£ Ax for any x E Q. This means that A~ c Ax u {O} for any x E Q. A Tuboid fiji = fijI(A) with profile A = A (Q) is a domain fiji c A of the form

fiji = {z = x + iy E em: x E Q, y E fijlJ with the following property: for any profile !\ <£ A there exists a complex neighborhood Q of the set Q such that Q n !\ c fiji (cf. Bros (1976), Bros, lagolnitzer (1974-75, 1976)). In other words, the set fiji near x E Q looks

Ax

Fig. 5

"asymptotically" like the tube cone over I\x (cf. Fig. 5). If, in particular, [2 = Q + iCR where CR = C n B(O, R) is the intersection of C with the ball B(O, R) = {Iyl < R} in [Rm we shall call the tuboid [2 a local tube over Q.

Many of the results valid for tube cones can be extended to tuboids. Fourier transform, which is crucial for complex analysis in tube cones, has been generalized to tuboids in the form of the local generalized Fourier transform now also called the FBI-transform due to the names of its inventors (and not to the known agency) the French mathematical physicists 1. Bros and D. Iagolnitzer (Bros-Iagolnitzer (1974-75,1976)).

We give two more examples. The first of them is an analogue of Bochner's Tube Theorem. It asserts that any function holomorphic in a tuboid [2(A) can be holomorphically extended to a tuboid [2'(1\) with profile 1\ = ch A (the tuboid [2' depends only on the tuboid [2(1\) and not on the function). This result can be considered as a microlocal version of Bochner's Theorem (cf. a local variant of Bochner's Theorem in Komatsu (1972)). The second result can be considered as a micro local version of Grauert's theorem on the hoi om orphic convexity of totally-real sets: for any tuboid [2 = [2(1\) there exists a tuboid [2' (1\) c [2 with the same profile which is a domain of holomorphy.

Bibliographical Notes

This chapter is of introductory character, so the exposition in the first four sections is rather detailed and many of the omitted proofs can be obtained by the reader using the given results. The material of the first two sections is known in general though it is hard to give the appropriate references for Sect. 1 and Sect. 2.1, 2.2. The transformations of Sect. 2.3 and further information on classical domains of the IVth type can be found in (Siegel (1949), Hua (1958), Piatetski-Shapiro (1961)). The twistor theory which is barely touched in Sect. 3.1 is studied in the collection of original papers of R. Penrose and his collaborators (Twistors and Gauge Fields (1983)) and in the books (Atiyah (1979), Manin (1984)) (cf. also the review article Sergeev (1991)). The exposition of Sect. 3.2 is based on (Sergeev-Vladimirov (1986)). Section 4 contains the results of (Sergeev

(1983, 1986) and Sergeev-Vladimirov (1986)). The last section is of expository character. Cf. further information on tube cones in (Vladimirov (1964, 1979)). Classical Cartan domains and corresponding homogeneous tube cones are considered in (Cart an (1935), Siegel (1949), Hua (1958), Piatetski-Shapiro (1961), Koranyi-Wolf(1965), Wolf (1972)). In Sect. 5.2 we formulate some of the results of (Bros (1976), Bros-Iagolnitzer (1974-75,1976)).

Chapter 2 Boundary Properties of Holomorphic Functions

Throughout this chapter, C will denote a convex open proper cone in ~m.

§ 1. Boundary Values in L P and ~

1.1. The Spaces HP(T C ). We define HP(TC ), ° < p ::s;; 00, as the space consisting of all functions f E (9(TC ) having the finite norm

Ilfllw = sup [f If(x + iy)iP dxJI/P for 0< p < 00, YEC IRm

Ilfllw = sup 1 f(z) 1 for p = 00. YE TC

The spaces HP(TC ) are Banach spaces for all p, 1 ::s;; p ::s;; 00.

Theorem 1 (Stein-Weiss-Weiss (1964), Stein-Weiss (1971)). Any function f E HP(TC ), 1 ::s;; p < 00, has boundary values in U for y --+ 0, y E C.

If C = ~~ we can assert moreover that for any function f E HP(T+) with ° < p < 00 there exists almost everywhere on ~m (with respect to Lebesgue measure on ~m) the limit

f(x + iy) --+ f(x) for y --+ 0, y E ~~,

so that

Ilf(x + iy) - f(x)IILP --+ ° for y --+ 0, y E ~~.

This result can be extended to general tube cones only in the following weak form.

Theorem 2 (Stein-Weiss-Weiss (1964), Stein-Weiss (1971)). Let f E HP(TC )

with 0< p < 00. Then for any compact sub cone C ~ C (cf. Chap. 1, Sect. 5.2) there exists the limit

f(x + iy) --+ f(x) for y --+ 0, y E c.


for almost all x E IRm and

IIf(x + iy) - f(x)lb -+ 0, for y -+ 0, y E C.

1.2. The Spaces H(6)(C). Denote by 2/, s E IR, the Hilbert space of functions g on IRm having the finite norm

IIgll(s) = [tm Ig(~W(1 + IWs d~ T/2,

and let the space J'f. consist of all functions f which are the Fourier transforms of functions g E 2/ : f = F[g]. The space J'f. is provided with the norm

Ilfll.'II5 = IIgll(s)' We define next the Banach space H(S)(C) (cr. Vladimirov (1979)) consisting of

all functions f holomorphic in T C and having the finite norm

Ilfll(S) = sup IIf(x + iy)IIx.· yee S

For s = ° the space H(O)(C) coincides with H 2 (TC ).

Theorem 3. Any function f E H(S)( C) has a boundary value f(x) in J'f.,

f(x + iy) -+ f(x) in J'f. for y -+ 0, y E C.

The function f belongs to H(S)(C) if and only if its spectral function g =

F- 1 [f(x)] belongs to 2/ with its support supp 9 c C* where C* is the cone dual to C (cf. Chap. 1, Sect. 5.1). Moreover, Ilfll(S) = IIf(x) II K. = IIgll(s) and the Laplace transform g -+ L[g] = f is an isomorphism between the space of functions in 2/ with supports in C* and the space H(S)( C).

This theorem is proved in Vladimirov (1979) (cf. also Tillmann (1961)). If a cone C is not proper and f E H(S)(C) then it follows that f == 0. This motivates our requirement that the cone C be proper.

§ 2. Boundary Values in Spaces of Distributions and Hyperfunctions

2.1. The Space H(C). Let us define the algebra H(C) of slowly increasing (or temperate) functions f E (!J(TC) as those satisfying the following growth condition

If(z)1 ~ M(1 + IzI2)"'/2[1 + .,rp(y)], 'E T C,

for some constants M > 0, IX, f3 ;:::: 0, where LI(y) is the distance from y to the boundary of the cone C (cr. Vladimirov (1979)). The topology in H (C) is defined by the system of seminorms

Ilfll(""P) = sup If(z)l/(1 + IzI2)"'/2(1 + LI-P(y)), IX, f3;:::: 0. ZE TC


Theorem 4 (Vladimirov (1960, 1961, 1979)). The Laplace transform g-)o L[g] = f is an isomorphism between the space 9"(C*) of tempered distributions with supports in C* and the space H(C). Any function f E H(C) has a boundary value f(x) in 9" for y -)0 0, Y E C.

The existence of boundary values offunctions from H(C) was proved also in Tillmann (1961). Generalizing this theorem to the case of non-convex tube cones we can obtain a variant of Bochner's Tube Theorem (with estimates) for the space H(C).

2.2. Boundary Values in the Sense of Hyperfunctions. Recall first the definition of hyperfunctions. Let Q be a domain in ~m and Q - a complex neighborhood thereof in em, Q n ~m = Q, which is a domain of holomorphy. The space of hyper functions fJ6(Q) is by definition the cohomology group Hm-l(Q\Q, (9) with coefficients in the sheaf (9 of hoi om orphic functions (this definition does not depend on the choice of the complex neighborhood Q) (Sato (1959-60)). Considering the covering of Q\Q by the domains of holomorphy

Ql={z=x+iYEQ:±Yj>O}, j=1, ... ,m,

we obtain, using the Leray Theorem, that any hyperfunction in Q is given by a collection of 2m functions fr. holomorphic in the domains

Q.={zEQ:t:jYj>O,j=I, ... ,m}, t:=(t:1 , ... ,t:m ), t:j= ±1 (cf. Fig. 6). (The collection {fr.} is defined up to the addition of an (m - 2)coboundary of the covering {Ql} with coefficients in (9). For us it is more convenient to use another representation of hyperfunctions which we shall obtain using the following covering of Q\Q. Consider half spaces in em of the form

Ej = {z = x + iy : (y, e) > O}, j = 0, ... , m,

where eo, ... , em are unit vectors in ~m such that U Ej = em \ ~m. Then the - - j -domains Qj = Q n Ej , j = 0, ... , m, form a covering of Q\Q by domains of

S2

--~r-~--~------+---~Y2

Fig. 6

Fig. 7

holomorphy. Using the Leray Theorem for this covering we can represent a hyperfunction on Q by a collection of m + 1 functions Ji. ... j~' 0 ~ j 1 < j 2 < ... < jm ~ m, holomorphic in the domains Qj ! ••• j~ = Qi! n ... n[}j~ (and defined up to the addition of an (m - 2)-coboundary of the covering {Qj } with coefficients in 0) (cf. Fig. 7).

Take now a function IE 0(TC n Q) and choose vectors eo, ... , em so that Q I ... m C T C n Q. Assign to the function I a collection of functions jj! ... j~ where II ... m = sgn(e l /\ ... /\ em)I (here sgn(el /\ ... /\ em) is the orientation of the polyvector (e l /\ ... /\ em) and the other components jj! ... j~ are set equal to zero. This collection defines a hyperfunction on Q which is called the boundary value 01 I and denoted by bv f The boundary value bv I does not depend on the choice of vectors eo, ... , em satisfying the above hypothesis nor on the complex neighborhood Q. Thus bv I can be computed using any domain of the form TC' n Q where C' c C, Q' c Q and Q' is a domain of holomorphy such that Q' n IRm = Q.

2.3. Distributional and Hyperfunctional Boundary Values in Tuboids. Let D be a tuboid with profile /\ over a domain Q c IRm (cf. Chap. 1, Sect. 5.2). We can assume (cf. loco cit.) that the profile /\ is fiber-convex and D is a domain of holomorphy. Let IE 0(D). Then for any point x E Q and a neighborhood U =

U(x) of x, we can take a local tube of the form f0u = U + iCR so that f0u c D and define the boundary value bVuI E .?4(U) as in the Sect. 2.2. These boundary values coincide on the intersection of neighborhoods U n U' and thus define the unique hyperfunction bv I E .?4(Q) which is called the boundary value of I (cf. Zharinov (1983)).

A function IE 0(D) is called a locally slowly growing (or tempered) Iunction, IE H,oc(/\)' iffor any point x E Q there exists a neighborhood U of x and a local tube U + iCR C D where the following estimate is satisfied: II(z) I ~ M/IYIN for some constants M, N > O. A function I of such type has by Theorem 4 a bound-


ary value on V in the sense of distributions. This boundary value coincides with the boundary value bVuf in the sense of hyperfunctions (Martineau (1964)). Thus, any function f E H1oc(/\) has a boundary value on Q in the sense of distributions (in £&'(Q» and this boundary value coincides with bv f. For functions of the class H1oc(/\) we have the microlocal analogue of Bochner's Tube Theorem (with estimates) from Chap. 1, Sect. 5.2. There is also the following interesting property: if a function f E (9(D) is locally tempered in a tuboid D' c D over the same domain Q then it is locally tempered also in the tuboid D.

§ 3. Boundary Values of Bounded Holomorphic Functions

3.1. Auxiliary Results. To study the boundary properties of holomorphic functions in tube cones T C the following two assertions formulated as lemmas may be useful.

Lemma 1. A tube cone T C is biholomorphically equivalent to a bounded domain £& contained in the polydisc. We can choose the biholomorphism in such a way that it maps the distinguished boundary of T C into the distinguished boundary of the polydisc.

To prove this lemma it is sufficient to consider a homogeneous linear transformation taking the cone C into the octant ~~. Its complexification maps T C

into T+ and, combined with the biholomorphism of T+ onto the polydisc, gives the required map. This lemma often allows one to restrict the proofs of statements for general tube cones to the case of T+.

To formulate the second assertion we introduce the following definition. An oriented C1-smooth hypersurface M in ~m will be called C-spacelike if the normal to M at any of its points belongs to C (cf. Fig. 8). Using the next lemma we can sometimes restrict the study of the boundary properties of holomorphic functions in T C to the case of strictly pseudoconvex domains.

c

Fig. 8


Lemma 2. Let M be a C-spacelike hypersuiface in IRm of class C2 • There exists a C2-smooth strictly pseudoconvex domain ~ c em such that ~ contains T C in a neighborhood of IRm and TxM c T:(o~) for any point x E M.

The last assertion means that M is an integral submanifold of p). We give an idea of the proof. Let us assume for simplicity that T C is the future tube r+ in en+1 (cf. Chap. 1, Sect. 1.1) and the hypersurface M is given by the equation X o = s(x') = s(x 1, ... , xn). If s is of class C3 we can define a domain p) as follows

{ n os(x') n}

~ = Z = x + iy E en+1 : Yo > L --Yj + L yJ .' j=l OXj j=O

Then in some neighborhood U of IRn+l in en+! we have: 1) ~ n U ::::> r+ n U; 2) o~ n U is C2-smooth and strictly pseudoconvex; 3) M is an integral submanifold of o~ (cf. Fig. 9). To prove the lemma when M is of class C2 we note that the defining function of o~ has 2nd derivatives at points where y = 0 so we can approximate this function by a C2-smooth function defining the required domain.

We introduce now the algebra A(TC ). Consider the one-point compactification em = em u { 00 } (where a base of neighbourhoods of 00 E em is given by the exteriors of balls: {I z I > R}) and denote by r C the closure of T C in em so that rC = f'c u {oo}. The algebra A(TC ) consists of all functions hoiomorphic in T C

and continuous in rc. It follows from Lemma 1 that the Shilov boundary of A(TC ) coincides with IRm = IRm u {oo}. There is an even stronger assertion.

Theorem 5 (Vladimirov (1979». If the boundary value off E H (C) is bounded, i.e. If(x)1 s M for almost all x E IRm then I f(z) I s M for all z ETC.

3.2. Fatou and Lindelijf Theorems. For functions f E HOO(TC ) we have the following analogue of the classical F atou Theorem: for almost all XO E IRm there exists a limit of f(z) when z -+ XO in the restricted admissible way (in the sense of the definition given in Chap. 4, Sect. 2.3). This assertion follows from the corresponding result for T+ (cf. Zygmund (1958), Stein-Weiss (1971» using Lemma 1. A stronger assertion is also true. We say that a function f E (!)(TC) is bounded at XO E IRm in the restricted sense if it is bounded in some approach set Ft-(xO) (cf. Chap. 4, Sect. 2.3). Let Q + iCR be a local tube over an open set Q c IRm (cf. Chap. 1, Sect. 5.2).

~TM

Fig. 9


Theorem 6 (Zygmund (1958), Stein-Weiss (1971)). If a function f E

(!)(Q + iCR ) is bounded in the restricted sense at each point x E Q then f has a restricted admissible limit almost everywhere on Q.

This result can be further strengthened using the following Theorem of Drozhzhinov-Zavialov (1982) and Khurumov (1983).

Theorem 7. Let f E (!)(Q + iCR ) where Q is open in ~m and let x E Q. Iff is bounded on some smooth totally real m-dimensional submanifold

going through x then f is bounded in the restricted sense at x.

Hence, if a function f E (!)(Q + iCR ) is bounded on some smooth (m + 1)dimensional submanifold of Q + iCR with edge Q then it has a restricted admissible limit almost everywhere on Q. What is proved by Drozhzhinov-Zavialov is in fact the following assertion implying Theorem 7. Let 1\ be a profile over Q (Chap. 1, Sect. 5.2) and

A = {z = x + iy E em : x E Q, y E Ax where Ax = tex, t > 0, ex E I\J be a one-dimensional smooth profile over Q contained in 1\. If a function f is holomorphic in a tuboid ~ = ~(I\) and bounded on A n ~ then it is bounded in some tuboid ~' = ~'(I\) with the same profile 1\ over Q (cf. Fig. 10). Note that, generally speaking, f is not bounded in the tuboid ~.

A result close to Theorem 7 (namely, a variant of the "two-constants" theorem for our situation) was proved in J6ricke (1982). A nice and short proof of Theorem 7 was proposed by Gonchar. It is based on his "boundary" variant of the theorem on separate analyticity (Gonchar (1985)).

The classical LindelOf Theorem does not have a direct extension to tube cones (Chirka (1973)). In fact, from the above formulation of the Fatou Theorem for tube cones we could expect that the following extension of the Linde16f Theorem is true. Let a function f E (!)(Q + iCR ), where Q + iCR is a local tube over an open set Q c ~m, be bounded in the restricted sense at XO E Q. Suppose that f

Fig. 10

has a limit along a continuous curve ro(t) lying in Q + iCR for 0 ~ t < 1 and approaching XO for t ~ 1. Then f has the same limit along any continuous curve r(t) such that r(t) ~ XO for t ~ 1 lying in Q + iCk for 0 ~ t < 1, where C' is a compact subcone of C. Unfortunately, this assertion is not true. To see this, it is sufficient to consider the tube cone T+ in C 2 and the function f(zo, zd = zdzo hoi om orphic in T+ and bounded at the origin in the restricted sense. However, f has different limits along distinct rays at the origin (it is not difficult to change this example in such a way that f would be bounded at the origin).

The correct extension of the Lindelof Theorem asserts that f has the same limit only for curves r(t) "tangential" to ro(t) for t ~ 1. More precisely, let r(t) be a continuous curve in Q + iCR of the same type as before with the endpoint xO. Denote by A/XO), Y E CR , the complex ray

Ay(XO) = {z = XO + (J(y: (J( E C, 1m (J( > O}

at XO with the direction y. Denote by y(t) the orthogonal projection of r(t) onto A/XO). We shall say that r(t) ~ XO alongside Ay(XO) if

Ir(t) - y(t)1 ~ 0 for t ~ 1. 11m y(t)1

Theorem 8 (Chirka (1973), Sergeev (1989)). Let f be holomorphic in a local tube Q + iCR over an open set Q c IRm and bounded in the restricted sense at XO E Q. Suppose that f has a limit along a continuous curve ro(t), 0 ~ t < 1, in Q + iCR such that ro(t) ~ XO alongside some complex ray Ay(XO), y E CR' Then f has the same limit along any continuous curve r(t), 0 ~ t < 1, in Q + iCR such that r(t) ~ XO alongside Ay(XO).

Again, according to Drozhzhinov-Zavialov and Khurumov (op. cit.) it is sufficient to require f to be bounded on some smooth totally-real n-dimensional submanifold going through the point xo.

3.3. Uniqueness Theorems. For functions bounded and holomorphic in tube cones we have the following well-known uniqueness theorem.

Theorem 9 (Zygmund (1958), Stein-Weiss (1971)). If a function f E HOO(TC) has restricted admissible limit 0 on a set E c IRm of positive measure then f == O.

A set E c IRm is called a uniqueness set for the algebra A(TC) (or a determining set in the terminology of Rudin (1969)) if for any function f E A (TC) the equality f(x) = 0, X E E, implies that f == O. According to Theorem 9 any set E of positive Lebesgue measure on IRm is a uniqueness set. On the other hand, not every set of (even infinite) (m - l)-dimensional Hausdorff measure on IRm is a uniqueness set (cf. Sect. 5.1).

Rudin (1969), Sect. 5.1, gives an example of a compact uniqueness set for A(T+) having finite linear measure. By the same methods as in Rudin (op. cit.) or using Lemma 1 it is easy to construct examples of compact uniqueness sets for A(TC) having finite linear measure. Note that a compact set of linear measure zero on IRm cannot be a uniqueness set for A(TC) (cf. Sect. 5.1).

§ 4. Inner Functions and Holomorphic Mappings

4.1. Rational Inner Functions. A function f E HOO(TC ) is inner if its limit boundary values on [Rm (which exist almost everywhere on [Rm by Theorem 6, Sect. 3.2) have modulus one almost everywhere on [Rm. The functions

(z, q) - IX + () where q E V, IX E C, 1m IX > 0, 2, q - IX

provide examples of inner functions in the future tube <+ (cf. Vladimirov (1983b)). Other examples can be generated by applying to these functions automorphisms of <+ and taking the product of different functions. In the case of the tube cone T+ which is biholomorphic to the polydisc, inner functions are given by Blaschke products (cf. Rudin (1969)). For the generalized unit disc Bm rational inner functions are given by the function det Z and functions obtained from this one by composing with automorphisms of Bm and taking products.

A complete description of rational inner functions in the polydisc was given in Rudin (1969). This result was extended also to general bounded symmetric domains in Koranyi-Vagi (1979). To formulate this extension we need to introduce some notation. Let:!JJ be a bounded symmetric domain (cf. Helgason (1978)). Denote by N the invariant norm on :!JJ. The general definition of N is given in Koranyi-Vagi (op. cit). We note here that for the generalized unit disc and classical domains of the IIIrd kind N is given by the determinant; for classical domains of the lInd kind - by the Pfaffian and for the Lie spheres - by

(z, z). Set r(z) = grad N(z), z E f0, and denote by Q(z) the polynomial obtained N(z)

from a polynomial Q(z) by the conjugation of coefficients.

Theorem 10 (Koranyi-Vagi (1979)). Let f0 be a bounded symmetric domain of tube type (Chap. 1, Sect. 5.1) and f a rational inner function on f0. Then f has the form

Q(r(z)) f(z) = M(z) Q(z) , z E f0,

where Q(z) is a polynomial having no zeros in :!JJ, and M(z) is a homogeneous polynomial having modulus one on the distinguished boundary of f0. If Q(O) = 1 then M and Q are uniquely defined.

If f0 is one ofthe classical domains then M(z) coincides, up to a constant with modulus 1, with a power of the norm N(z). The proof of Theorem 10, as in Rudin (1969), is based on the following Lemma which is interesting in itself.

Lemma 3 (Koranyi-Vagi (1979)). Let E be a compact uniqueness set for polynomials lying on the distinguished boundary of f0 and let f E (I)(f0). If for each x E E the function fx(A) = f(xA), IAI < 1, is a rational function of degree k (i.e. the maximum of the degrees of the numerator and denominator is equal to k) which is


continuous for lA-I ~ 1 then f = P/Q where P, Q are polynomials without common divisors and the degree of f is k. If, in addition, the function fx is inner for each x E E then Q has no zeros in ~ u E and P(x) = M(x)Q(x) for x E E where M is a homogeneous polynomial having modulus one on E.

Note that all inner functions in A(~) are rational; the same is true for inner functions meromorphic in a neighborhood of {fj (Korailyi-Vagi (op. cit.)).

4.2. General Inner Functions. Infinite Blaschke products provide examples of nonrational inner functions in the tube cone T+. The analogues of the Blaschke products in the future tube r+ are given by the functions

fI [(z, qk) - fXk • iik - i .lfXk - ilJnk

k;l (z, qk) - fXk fXk - i iik - i

where fXk are complex numbers with 1m fXk > 0, fXk =f. fXl for k =f. 1; qk = (1, qD E

V+ and nk are positive integers. This infinite product converges uniformly on compacta in r+ if and only if the following series is finite

This is proved in Vladimirov (1983b) where Blaschke products for the generalized unit disc are also described.

The general results of Aleksandrov and Low on inner functions (cf. Aleksandrov (1984, 1983), Low (1984) and Aleksandrov's article in this volume) are true also for inner functions in tube cones. We give here several of their results.

Theorem 11. Let cp be a positive lower semicontinuous function on [Rm with cp E U([Rm), 1 ~ p ~ 00. Then there exists a function f E HP(Tc) such that I f(x) I = cp(x) almost everywhere on [Rm.

Theorem 12. Let cp be a positive lower semicontinuous function on [Rm

with cp E U([Rm). Then for any B > ° there exists a function f E A(TC) such that I f(x) I ~ cp(x) almost everywhere on [Rm and the measure of the set {x E [Rm : If(x)1 =f. cp(x)} is less than B.

It follows that inner functions are dense in the unit ball of HCO(TC ) in the topology of uniform convergence on compacta in T C• Also, for any function f belonging to the unit ball of A(TC) there exists an inner function in T C having the same zeros as f.

4.3. Holomorphic Mappings. Let ~ = T C and ~' = TC' be tube cones in em with m ~ 2. The domains ~ and ~' are biholomorphically equivalent if and only if the cones C and C' are affinely equivalent; any biholomorphic mapping F: ~ -)- ~' is necessarily rational (cf. Matsushima (1972), Murakami (1972), Yang (1982)). The last assertion is also true for proper mappings F (recall that

F is proper if the preimage of any compact subset of ~' under this mapping is compact in ~).

Theorem 13. Let F: ~ ~~' be a proper holomorphic mapping. Then F is rational. If the cones C and C' are irreducible domains of positivity (Chap. 1, Sect. 5.1) then F is biholomorphic.

This theorem was proved by Khenkin and Tumanov (cf. Khenkin-Tumanov (1983)) using a result of Bell (1982) (it is also true for Siegel domains of the lInd kind). In the case of the polydisc it was proved in Rudin (1969).

Let ~ denote now a bounded symmetric domain. A holomorphic mapping F : ~ ~ ~ is called inner if its boundary value F* on the distinguished boundary S of ~ has the property: F*(x) E S for almost all XES. The importance of inner mappings is due to the following theorem of Koranyi-Vagi (1976).

Theorem 14. Let T be a linear isometry of the space HP(~), 0 < P < 00, p #- 2, into itself and let 9 = T(1). Then, there exists an inner mapping F : ~ ~ ~ such that Tf = g(f 0 F) for any f E HP(~). Moreover

1 (h 0 F*)lg*IP dJ1 = 1 h dJ1

for any hE L OO(S) where J1 is the invariant measure on S. Conversely, if F is an inner mapping and the last equality is true for spme function 9 E HP(~) and for any continuous function h on S then the operator Tf:= g(f 0 F) defines an isometry of HP(~). The isometry T maps HP(~) onto HP(~) if and only if F is an isomorphism of ~ and the function 9 is given by

_ (f,,2 (z)) lip g(z) - rx f,,(u)

where f,,(z) = %(u, z) is the Cauchy kernel of~, rx E 1[, Irxl = 1 and u = F-l(O).

§ 5. Interpolation Sets

5.1. Properties of Interpolation Sets. Let K be a compact subset of the distinguished boundary IRm of yC (cf. Sect. 3.1). We call K a zero set for the algebra A(TC) if there exists a function f E A(TC) equal to zero on K such that f(z) #- 0 on YC\K. K is called a peak set for A(TC) if there exists a peak function f E A(TC) equal to 1 on K such that 1 f(z) 1 < 1 for z E YC\K. We call K an interpolation set for A(TC) if any continuous function on K can be extended to a function in A(TC). Finally, K is called a peak interpolation set for A(TC) if for any continuous function 9 of. 0 on K there exists a function f E A(TC) such that f(x) = g(x) for all x E K and If(z) 1 < sup Ig(x)1 for any z E YC\K.

K

Fig. II

Theorem 15. All the properties of a compact K c [ijm listed above are equivalent to each other and to the following Bishop's property: for any finite Borel measure /l on [ijm orthogonal to A(TC) (i.e. J f d/l = 0 for any f E A(TC)), we have

JK Id/ll = O.

This theorem follows from the corresponding theorem of Rudin (1969) for T+ using Lemma 1. We shall give one more result on interpolation sets, well known for the tube cone T+. Let us say that a compact K is of zero width with respect to a set N consisting of unit vectors in IRm if for any t; > 0 there exist a collection of vectors {nd c N and a collection of Borel subsets {ed of the real line IR such that I led < t; (where led is the Lebesgue measure of ei) and K is contained in the union U Ei where Ei = {x E IRm: (x, ni) E ed (cf. Fig. 11).

Theorem 16. Let N be a compact set of unit C-like vectors (i.e. vectors belonging to C) in IRm. If a compact set K has zero width with respect to N then it is an interpolation set for A(TC).

This theorem follows from the Forelli Theorem for T+ (cf. Rudin (1969)) using Lemma 1. As a corollary of this theorem we obtain that compact sets K oflinear measure zero are interpolation sets.

5.2. Interpolation Manifolds. A C1-smooth submanifold M in IRm is called an interpolation manifold if any compact K c M is an interpolation set for A(TC). It follows from Theorem 16 that any C1-smooth C-spacelike curve (i.e. a curve such that its tangent vector at any of its points lies outside e u ( - e)) is an interpolation manifold (the smoothness condition here can be weakened, cf. Rudin (1969, 1971)). This result can be extended to submanifolds of IRm of arbitrary dimension :s; (m - 1). We formulate here (for the sake of simplicity) this extension for the case of hypersurfaces in IRm.

Theorem 17. All C1-smooth C-spacelike hypersurfaces in IRm are interpolation manifolds.


Fig. 12

This theorem follows from the corresponding assertion for smooth strictly pseudoconvex domains (cf. Khenkin-Tumanov (1976), Nagel (1976), Rudin (1978)) using Lemma 2. A similar proof was proposed by Saerens (1984). There is a simple proof of the theorem in the real analytic case proposed by Bums-Stout (1976). Let M be a real analytic C-spacelike hypersurface in [Rm and !VI its complexification. By the hypothesis on M there exists a neighborhood W of M in !VI such that W n fC = M (cf. Fig. 12). Let J be a holomorphic function on W equal to a given real analytic function f on M. Since W is a complex submanifold in a neighborhood of fC in em, by Cart an's Theorem J can be extended to a function holomorphic in a neighborhood of fC. We have proved that any real analytic function f on M can be extended to a function holomorphic in a neighborhood of fC. Using Theorem 15 it is easy to prove now that M is an interpolation manifold. Conversely, if M is a real-analytic interpolation submanifold of [Rm then it has no tangent C-like vectors. Indeed, assume the opposite and consider an arc of a real-analytic C-like curve on M. Then (by Theorem 15) there exists a function f E A(TC) which is equal to zero on this arc and f(z) oF 0 at other points of fC. But the complexification of the curve (because it has C-like tangent vectors) has non-void intersection with T C• Thus the zero set of f intersects T C• Contradiction.

A partial converse to Theorem 17 can be proved also in the smooth case. Namely, if a C2-smooth hypersurface M on [Rm is an interpolation manifold for A(Tc) then it has no C-like tangent vectors (Saerens (1984)).

A result combining the Forelli Theorem with Theorem 17 was proved by Labonde (1985).


Boundary values in the space H2(TC) were studied by Bochner (1944), and further results on boundary values in Hardy spaces HP(TC ) were given in SteinWeiss (1971). The assertions on the spaces H(S)(C) and H(C) given in Sect. 1.2, 2.1 are contained in Vladimirov (1979) (where also more general spaces mS)(C),


and Ha( C) with exponential scale of type a at infinity along the imaginary space are considered; we restricted ourselves for the sake of simplicity to the case a = 0). For further information on hyperfunctions and their boundary values cf. Schapira (1970), Morimoto (1973), Sato-Kawai-Kashiwara (1973). Hyperfunctional boundary values in tuboids were considered in Zharinov (1983). Fatou and Lindeloftheorems were considered in many papers (cf. Chirka (1973), Chirka-Khenkin (1975), other references are given in the Notes to Chap. 4). New variants of Fatou and Lindelof theorems formulated in Sect. 3.2 were proved in Drozhzhinov-Zav'ialov (1982) and Khurumov (1983). The results of Sect. 3.3,4.1 are parallel to those for the polydisc (Rudin (1969». The general properties of inner functions were studied in Aleksandrov (1984, 1983), Low (1984), Rudin (1980). The results of Sect. 5.1 are analogous to those for the polydisc (Rudin (1969». For the description of interpolation manifolds in Sect. 5.2 see BurnsStout (1976), Stout (1981), Saerens (1984), Sergeev-Vladimirov (1985), and Sergeev (1989).

Chapter 3 "Edge-of-the-Wedge" Theorem and Related Problems

§ 1. "Edge-of-the-Wedge" Theorem

1.1. Theorem of Bogolubov. This theorem was announced by Bogolubov at the International Conference in Seattle (September 1956) (the first detailed proof was published in Bogolubov-Medvedev-Polivanov (1958». We formulate it in the form convenient for our goals. Denote temporarily a cone C by C+, the opposite cone - C by C- and by Q a domain in [Rm.

Theorem of Bogolubov. Let f+ E (9(.@+) be functions of locally slow growth (cf. Chap. 2, Sect. 2.3) in local tubes.@± =-Q + iCk'. Suppose that their boundary values (in the distributional sense) coincide on Q. Then there exists a complex neighborhood Q of Q (cf. Fig. 13) and a function f which is holomorphic and has locally slow growth in Q u.@+ u.@_ equal to f± on .@±.

Note that the neighborhood Q does not depend on f and is described rather explicitly (cf. Vladimirov (1964». The "Edge-of-the-Wedge" Theorem of Bogolubov has generated many generalizations, first in quantum field theory and then in several complex variables. Now (along with different variants and generalizations) it constitutes, in fact, a separate chapter in the theory of functions of several complex variables. We wish to emphasize, in particular, its close relation to the local Bochner Tube Theorem (cf. Chap. 1, Sect. 5.2) and the theorem on separate analyticity (Siciak (1969), Zakharyuta (1976».


Fig. 13

Fig. 14

1.2. Theorem of Martineau. We formulate here one of the generalizations of Bogolubov's Theorem proved by Martineau (1970).

Theorem of Martineau. Let fk E (!)(E&k) be functions of locally slow growth given in local tubes E&k = Q + iC~, k = 1, ... , N, over a domain Q in IRm. Suppose that the boundary values fk(x) of fk(z) on Q (in the distributional sense) satisfy the following condition

N

L fk(x) = 0, X E Q. k=1

Then there exists a complex neighborhood {j of Q and functions !jk, j, k = 1, ... , N, which are holomorphic and of locally slow growth in the domains

. jk -E&jk = (Q + zCR ) n Q,

where Cjk = ch(Cj U Ck) is the convex hull of the cones cj and Ck (cf. Fig. 14), and satisfy the following conditions:

1) !jk = -fkj,j, k = 1, ... , N; N

2) J;,(z) = L J;,iz), z E E&k n {j. j=1


Other generalizations of the "edge-of-the-wedge" theorem will be given in Sect. 3.l.

§ 2. "C-convex Hull" Theorem

2.1. "C-convex Hull" Theorem. Consider again Bogolubov's Theorem which we reformulate in another form. Consider a "unified" function f in the domain ~ = ~+ u ~_ u Q which is holomorphic in ~+ u ~_ and belongs to the space ~'(Q) on Q (in other words, the boundary values f±(x) of f on ~m from ~+ and ~ _ exist in the distributuional sense and coincide on Q). Bogolubov's Theorem provides a holomorphic extension of any such function along "imaginary directions" into the domain ?J = ~+ u ~_ u ii. From this point of view, Bogolubov's Theorem gives an estimate of the holomorphic hull of ~ with respect to functions of locally slow growth near ~m. It appears that sometimes we can considerably improve this estimate using the extension along "real directions".

Namely, denote temporarily a cone C+ u C- by C and call a C1-smooth curve in ~m C-like if its tangent vectors at each of its points x belong to the cone x + c. The C-convex hull BdQ) of Q is the convex hull of Q with respect to C-like curves, more precisely, the smallest open neighborhood of Q in ~m satisfying the following condition: along with any arc [x', x"] of a C-like curve it contains also the "diamond" (x' + C+) n (x" + C-) (we suppose that the parameter on a curve is chosen in such a way that a tangent vector at an arbitrary point x "points to the future", i.e. belongs to x + C+) (cf. Fig. 15).

"C-convex Hull" Theorem. Let a function f be holomorphic in a domain of the form ~+ u ~_ u ii where ~± = Q + iCi and ii is a complex neighborhood of Q. Then f is extends to a holomorphic function in a domain

~

~+ u~_ uBdQ) ~

where BdQ) is a complex neighborhood of the C-convex Hull BdQ) of Q.

This theorem was proved by Vladimirov (1960, 1961). Other proofs and extensions for the case of the light cone C = V can be found in Vladimirov

Fig. 15


(1964), Borchers (1961), Araki (1963); variants of this theorem for classes of ultradistributions and hyperfunctions were obtained by Beuding (1972) and Morimoto (1973). From the "C-convex Hull" Theorem we can deduce an interesting quasi analytic property of distributions. Namely, denote by L(C) the class of distributions f E g"(lRm ) represented as the jump f(x) = f+(x) - f-(x) of boundary values of functions f± E H(C±). This class consists precisely of functions whose Fourier transforms vanish outside the cone C* = (C+)* u (C-)* (cf. Vladimirov (1964)).

Theorem I (Vladimirov (1964)). If a function f E L(C) vanishes on an open set Q c IRm then it vanishes also on its C-convex hull BdQ).

2.2. Holomorphic Hulls and Dyson Domains. It is natural to consider, in connection with the "Edge-of-the-Wedge" and "C-convex Hull" theorems, the problem of describing holomorphic hulls of domains of the form f12+ u f12_ u Q. This problem is not solved in general but there is one particular case, important for physical applications, when it is possible to obtain a simple description of the above holomorphic hull. Namely, consider domains of the form f12 = f12(Q) = r+ u r- u Q where the domain Q lies between two spacelike hypersurfaces (cf. Chap. 2, Sect. 3.1). For any domain f12 of this type we can construct its holomorphic hull .@(Q) in en+1 in the following way. We call a complex hyperboloid {z E I[:n+1 : (z - U)2 = A.2} where u E IRn+l, it E IR, admissible for Q if its real section does not intersect Q (cf. Fig. 16). Denote by .@ = .@(Q) the domain in en+1 obtained by deleting all complex hyperboloids admissible for Q. Then.@ is a domain of holomorphy which is called the Dyson domain associated with Q. We show that.@ ::> f12. It is sufficient to prove that.@ ::> r = r+ u r-. If this is not so then (z - U)2 = it 2 for z E r and some u E IRn+1, it E IR. This equation is equivalent to two equations (x - uf = y2 + it 2 , (x - u)· y = O. It follows from the first equation that (x - U)2 > 0 which contradicts the second equation. Thus .@ is a domain of holomorphy (in fact, a polynomially convex domain) containing f12. The natural question is whether it coincides with the holomorphic hull of f12. The positive answer to this question follows from a theorem proved in Vladimirov

Fig. 16

" / " / 11)( / "

/ "

(1964), § 33, with the help of the Jost-Lehmann-Dyson integral representation (cf. Chap. 4, Sect. 3.1) and a theorem of Pflug (1974) (cf. also Bros-Messiah-Stora (1961)). The Jost-Lehmann-Dyson representation allows one also to describe the holomorphic hull of domains!!) = TC+ u TC- u 12 where C+ = C, C- = - C and 12 = (C+ + a) u (C- - b) where a, be C+ (or, more generally, 12 is an nseparated set in the sense of Vladimirov-Zharinov (1970)). In this case the holomorphic hull fij is described as above by using admissible complex hyperplanes defined in analogy with admissible hyperboloids (cf. Vladimirov-Zharinov, op. cit.)

There is one more interesting result connected with Dyson domains. It is the "Finite Covariance" Theorem proved by Bogolubov-Vladimirov (1958) for so called I-point functions. Let f be a holomorphic function in the Dyson domain ,+(3) u ,-(3) u J associated with the domain J = g E ~4: x2 < O} and let f E

k

H(V+ u V-). Then f(z) = L &'v(Z)fv(Z2) where &'V are polynomials, fv(O-v=1

functions of a single complex variable , holomorphic and of slow growth on the complex plane 1(;1 slit along the positive real half-line. This theorem was extended in Bros-Epstein-Glaser (1967) and Bogolubov-Vladimirov (1971) to so called N-point functions J, when the tube ,± is replaced by the direct product '& = ,± x ... x ,± (N times) in 1(;4N, assuming that the "extended future tube conjecture" is true. This conjecture asserts that the extended future tube ,~ (to be defined) is a domain of holomorphy. The domain ,~ consists of points in 1(;4N

which can be represented in the form (/\ Z1, ... , /\ ZN) where (Z1, ... , ZN) E ,~, /\ is a transformation from L+(C), the proper complex Lorentz group (or the component of the identity of the complex Lorentz group). The extended future tube conjecture still remains unproved for N ~ 3 (cf. review articles of Vladimirov (1970,1982, 1983a)). The compact version of this conjecture (where ,+ is replaced by the generalized unit disc B2 and the Lorentz group L+(C) - by the group SL(2, C) x SL(2, C)) is proved in Heinzner-Sergeev (1991).

§ 3. Analytic Representations

3.1. Decomposition of Hyperfunctions in Tuboids. Extensions of the "Edge-ofthe-Wedge" Theorem. A decomposition theorem was already formulated in Sect. 2.l: a function f E 9"(~m) can be represented as the jump f{x) = f+{x) - f-{x) of boundary values of functions f± E H(C±) if its Fourier transform vanishes outside (C+)* u (C-)* (this assertion can be extended also to the RiemannHilbert problem in T C+, cf. Vladimirov (1965)). We give here some generalizations of this result.

We define the microlocal singular support SS(f) (the singular spectrum in the terminology of Sato-Kawai-Kashiwara (1973)) of a hyperfunction f E [11(12) where 12 is open in ~m as the complement of the set of points (x, 0) E 12 x sm-1


having the following property. A point (x, cr) f!. SS(f) if there exists a neighborhood U of x such that for some collection of local tubes flfiv = U + iC;', v =

k

1, ... , k, such that cr f!. U (C)* (cf. Fig. 17) there exist functions fv E (o(flfiv) such v=l

that we have the representation

k

f= L bv fv· v=l

Let now flfi be a tuboid with profile /\ over Q. Denote by /\* the profile dual to /\, i.e. /\~ := (/\x)* for x E Q and define pr /\ to be the subset of Q x sm-1

of the form

Theorem 2 (Sato-Kawai-Kashiwara (1973), Zharinov (1983), Morimoto (1973)). A hyperfunction f E 31(Q) can be represented as the sum of functions Iv E (o(flfiv) holomorphic in tuboids flfiv = flfi.(/\v), v = 1, ... , k, over Q

k

f = L bv fv v=l

if and only if its microlocal singular support SS(f) is contained in pr /\~ U··· u pr /\t.

The space of hyperfunctions on Q having microlocal singular support contained in the projection pr /\ of a profile /\ = /\ (Q) is denoted by fJ6(Q, /\).

Fig. 17

The microlocal variants of Bochner's Tube Theorem (Chap. 1, Sect. 5.2) and of Bogolubov's "Edge-of-the-Wedge" Theorem (Sect. 1.1) follow immediately from Theorem 2. We formulate now a microlocal version of Martineau's Theorem (Sect. 1.2).

Theorem 3 (Zharinov (1983), Morimoto (1973)). Let fv, v = 1, ... , k, be functions holomorphic in tuboids ~v = ~vV\v) over Q. If

k

f= L bv fv = 0 v=1

on Q then there exist functions fl'v = - fvl' holomorphic in some tuboids ~I'V = ~Vl' with profiles /\I'V = ch(/\I' U /\v) such that

k

fv(z) = L fl'v(z), z E ~v n ~~, v = 1, ... , k, 1'=1

k

where ~~ = n ~I'v' 1'=1

There is a more general formulation of the "Edge-of-the-Wedge" Theorem due to Zharinov (1980, 1983) which implies the Theorems 2 and 3. Namely, let /\ be a fiber convex profile over an open set Q in [Rm (Chap. 1, Sect. 5.2). Denote by CD(Q, /\) = lim CD(~(/\)) the inductive limit of the spaces CD(~(/\)) with respect to all tuboids fiJ(/\) with profile /\ over Q. In other words, CD(Q, /\) consists of functions "holomorphic in directions from /\". In Chap. 2, Sect. 2.2 we defined the boundary value map bv : CD(Q, A) -+ e4(Q) assigning to a function f E CD(~(/\)) a hyperfunction bv f. By Theorem 2 we have bv CD(Q, /\) =

e4(Q, /\*). Consider now a more general situation. Let /\1' ... , /\N be a collection of fiber convex and fiber proper profiles over Q. Denote by /\v, ... vp ' 1 ~ p ~ N, the profile ch (/\v, U··· U /\v,), 1 ~ VI' ••• , Vp ~ N. Introduce the space CDp(Q, {/\v}) of p-chains with respect to {/\v} consisting of collections f =

{fv, ... vp } of functions fv, ... Vp E CD(Q, /\ v, ... v) skew-symmetric with respect to permutations of the indices VI' ••• , vp- Define now the boundary operator bp: CDp(Q, {/\v}) -+ CDp- 1(Q, {Av}) by theformula

N

(bpf)v, ... vp _, = L fvv, ... vp ., for f = {Jv, ... vp } E CDp(Q, {/\v})' v=l

Using, as above, the boundary value map bv, we introduce the spaces of pchains Ap(Q, {/\v}) = bv CDp(Q, {/\v}) and extend in the natural way the action of bp to these spaces. Then the following generalized "Edge-of-the-Wedge" Theorem is true.

Theorem 4 (Zharinov (1983)). The homology sequence

0<- e4 ( Q, vV1 /\~) ti. Al (Q, {/\v}) ~ ... ~ AN(Q, {/\v}) <- 0

is exact.

Theorem 2 is equivalent to the exactness of this sequence in the term !!4, Theorem 3 - in the term A 1 .

3.2. Generalized Fourier and Radon Transforms. The micro local singular support of a distribution can be defined as for hyperfunctions, however we prefer to give here another equivalent definition in terms of the generalized Fourier transform which has proven to be important for applications in several complex variables.

Let f E 9"(lRm). The generalized Fourier transform or FBI-transform of f is defined at a point XO E IRm by the formula (Bros-Iagolnitzer (1974-75))

Fxo(~, ~O) = (2nrm/2 f f(x)e-i(~,X)-~O'X-xo'2 dx D;lm

where ~ E [Rm, ~o > O. The function FA~, ~O) is defined for ~o > 0 and satisfies the estimate

for some constants M > 0, 0(, /3:2: O. Its boundary value in 9" for ~o --+ 0 coincides with the usual Fourier transform of f. (We can generalize this definition replacing the function Ix - xOl 2 in the exponent by a function of more general form, cf. Bros-Iagolnitzer, op. cit). The FBI-transform has many properties of the usual Fourier transform, in particular there is an inversion formula for this transform.

We define now the microlocal singular support SS(f) of a distribution f E

9"(lRm) (the essential support in the terminology of Bros-Iagolnitzer, op. cit.) as the complement of the set of points (x, 0") E IRm x sm-l having the following property. A point (x, 0") ¢ SS(f) if there exists a conical neighborhood C of 0" such that for some A, 0(, /3, y > 0 the following estuimate is true

for ~ E C, 0 < ~o ::;; AI~I. Setting here ~o = AI~I we obtain that a point (x, 0") ¢ SS(f) if FJ~, AIW decreases exponentially for I~I--+ 00 in a conical neighborhood of 0". (This definition can be extended to functions of the class .s&'(Q), Q c IRm, using a "cut-function", i.e. a COO-function with compact support which is real-analytic in a neighborhood of the considered point x).

Thus defined, the microlocal singular support of a distribution coincides with its microlocal singular support in the sense of the boundary value mapping defined in Sect. 3.1, and with its analytic wave front set in the sense of Hormander (1971) (The equivalence of these three definitions is proved in Bony (1976)). The decomposition theorems for hyperfunctions given in Sect. 3.1 have their analogues for distributions and can be reformulated in terms of the FBItransform (cf. Bros-Iagolnitzer, op. cit, and Zharinov (1983)).


With the generalized Fourier transform, as in the case of the usual Fourier transform, is closely connected the generalized Radon transform. Just as the usual Radon transform is based on the decomposition of the b-function through "plane waves", the generalized Radon transform is based on the decomposition of the b-function through "curvilinear waves"

b(x) = (m - 1)! r [1 + (x, e)]w'(e)

(-2ni)m Jsm-' [(x, e) + iJxJ 2 + io]m

where the kernel [(x, e) + iJxJ 2 + iOrm is defined as the limit in the sense of distributions of the functions

[(x + iy, e + i1]) + i(x + iy, x + iy)rm

as y ~ 0, 1] ~ 0 and

This formula was proved in Bony (1976); a similar formula was proved earlier in Sato-Kawai-Kashiwara (1973), p. 473. It is possible to reformulate the definition of the microlocal singular support and the decomposition theorems of Sect. 3.1 using the generalized Radon transform (cf. Kataoka (1981)).

3.3. Factorization of Hyperfunctions. The multidimensional factorization problem (cf. Sergeev (1978)) is a multiplicative analogue of the decomposition problem for hyperfunctions considered in Sect. 3.1. The results of Sect. 3.1 can be partially extended to this problem. Let U be a convex open subset of IRm and o a complex convex neighborhood thereof in en. The space of multiplicative hyper functions ~*(U) (cf. Sergeev (1975)) is by definition the cohomology group Hm - 1(0\ U, 0*) with coefficients in the sheaf 0* of holomorphic functions without zeros. (This definition does not depend on the choice of 0 and defines a sheaf ~* of multiplicative hyperfunctions on IRm. The space ~*(.Q) for any open subset Q of IRm consists of sections of the sheaf ~* over Q).

Using, as in Chap. 2, Sect. 2.2, different special coverings of 0\ U, we can represent a multiplicative hyperfunction by a collection of 2m functions f. E

0*(0.) or by a collection of m + 1 functions !j, ... im E 0*(~' ... iJ (defined up to multiplication by an (m - 2)-coboundary). This allows us to define the boundary value map bv*(f) for functions from 0*(Tc nO). The exponential map o ~ 0* induces an exponential map ~ ~ ~* which can be included in the exact sequence

o ~ 71. ~ ~(U) ~ ~*(U) ~ 1

for convex U. Hence the assertions of Sect. 3.1 being of microlocal character can be extended to the multiplicative case using the above exact sequence and its analogues.



Problems related to Bogolubov's "Edge-of-the-Wedge" Theorem are considered, for example, in the books and review articles: Vladimirov (1965, 1969, 1983a, 1971, 1982), Zharinov (1983), Morimoto (1973). On the "C-convex Hull" Theorem cf. Vladimirov (1964, 1983a, 1971, 1982), Morimoto (1973). In the exposition of Sect. 3.1 we followed Zharinov (1983). The generalized Fourier transform and its properties were studied in Bros-Iagolnitzer (1974-75), the generalized Radon transform - in Kataoka (1981). The multiplicative theory of hyperfunctions was considered in Sergeev (1975).

Chapter 4 Integral Representations

§ 1. Cauchy-Bochner Integral Representation

1.1. Cauchy-Bochner Integral in Tube Cones. The Cauchy kernel of a tube cone T C is defined as the Laplace transform of the characteristic function (}c* of the dual cone C*, i.e.

The Cauchy kernel is evidently holomorphic in T C• There is an other representation for this kernel (cf. Vladimirov (1979))

f da Jfdz) = im F(m) ~)m' Z ETC,

prC' (a, z (1)

where pr C* = C* n sm-l (cf. Chap. 3, Sect. 3.1). As both sides of (1) are holomorphic in TC, to prove (1) it is sufficient to prove it, say, for z = iy, Y E C. In this case we have

Jfdiy) = f e-(Y.~) d~ = r dO' foo e-p(y,G) pm-l dp c· Jprc. 0

= r ~ foo e-uu m - 1 du = im r(m) r ~ Jprc* (y, at 0 Jprc* (iy, a)m'

q.e.d. It follows from (1) that the Cauchy kernel fdz) is in fact holomorphic in a

larger domain containing T C and T-c, namely in the domain

Cm \ U {ZECm:(z,a)=O}. O'EprC*

This domain contains, besides T C and T-c, also real points belonging to the cones C and - C in IRm.

In the case of the future tube r+ = r+(n) the Cauchy kernel has the form (cf. Bochner (1944), Vladimirov (1979))

2nn(n-1)/2 r( n ; 1) .)f; (z) - Z2 = Z02 - Z21 - ••• - zn2 • (2) v+ - (_z2)(n+1)/2

In the case of the tube cone T+ over the octant IR~ the Cauchy kernel is the direct product of the usual Cauchy kernels

im r~,;,,(z)=---

Zl··· Zm

The Cauchy kernel for a general tube cone TC satisfies the following estimate

IDa rdz) I ::s; Am~~(y)' z E T C

where A (y) is the distance from y to the boundary of the cone C

A(y) = inf (a, y). aeprC*

In the norm of the space ~ (cf. Chap. 2, Sect. 1.2) the Cauchy kernel is estimated by

1 + A-S(y) \IDa rdz) \I ~ ::S; Ma.s Am/2 +lal(y) ' Z E T C

with the usual multi-index notations. It follows from the last estimate that rdz) has a boundary value, as y -+ 0, y E C, in each space ~ with s < - m12, which coincides with the Fourier transform of the characteristic function ec" Using the representation (1) we obtain the following formula for rdx)

rdx) = n( - i)m-1 f (j(m-1)((x, a)) da - (- ir f gp(m-l) (_1_) da (3) prC' prC' (x, a)

where gpG) is the principal value in the Cauchy sense (cf. Vladimirov (1979)).

The importance of the Cauchy kernel is explained, in particular, by the following theorem.

Theorem 1. A function f is in H(S)(C) (cf. Chap. 2, Sect. 1.2) if and only if it is given by the Cauchy-Bochner integral representation

f(z) = (2~)m f ~m rdz - t)f(t) dt = (2nrm rc * f(z), z E T C (4)

where f(t) is the boundary value off in £S.

226 A.G. Sergeev, V.s. Vladimirov

This theorem was proved for s = 0 in Bochner (1944), the general case was considered in Vladimirov (1969a, 1979).

1.2. Cauchy-Bochner Integral for Classical Domains. We give here explicit formulas for the Cauchy kernel of some classical domains. In the case of the cone Yfl of positive definite Hermitian I x I-matrices (Chap. 1, Sect. 5.1) the Cauchy kernel has the form (Bochner (1944))

f (Z) = 1(1-1)/2 ,,21 ! ... (1- 1)! .1!'i n I (det Z)I

The Cauchy-Bochner integral in the representation (4) in this case is taken over the space of Hermitian I x I-matrices. Analogous explicit formulas are known for the cones [lJ>1 and QI from Chap. 1, Sect. 5.l.

We present also Cauchy-Bochner representations for the generalized unit disc (Chap. 1, Sect. 2.2) and the Lie ball (Chap. 1, Sect. 2.3). In the case of the generalized unit disc Bm the Cauchy-Bochner representation for holomorphic functions has the form (Bochner (1944), Hua (1958))

Z _1!·2!·····(m-1)! r f(X)w(X) f( ) - (2n)m(m+1)/2 J Un [det(l _ zx*)]m' Z E Bm

where w(X) is the volume form on the space of unitary m x m-matrices Um, w(X) = [Tr(dX /\ dX*)] Am, i.e. the m-th exterior power of the form Tr(dX /\ dX*).

For the Lie ball this representation takes the form (Hua (1958))

r(~) r f(u)e- i(n+1)9 w(u) f(Z) = 2n(n+3)/2 J SL [(u - z, u - z)]<n+1)/2

= .r(~) i" dO i f(x,O)w(x) z E BL I 2n(n+3)/2 [(xei9 _ Z xei9 _ z)](n+1)/2' o Ixl=l ,

where u = xe i9 E SL' w(u) and w(x) are the volume forms on SL and sn respectively.

1.3. Hilbert Transform. The Hilbert transform of a function f E ~ is a function of the form

(Hf)(x) = - (2~)m fRm 1m fdx - t)f(t) dt

2 = - (2nr 1m fc * f(x), x E ~m.

(Here the integral should be considered as the convolution of the distributions 1m fdx) and f(x)) (cf. Vladimirov (1979)). The explicit expression for the kernel

1m .Jf"dx) is given by formula (3). The Hilbert transform of f E ~ belongs again to ~ and its support is contained in C* u ( - C*). The function f is expressed through Hf by the same integral as above but with the" + " sign before it.

Theorem 2 (Vladimirov (1969a, 1979)). If f(x) is the boundary value of a function f E H(S)(C) then its real and imaginary parts are connected by the Hilbert transform

1m f(x) = (2~)m fIRm 1m .Jf"dx - t) Re f(t) dt = (2~)m 1m .Jf"c * Re f(x), x E IRm.

Such formulas are also called dispersion relations.

1.4. Estimates of the Cauchy-Bochner Integral. Denote by Kcf(z) the Cauchy-Bochner integral given by the right hand side of (4) and consider it as an integral operator acting on functions f(t) defined on IRm. Then the following Theorem holds.

Theorem 3 (Vladimirov (1979)). The Cauchy-Bochner integral Kcf(z) of a function f E ~ satisfies the following estimate in the ~-norm

In particular, the Cauchy-Bochner integral operator Kc is bounded as an operator

Kc:~--+~·

This result is true for any tube cone T C (with an open convex proper cone C). But if we pass from the Hilbert spaces ~ to the Banach spaces Lp (or the Lipschitz spaces A~) the estimates will depend essentially upon the tube cone considered. To see this, we give some estimates for the Cauchy-Bochner integral in the future tube and compare them with the corresponding results for the tube cone T+ = TIR'J' over the octant. Denote temporarily the Cauchy-Bochner integral for the future tube r+ = r+(n) in !C"+1 by K and the Cauchy integral for T+ in Cn+1 - by K o.

First, we consider estimates in Lp spaces with 1 < p < 00. For p = 2, as we know from Theorem 3, K and Ko are bounded in L 2 (lRn+1 ). For 1 < p < 00

the same assertion is true for K o but not for K. It follows from a theorem of Fefferman (1970), that the Cauchy-Bochner integral K is unbounded in Lp(lRn+1) for any p i= 2, 1 < p < 00 (cf. Stein (1971)).

The difference between K and K o becomes perhaps even more clear if we consider the behavior of these operators on the space L;'(lRn+1), n ;;::: 2, of essentially bounded functions on IRn+1 with compact supports. But first we need to define the types of estimates we shall consider. Usually we prove estimates for Kf(z) = Kf(x + iy) at points x of the distinguished boundary when y belongs to a compact subcone C of V+. We call such estimates conical. More precisely, we say that a function g(z) holomorphic in r+ satisfies some estimate (*) at a point Xo E IRn+1 if for any compact subcone C of V+ and for any r, 0 < r :::; ro, the

estimate (*) is satisfied for z = x + iy E r+ such that Ix - xOI < r, Iyl < r, y E C, with a constant depending on rand C. At points ZO = XO + iyo of the smooth part S of the boundary or+ we consider estimates of another type. We say that a function g satisfies some local estimate (*) at a point ZO E S if for any r, 0< r s ro, the estimate (*) is satisfied for z = x + iy E r+ such that Ix - xOI < r, Iy - yOI < r, with a constant depending on r. Now we can formulate the estimates for K in L~(/Rn+1 ).

Theorem 4. The Cauchy-Bochner integral Kf = %v+f of a function f E

L;'(/Rn+1) given by formulas (4), (2) satisfies at any point XO E /Rn+1 the following conical estimate

f IlfllLoo I K (z)1 s M (n-1)/2'

Yo

At points ZO E S it satisfies the local estimate

I Kf(z) I s MllfIILoo·llnly _ yOII·

These estimates are sharp.

(5)

(6)

The estimate (5) was proved in Joricke (1983), estimate (6) - in Sergeev (1986). Estimates analogous to (5) were proved also for the Lipschitz spaces Aa and for some classical domains (cr. Joricke, op. cit., and Mitchell-Sampson (1982).

We see from (5) that I Kf(z) I could grow like a power of Yo when z = x + iy-+ /Rn+1. For K o it is well known that I Kof(z) I can grow only logarithmically when z -+ /Rn+1•

1.5. Schwartz Representation. We call an open convex proper cone C regular if its Cauchy kernel %dz) is a divisor in the algebra H( C), i.e. 1/ %dz) E H( C). All cones from the examples given in Chap. 1, Sect. 5.1 are regular. It can be proved also that any open proper convex cone C in /Rm with m s 3 is regular; for m > 3 this is not true (Danilov (1985». The Schwartz kernel of a tube cone T C with regular C is the function

° 2%dz)%d -ZO) ° ° ° C Y'dz, z ) = (2n)m %dz _ ZO) - 2Pdx , y), z, Z E T (7)

where 2Pdxo, yO) is the Poisson kernel for T C (cf. Sect. 2.1 below). For ZO = z the Schwartz kernel coincides with the Poisson kernel.

For the future tube r+ = r+(n) the Schwartz kernel is given by (Vladimirov (1979»

r(n + 1) [ -(z _ zO)2Jn+1)/2

° 2 ° ° Y'y+(z, z ) = n(n+3)/2( _z2)(n+1)/2[ _(zO)2](n+1)/2 - £i7lv+(x , y )

(The explicit formula for 2Py+ is given in Sect. 2.1). In the case of the octant C = /R';' the Schwartz kernel is given by

° 2im (1 1 ) ( 1 1 ) ° ° Y'Qlm(Z, z ) = -- - - -::::- ..... - - =- - 2PQlm(X , y ) + (2nt Zl z? zm z~ +


(for Y'~,:, cf. Sect. 2.1). In particular, for m = 1 we have

Y'~ (z, ZO) = ~(! -~) + n z \ZO\2

and Re Y'~Jz, ZO) coincides with the usual Poisson kernel Y'(x, y) on the plane.

Theorem 5 (Vladimirov (1979)). Let a function f E H(c),for a regular cone C, satisfy the condition: f(x)fdx - ZO) E £. for some s and all ZO E T C (where f(x) is the boundary value of f in 51"). Then f has the Schwarz representation

f(z) = if Y'dz - t, ZO - t) 1m f(t) dt + Re f(zO), z, ZO E T C

~m

where the integral should be considered as the value of the distribution functional 1m f(t) on the function Y'dz - t, ZO - t).

§ 2. Poisson Integral Representation

2.1. Poisson Integral in Tube Cones. Let C be a convex open cone in [Rm. The Poisson kernel of the tube cone T C is the function

\fdx + iyW Y'dx, y) = (2nt fd2iy) ' x + iy E T C (8)

where fdz) is the Cauchy kernel (cf. Sect. 1.1). The ,Poisson kernel &'dx, y) is non-negative in T C and satisfies:

f Y'dx, y) dx = 1, y E C, r &'dx, y) dx ~ 0, as y ~ 0, Y E C ~m Jlxl>o

for any (j > 0; i.e. it has properties analogous to the ones for the usual Poisson kernel (cf. Vladimirov (1979), Stein-Weiss (1971)).

The Poisson kernel for the future tube ,+ = ,+(n) is given by the formula (Vladimirov, op. cit.)

In the case of C = [R~ the Poisson kernel is the product of the usual Poisson kernels

The Poisson kernel for a general tube cone satisfies the estimates (Vladimirov, op. cit.)

IIDlXmJ ( )11. < M [1 + ,rS(y)] [1 + A-P(y)] I 1m TC ;7C x, Y X. _ IX,S,p Am+llXl(y) y , X + iy E

for some constants MIX' MIX,s,p > 0 and for all s ~ 0, p > s + ~, using the usual

multi-index notations (A (y) is the distance of y from cC).

Theorem 6. A function f is in H(S)(C) if and only if it can be given by the Poisson integral representation

f(z) = f f!J.dx - t, y)g(t) dt = f!J.c * g(z), z = x + iy E T C (9) H;lrn

where g E £. and the support of the Fourier transform F- 1 [g] belongs to C*. Iff is given by formula (9) then g(t) coincides with the boundary value f(t) off in £.. Accordingly, Re f(z) (resp. 1m f(z)) is given by formula (9) with g(t) = Re f(t) (resp. 1m f(t)).

This theorem was proved in Vladimirov (1979), the case s = 0 - in SteinWeiss (1971).

2.2. Poisson Integral in Classical Domains. We shall now give explicit formulas for the Poisson kernel of some classical domains. In the case of the cone £; of positive definite Hermitian I x I-matrices (Chap. 1, Sect. 5.1) the Poisson kernel has the form

[ det Y J' ..m f!J.Jfl(X, Y) = c, Idet Zl2 ' Z = X + lY E T I

where

The Poisson integral in the representation (9), in this case, is taken over the space of Hermitian I x I-matrices. Analogous formulas are known for the cones f!J., and Q, from Chap. 1, Sect. 5.1.

In the case of the generalized unit disc Bn (Chap. 1, Sect. 2.2) the Poisson representation takes the form (Hua (1958))

1! .... '(m - 1)! r [det(l- zz*)]m f(Z) = (2nr(m+1)/2 J Urn f(X) Idet(Z _ X)12m w(X), Z E Bm

where w(X) is the volume form on Um (cf. Sect. 1.2). For the Lie ball (Chap. 1, Sect. 2.3) this representation is given by (Hua (1958))

( n + 1) _ r -2- i [1 + I(z, zW - 2IzI2]<n+l)/2

fez) - 2 (n+3)/2 feu) I( w+1 w(u), n SL u - z, u - z

where w(u) is the volume form on the Lie sphere SL (cf. Sect. 1.2).

2.3. Boundary Properties of the Poisson Integral. Denote by Pcf(z) the Poisson integral given by the right hand side of (9) and also denote by Pc[dJ.L](z) the Poisson integral of a complex-valued Borel measure J.L on IRm given by the same formula (9). We formulate in the next theorem some boundary properties and estimates for the Poisson integral in different spaces.

Theorem 7. 1) Iff E ~ then

IlPdx + iy)11 Jfs ::; Ilfll Jfs' y E C.

Moreover,

Pcf(x + iy) - f(x) in ~ for y - 0, Y E C.

2) If f E U(lRm), 1 ::; p < 00, then

IlPcf(x + iy)IILP ::; IIflb for any y E C

and

Pcf(x + iy) - f(x) in U for y - 0, Y E C.

3) If f E L oo(lRm) then

IPcf(z) I ::; IlfIILoo for z E T C

and

Pcf(x + iy) - f(x) for y - 0, Y E C

in the weak* topology of the space L 00. If, in addition, the function f is continuous at a point x then

Pcf(x + iy) - f(x) for y - 0, Y E C

4) For a finite Borel measure J.L on IRm

IlPc[dJ.L] (x + iy)llL' ::; IIJ.LII = f IdJ.Ll. IJ!"'

Moreover,

Pc [dJ.L] (x + iy) - J.L(x) for y - 0, Y E C

in the weak* topology of the measure space.


The assertions of this Theorem follow from the basic properties of the Poisson kernel given in Sect. 2.1. Assertion 1) was proved in Vladimirov (1979); assertion 2) - in Koranyi (1965), Stein-Weiss (1971); assertion 3) - in Vladimirov (op. cit.), Koranyi (op. cit.); assertion 4) - in Koranyi (op. cit.).

The problem of pointwise convergence of Poisson integrals is much more complicated. Let us first define the sets of approach. We say that z = x + iy E

T C tends to XO in the restricted admissible sense if z -+ XO staying inside the set

for some constant IX > 0 and some compact subcone C' in C. Most results on the pointwise convergence of Poisson integrals have been

proved in the homogeneous case so we restrict our attention now to tube cones T C corresponding to the classical domains (cf. Chap. 1, Sect. 5.1).

Theorem 8 (Stein-Weiss (1969), Weiss (1972), Stein-Weiss (1971)). If f E

U(lRm), 1 ~ p ~ 00, then its Poisson integral Pcf(z) converges to f(xO) for almost all XO E IRm when z -+ XO in the restricted admissible sense. If p, is a finite Borel measure on IRm such that its absolutely continuous part (with respect to dx) is equal to f(x) then Pc [dp,] [z] converges to f(xO) for almost all XO E IRm when z -+ XO in the restricted admissible sense.

A further generalization of the notion of restricted admissible convergence was proposed by Koranyi (1969, 1972). His definition uses, in full strength, the theory of semisimple Lie groups so for the precise formulations we should require much background material. To avoid this, we prefer to give here only a sketch of his results referring for the details to the papers of Koranyi (1969,1972, 1976, 1979) and Stein (1983).

For a bounded symmetric domain it is possible to construct several (in general) different compactifications. To each ofthese compactifications corresponds its own distinguished boundary and Poisson integral and the notion of the restricted admissible limit can be introduced at points of the distinguished boundaries of these compactifications in an invariant way using approaching sets of the type Fc.(xO). Theorem 8 remains true for any compactification of a bounded symmetric domain as was proved by Stein (1983). The notion of the restricted admissible limit can be formulated also for the other boundary points of these compactifications outside their distinguished boundaries. An analogue of Theorem 8 for these points was proved for functions of class U, 1 ~ p ~ 00,

in Koranyi (1979), Stein (1983). The notion of the restricted admissible limit is a natural extension of the

notion of non tangential limit to domains of tube type. But for some tube cones we can assert the existence of a limit in a stronger sense. In particular, if C = IR~ we can replace the approach sets Fc.(xO) by the approach sets

Then for any function f E U(l~m), p > 1, its Poisson integral PfR':.f(z) converges to f(xO) almost everywhere on IRm when z ~ Xo satying within some approaching set FIZ(xO) (cf. Stein-Weiss (1971)). However, if we translate this notion of unrestricted limit directly to general tube cones then the last assertion fails. The counterexample given in Stein-Weiss (1969) shows that for the future tube .+(n), n :?: 2, this unrestricted limit does not exist in any U, 1 :::;; p :::;; 00. The correct extension of the restricted admissible limit to homogeneous tube cones was found in Koranyi (1969, 1972). This notion, called the admissible (or semi restricted admissible) limit, can be defined, as in the restricted case, for each of the compactifications of a bounded symmetric domain in an invariant way and for all boundary points. It was shown in Lindahl (1972) (cf. also Knapp-Williamson (1971)) that Theorem 8 remains valid for any compactification of a bounded symmetric domain of tube type for functions f E LP when z ~ XO in the admissible way and Po 1). An analogue of this result for the other points of the boundaries was proved in Koranyi (1979), Stein (1983).

Let us illustrate now the notion of the admissible limit, considering the case of the future tube .+(2) in 1C 3 (cf. Koranyi (1976)). For the standard compactification of .+(2) analogous to the one considered in Chap. 1, Sect. 3.2 we obtain an admissible limit which coincides with the restricted admissible limit at points of the distinguished boundary 1R3. At other points of the boundary of .+(2) the admissible limit is non tangential in some directions while in other directions (e.g. on the plane (YI' Y2)) contact of the 1st order with the boundary is allowed. Another (so called, maximal) compactification of .+(2) provides us with another notion of admissible limit at points of 1R3 for which tangential approach (of any order) is allowed along almost all real light rays.

2.4. Pluriharmonic Functions. Denote by RP(TC ) the space of pluriharmonic functions, i.e. functions on T C which are the real parts of holomorphic functions. It follows from Theorem 6 (Sect. 2.1) that the Poisson integral Pcg(z) of a real function g E £. belongs to RP(TC ) if and only if the Fourier transform F- 1 [gJ vanishes outside C* u ( - C*). The discrete analogue of this assertion for the generalized unit disc B2 was proved in Vladimirov (1974) for slowly growing functions; its analogue for bounded symmetric domains and L 2 functions is contained in Schmid (1969).

For functions of the class RP(TC ) we have the following generalization of Rudin's "Correction" Theorem (Rudin (1969)).

Theorem 9. Let g be a lower semicontinuous positive function on IRm, g E

U(lRm). Then there exists a positive singular (with respect to Lebesgue measure) measure (J on IRm such that the Poisson integral

belongs to RP(TC ).


This theorem was proved in a more general situation in Alexandrov (1984). The following "Localization" Theorem of Rudin is also related to the class RP(TC ).

Theorem 10. Let Q be an open subset of IRm. There exists an open set f0 = T C+ u T C- u Q where C+ = C, c- = - c, Q is a complex neighborhood of Q,

having the following property. If the Poisson integral Pc[dJ.l] of a measure J.l on IRm belongs to RP(TC ±) and the support of J.l does not intersect Q then PddJ.l] belongs to RP(f0) and vanishes on Q.

This theorem was proved in Rudin (1970) for the polydisc but its proof, based on the "Edge-of-the-Wedge" Theorem, is valid for general tube cones. Note that the theorem is not true for arbitrary Borel measures (a counterexample is given in Rudin (1969), Sect. 2.3).

For other results on pluriharmonic functions cf. Vladimirov (1979), Stoll (1974) and Sect. 4.2, Chap. 4.

2.5. Functions given by Poisson Integrals. In the case when the cone C is the octant IR~ the class of functions given by Poisson integrals coincides with the class of m-harmonic functions, i.e. functions which are harmonic with respect to each of the variables separately. The class of real m-harmonic functions strictly contains the class RP(T~':') (cf. Rudin (1969». What is the characterization of functions given by Poisson integrals in the case of a general tube cone? We consider first this question for the generalized unit disc Bm (Chap. 1, Sect. 2.2). Let us introduce a matrix operator Az whose components are differential operators of the 2nd order

This operator can be written symbolically in the form (cf. Hua (1958»

Az = (I - ZZ*)(jz·(I - Z*ZYoz

where

Z = (zij), Oz = %Z = (%zij), 1::;; i,j ::;; m.

The trace Tr Az of Az is the invariant (with respect to automorphisms of Bm) Laplacian of Bm so that functions u given by Poisson integrals in Bm are harmonic with respect to Tr A z

(Tr Az}u = O.

This result was proved by Hua (1958) who noted, moreover, that functions u given by Poisson integrals in Bm satisfy in fact the system of differential equations

Azu = 0

(this assertion was proved in Hua (op. cit.) for classical Cartan domains of the 1st type). E.M. Stein has conjectured that the equations found by Hua Loo-keng completely characterize functions given by Poisson integrals. This conjecture was proved for bounded symmetric domains ~ of tube type in JohnsonKoranyi (1980), Berline-Vergne (1981) (cf. also Lasalle (1984a, b), Johnson (1984a, b» (partial results in the same direction were proved in KoranyiMalliavin (1975), Johnson (1978». We formulate here the result of BerlineVergne (op. cit.). They constructed a system of differential operators of the 2nd order called Hua operators, which coincides with L1z in the case of the generalized unit disc, and proved the following assertion. A function F in a domain ~ is the Poisson integral of some hyperfunction over the distinguished boundary (Shilov boundary) of ~ if and only if it satisfies the Hua equations. It is interesting to compare this result with the theorem of Furstenberg (1963) which asserts that any bounded function in a bounded symmetric domain ~ which is annihilated by all invariant differential operators without a constant term in ~ is in fact the Poisson integral of some bounded function over the distinguished boundary of~. Here we take the Poisson integral and the distinguished boundary with respect to the maximal compactification of ~ mentioned before in Sect. 2.3.

§ 3. Other Integral Representations

3.1. Bergman Representation. The Bergman representation for classical domains and Siegel domains was constructed in Hua (1958), Rothaus (1960), Gindikin (1964), (for the general properties of the Bergman representation cf., e.g. Fuks (1963». In the case of the future tube r+ = r+(n), n ~ 2, it has the form (cf. Sergeev (1985»

(i)n+l (n + 1)! f few) dw 1\ dw +

J(z) ~:1 2(2n)"H" [(z ;i ''Yr' , z E< (10)

for functions f E L2(r+) (\ CD(r+). We see that, in contrast with the Cauchy and Poisson integral representations considered above, the Bergman representation involves the integration over the entire domain r+. The Bergman operator Kf(z) given by the right hand side of(10) is an orthogonal projector of L2 (r+) onto the space L 2(r+) (\ CD(r+). We can estimate the operator Kf on functions f E L;'(r+) vanishing outside some ball {Izl < R} as follows (cf. the definition of conical and local estimates in Sect. 1.4).

Theorem 11 (Sergeev (1985». The Bergman operator Kf(z), f E L;'(r+), given by the right hand side of (10), has the following conical estimate at any point X O E [Rn+1

f( ) IlfllLoo IK z I::::; C (n-I){2'

Yo

At points Zo = Xo + iyo E S we have the following local estimate

I Kf(z) I ~ Cllfllv" ·Iln Iy - yOII· Very little is known about estimates of the Bergman operator in U norms

with 1 < P < 00, p =1= 2. D. Bekolle (1984) has considered this operator in the case of the future tube .+(2) in e3 • He proved that it is bounded in U for p "close" to 2 and is unbounded for p "close" to 1 and infinity. Still some gap in between these two subsets of the p-axis remains where it is unknown whether the operator is bounded.

3.2. Cauchy-Fantappie Type Representations. We begin with a general scheme for the construction of Cauchy-Fantappie type integral representations for holomorphic and smooth functions in smooth domains. After that we shall show how these representations are modified when applied to the future tube and Dyson domains. A detailed exposition of Cauchy-Fantappie integral representations and further references can be found in Aizenberg-Yuzhakov (1979), Chirka-Khenkin (1975), Khenkin-Leiterer (1984), Leray (1959) and this series, vol. 7, part II.

Let ~ be a C1-smooth domain in em and qJ«(, z) a C1-smooth function on a~ x ~ which is holomorphic in z E~. Such a function qJ is called a barrier function if qJ«(, n = 0 and qJ«(, z) =1= 0 for «(, z) E a~ x ~. For example, the func-

tion i~ U(i - zJ is a barrier function for the ball t~ Iz;l2 < 1} in em.

By the Hefer representation of a barrier function qJ we mean the representation of qJ in the form

m

qJ«(, z) = L Pi«(, Z)«(i - z), «(, z) E a~ x ~ i;1

where Pi are C1-smooth in their domain of definition and holomorphic in z E ~. Denote by P the column vector t(P1 , ... , Pm) so that qJ«(, z) = (P«(, z), (- z) and define the vector-function

P«(, z) w«(, z) = qJ«(, z)' «(, z) E a~ x ~.

Consider also the universal barrier function (non-holomorphic in z) of Martinelli-Bochner

qJo«(, z) = I( - Zl2 = (pO«(, z), ( - z), p O«(, z) = ~(1 - Z1"'" (m - zm),

and set

Denote, at last, by w«(, z, 2) the linear combination of the vector-functions w and wo:

w«(, z, 2) = (1 - 2)wo«(, z) + 2w«(, z), «(, z) E a~ x~, 0 ~ 2 ~ 1.


The kernel of the Cauchy-Fantappie representation associated with a barrier function cp is given by the differential form

D(" z, A) = det(w, dw, ... , dw) 1\ d" (', z) E ofi) x fi), 0:::;; A :::;; 1

where w = w(" z, A), d, = d'l 1\ ••• 1\ d'm. The differential d is taken with respect to the variables " A and the determinant is expanded with respect to columns replacing the multiplication by exterior multiplication.

Let us introduce the space E(fi)) consisting offunctions v E Cl(~) decreasing sufficiently rapidly at infinity along with their a-derivatives (the rate of decay depends on the kernel of the representation). The Cauchy-Fantappie integral representation (associated with a barrier function cp) for functions v E E(fi)) has the form

-(2ni)mv(z) = f av 1\ Do + f av 1\ D - f vD l , Z E fi) (11) f'fi of'fix[O,lJ of'fi

where Do (respectively Dd denotes the restriction of D to the set {A = O} (respectively {A = I}). For holomorphic functions v = f E E(fi)) this representation takes the form

f(z) = (2:i)m If'fi vD l , Z E fi). (12)

By slightly modifying this construction, let us show now how to obtain the Cauchy-Bochner integral representation in ,+ in en +1 with n ~ 2. We define first the natural barrier function for ,+ using the convexity of ,+. For , = ¢ + iIJ E S (cf. Chap. 1, Sect. 1.1), Z E ,+ we put

cp(C z) = 1 - f. IJj Zj - ¢j . j=l IJo Zo - ¢o

(13)

Note that the equation cp(', z) = 0 (with, fixed) defines the complex tangent space 7;;c,+ at, (cf. Fig. 18) so the function cp is a barrier. Taking the limit in (13) for IJo -+ 0, IJ)IJo -+ (Jj we obtain the function

(14)

Fig. 18

where (J = ((JI' ••• , (In), I (JI = 1. Note that for any (J, I(JI ~ 1, the function epu(~' z) is a barrier function on ~n+1 X r+ (i.e. epu(~' z) -# 0 for (~, z) E ~n+1 X r+). The equation epu(~, z) = 0 (with ~ fixed) defines a support space of r+ at the point ~ (cf. Fig. 18). Thus we can say that the functions ep((, z), ep<T(~' z) form a family of barrier functions for r +. We can consider this family as one barrier function (in the sense of the definition given in the beginning of this section) for the domain ~ obtained by applying a real monoidal transformation or (J-process to r+ along ~n+l. The Cauchy-Fantappie representation (12) for holomorphic functions associated with this barrier function coincides with the Cauchy-Bochner representation (1). For E(r+), in this case, we may take the space Elr+) of functions v decreasing at infinity along with their a-derivatives faster than 1/1zln, i.e.lv(z)1 ~ C/lzln+e, lav(z)1 ~ C/lzln+e for Izl > R and for some positive constants C,B >0.

Another natural integral representation on r+ can be obtained by considering the Cauchy-Fantappie representation associated with the barrier function for r+ given by the Levi polynomial for r+

(( _ Z)2 ep(C z) = il]' (( - z) - 4 ' ((, z) E ar+ x r+. (16)

The integral operator Kf(z) for f E L;:,(ar+) defined by the right hand side of (12) with the barrier function (16) satisfies the following estimates (cf. Sergeev (1986))

1) conical estimate at points X O E ~n+1:

lIn Yol I Kf(z) I ~ CIIfilLro (n-l)/2'

Yo

2) local estimate at points ZO = X O + iyo E S:

I Kf(z) I ~ ClllfIIL~I' y-y

Another application of Cauchy-Fantappie representations is considered in the next section.

3.3. The Jost-Lehmann-Dyson Representation. Let us consider a Dyson domain (cf. Chap. 3, Sect. 2.2.) § = r+ u r- u Q associated with a domain Q in ~n+1 bounded by two spacelike hypersurfaces. Let I be a spacelike hypersurface in Q. We shall construct an integral representation for holomorphic functions in § using the Cauchy-Fantappie representation. We define first a barrier function for §. Through any non-real point ( E a§ there passes the unique admissible hyperboloid (z - U)2 = 22 where U E I; we denote by u((), 2(0 the parameters of this hyperboloid. On the other hand, through any real point ~ of the set ~n+l \Q, which is the union of non-intersecting domains ~+ and ~_, there passes a whole family of complex hyperboloids (z - U)2 = 22; their centers u fill up a domain I ~ on the hypersurface I; the parameter 2 of any such hyperboloid is uniquely determined by u. We define a barrier function epu in § by the

formula

{ u = u«() C E a.@\~n+1 ({Ju(C, Z) = (C - U)2 - (Z - U)2 where U E E~,' C = e E ~n+1 \Q. (17)

We obtain again a family ({Ju of barrier functions which can be interpreted as in Sect. 3.2 as a single barrier function of a domain obtained from .@ by application of a real monoidal transformation. The Cauchy-Fantappie representation (12) for holomorphic functions associated with the barrier function (17) has the form

f(z) = (2 \n+l f e(e) [f+(e) - f-(e)] f Ql, z E.@, (18) 1lZ !')+u!')_ I~

where e(e) = ± 1 for e E P}±, f±(x) is the boundary value of f(x + iy) as y -+ 0, yE V±.

The integral representation (18) was obtained in Bros-Itzykson-Pham (1966) and, as was shown in that paper, coincides with the Jost-Lehmann-Dyson representation introduced in Jost-Lehmann (1957), Dyson (1958). It makes sense for functions f E H(S)(V+ u V-) u (9(§) (cf. Vladimirov (1964». An integral representation of the Jost-Lehmann-Dyson type for domains Q = (C+ + a) u (C- - b), a, b E C+, and for similar domains of more general type (cf. Chap. 3, Sect. 2.2) was obtained in Vladimirov-Zharinov (1970).

In the limit case when the domain Q degenerates to a spacelike hypersurface 1: we obtain from (18) an integral representation for functions f holomorphic in the future tube T+. In this case E~ is the intersection of E with the light cone {<e - U)2 ~ O} (cf. Fig. 19). In particular, if E = {u: Uo = O} we can compute explicitly the kernel of the corresponding integral representation (18) (cf. Sergeev (1986»

22n1l(n-l)/2 r( n ; 1) f(z) = (21li)n+l

f e(e)e8(eo + zo)f(e) de x ~+ u!')_ {[e~ _ z~ _ (C' _ Z')2]2 _ 4z~(f _ Z,)2 }(n+l)/2' (19)

where Z E T+, K - Z')2 = (el - Zl)2 + ... + (en - zn)2. Certain estimates for the integral operator defined by the right hand side of (19) in the space L:,(~n+1) were given in Sergeev (op. cit.).

{ ~

/ , ID+ / ,

/ , / ,

/ , , ! /

/ !~ , / ,

/ " fJ)_

Fig. 19

3.4. Representations for Solutions of the a-equation. The Cauchy-Fantappie representation (11) can be used to represent smooth solutions of the a-equation

av = u (20)

in a domain f0 where u is a smooth a-closed (0, I)-form, with coefficients in E(f0). Indeed, the function

-(2nW+1v(z) = f u 1\ Do + f U 1\ D, 'E f0 (21) !') B!')x[O,l]

is a solution of (20) singled out by the property that the integral

f VD1 = O. B!,)

Using the formula (21) for the different barrier functions defined in Sect. 3.2, 3.3 we obtain different solutions of the a-equation in r+. We shall give here some L oo-estimates of these solutions assuming that the form u is bounded in r+ and has a finite support in the sense that it vanishes outside some ball {I z 1 < R} (depending on u).

Consider first the Cauchy-Bochner solution, i.e. the solution of (20) given by formula (21) with the barrier function (13), (14). This solution satisfies the following estimate:

Iv(z)1 ::; c IlullLoo (jY2r 1

(22)

for z E r+, Izl < r. This estimate was obtained in Sergeev (1986); the analogous estimate for the generalized unit disc Bm was given in Khenkin-Leiterer (1984), Probl. 4.7. It is not known whether this estimate is sharp. It is known (cf. Sergeev (1991» that there exists a right hand side u of the a-equation (20) in r+ in Cn+1

with n ;:::: 3, which is bounded in r+ and has finite support, such that the following is true: any solution v(z) of (20) annihilated by the Cauchy-Bochner integral operator (cf. Sect. 1.4) grows like a power Cjy~-3)/2 for some sequence z-+ XO E IRn+l. In particular, the rate of the growth is increasing with n.

For the solution given by formula (21) with the barrier function (16) given by the Levi polynomial of r+ there is the following estimate

1) conical estimate at points XO E IRn+1:

Ilull oo 1 v(z)1 ::; C (n-1)/2;

Yo

2) local estimate at points ZO = XO + iyo E S:

This estimate is proved in Polyakov (1985) (cf. also Sergeev (1986». In the last paper some other solutions of (20) with estimates in r+ are also considered (e.g. the solution associated with the 10st-Lehmann-Dyson barrier function).

All the solutions of (20) in r+ in e+1 so far constructed are estimated by terms growing like a power (increasing) when z approaches IRn+l. In KhenkinSergeev (1980) it was conjectured that the a-equation in r+ (for n ~ 2) has no uniformly bounded estimate, i.e. there exists a bounded smooth (0, 1)-form U in r+ having finite support such that any solution of (20), with the right hand side given by u, grows (like a power?) when z approaches the distinguished boundary. This conjecture is still unproved. It is proved only (cf. Sergeev (1988, 1991)) that the boundary L+ contains a Sibony-type compact set, i.e. a polynomially convex compact subset Xc or+ having the following properties:

1) there exists a system of smooth strictly pseudo convex neighborhoods

U 1 ~ Uz ~ .•. ~ Uk ~ •.. ~ X

of the compact set X such that 00

X = n Uk; k=1

2) there exists a system of right hand sides Uk of the a-equations

aVk = Uk (23)

in Uk such that Uk is a smooth a-closed (0, 1)-form with finite support in e+1 for k = 1,2, ... and the norms of Uk are uniformly bounded

IlukIILOO(Uk) ::::; C < 00, k = 1,2, ... ,

where C does not depend on k; 3) for any system Vk of solutions of (23) we have

Ilvkll x -+ 00 as k -+ 00,

i.e. the sequence of norms of Vk is unbounded. For a further discussion and the proof of this assertion, see Sergeev (op. cit.).

§4. Functions with Nonnegative Imaginary Part

4.1. Properties of Functions with Nonnegative Imaginary Part in Tube Cones. Denote by H+(TC ) the class of all holomorphic functions with nonnegative imaginary part in T C• Functions of this class satisfy the estimate (Vladimirov (1969b, 1979))

for any cone C ~ C where the constant M depends on c. In particular, H+(TC ) c H(C). The boundary value of the imaginary part fl(X) = 1m f(x) is a nonnegative measure of slow growth. The Poisson integral of this measure satisfies the estimate (Vladimirov (1978a))

Pc [dfl] (z) ::::; 1m f(z), z ETc.


We note that ,u(x) is the boundary value of 1m f(z) not only in the space [/' but also in the following "weak" sense

f 1m f(x + iyO)Pdx, y)cp(x) dx -+ f Pdx, y)cp(x) d,u(x) ~m ~m

as yO -+ 0, yO E C' (£ C for any bounded continuous function cp on IRm. For functions f E H+(TC ) we have the following uniqueness theorem

(Vladimirov (1979)): if ,u(x) = 1m f(x) = 0 then f(z) = (a, z) + b where a E C*, 1m b = 0 (here C* is the dual cone of C, cf. Chap. 1, Sect. 5.1).

Denote by h(lm f; y) the growth indicator of a function f E H+(TC ), i.e.

. 1m f(zO + ity) h(lm f; y) = hm , ZO ETc, y E C.

t-+oo t

This limit does not depend on ZO and defines a nonnegative concave function homogeneous of degree 1 in C, moreover

h(lm f; y) ~ 1m f(zO + iy)

for ZO ETc, y E C (cf. Vladimirov (op. cit.)).

4.2. Integral Representation. The following theorem is an extension to H+(TC ) of the well-known (in the one-dimensional case) integral representation of Herglotz-Nevanlinna.

Theorem 12 (Vladimirov (1978a, 1979)). The following conditions for a function f E H+(TC ), ,u(x) = 1m f(x), are equivalent:

(i) the Poisson integral Pc [d,uJ is pluriharmonic in T C;

(ii) the function 1m f(z) is represented by the Poisson formula

1m f(z) = Pc [d,uJ (z) + (a, y), z E T C (24)

for some a E C*; (iii) for all ZO E T C (and under the assumption that the cone C is regular, cf

Sect. 1.5) the Schwarz representation holds

f(z) = if [/dz - t, ZO - t) d,u(t) + (a, z) + b, z E T C (25) IRm

where b = b(zO) = Re f(zO) - (a, XO). Moreover, (a, y) is the best linear minorant of the indicator h (1m f; y) in the cone C.

In connection with Theorem 12 we may pose the following question: when is the Poisson transform Pc[d,uJ for ,u(x) = 1m f(x), f E H+(TC ) pluriharmonic in T C? The explicit answer to this question is obtained in the two cases: for C = IR~ and for C = V+ = V+(3).

In the case of the octant C = IR~ the Poisson integral PIR'."[d,uJ is always pluriharmonic in TIR'." so the other assertions of the Theorem 12 are also true in this case Korailyi-Pukansky (1963), Vladimirov (1969b, 1979)). The vector a

from the assertion (iii) of the Theorem is defined in this case by the equations

. 1m I(iy) . aj = hm , ] = 1, ... , m.

Yj-OO Yj

In Vladimirov (1969b) also the following criterion of pluriharmonicity for an arbitrary nonnegative measure p. is proved: such a measure p. is the imaginary part of the boundary value of a function I E H + (T ~'J') if and only if the following two conditions are satisfied:

f dp.(x) ~m (l + xi)' ... ·(1 + x;') < 00,

f (Xl - i)a, (Xm - i)am dp.(x) ~m x 1 +i ..... xm+i (l+xi)·····(I+x;,)=O

for any rx = (rx1' ... , rxm) E 7l.m such that rx ¢; 7l.~ u 7l.~. For the future tube ,+ = ,+(3) the assertions of Theorem 12 are satisfied

for functions IE H+(,+) whose indicators have the following properties (Vladimirov (1974)).

h(lm I; y) = ho(y) + (a, y), ho(Y) ~ 0, a E Y+, Y E V+,

lim r ho(l, sly'1) ds = 0, y' = (Y1' Y2' Y3)' ly'I-1-0 J Isl=l

In this case the formulas (24), (25) take the form

( 2)3 f (y2)2 dp.(t) + 1m I(z) = ~ ~. I(z _ t)214 + (a, y), z E, ,

i • 2 2 f dp.(t) . (2)3 f dp.(t) I(z) = n3[(z + I) ] ~. [(z _ t)2(t + i)2J2 - I ~ ~'I(t + ifl4 + (a, z) + b

where a E Y+, b = Re I(i), i = (i, 0, 0, 0), (a, y) is the best linear minorant of the indicator h(lm I; y) in the cone V+.

Using the criterion for pluriharmonicity of the Poisson integrals of distributions in the generalized unit disc B2 (Vladimirov (op. cit.)), we obtain the following condition for pluriharmonicity of a nonnegative measure on [R4: a measure p. is the imaginary part of the boundary value of a function IE H+(,+) if and only if the following conditions are satisfied

f dp.(x) ~.I(x + i)214 < 00,

t. L1~~q2[X(X)J I(Xd:(~214 = 0, 2j = 2,3, ... , [= -1, -2, ... , -2j + 1;

-j::; Q1' Q2 ::;j

where L1~~q2(X) for [=0, ±1, ... , 2j=0, 1, ... ; -j::;Q1' Q2::;j are spherical functions on the group U(2) and x --+ X(x) is the mapping [R4 --+ U(2) given by formula (3) from Chap. 1, Sect. 2.1.


The description of hoI om orphic functions with nonnegative imaginary part in bounded symmetric domains was obtained in Aizenberg-Dautov (1976).

4.3. Tauberian Theorems. Vladimirov (1976) gave a multidimensional generalization of the Hardy-Littlewood Tauberian Theorem for measures. These results were then extended to the case of temperate distributions having support in a proper convex cone in Drozhzhinov-Vladimirov-Zavialov (1984), Drozhzhinov (1982), Drozhzhinov-Zavialov (1979, 1985). We shall present here one of the results of Drozhzhinov-Zavialov (1985). Let r be a proper convex cone in IRm and ~, k = 0, 1, ... , a semigroup oflinear nondegenerate transformations of IRm preserving the cone r. We can take, for example, for ~ the dilatation of the cone r with ratio equal to k. We say that a function 9 on r has an asymptotic with respect to the semigroup {~} and a nonnegative function p(k) depending on k iff

lim g(~~) = fl(~) i= 0 k~oo p(k)

(26)

for any ~ E r. If 9 E fI"(r) and the limit (26) exists in the space [j>'(r) we shall say that the function 9 has a quasiasymptotic with respect to the semigroup {~} and the function p. An equivalent formulation asserts that some primitive function g(-N)(~) of 9 has a "usual" asymptotic with respect to the semi group {~} and the function (det ~t· p(k). If a function 9 has a quasi asymptotic then the functions p and fl cannot be arbitrary. The function p is necessarily automodel (or regularly varying) which means that for any a> 0 the limit of p(ak)/p(k) exists as k -+ 00 (uniformly on compacta in IR+) equal to aY• The function fl is necessarily homogeneous of degree y with respect to ~, i.e. fl(~~) = k Y fl(~)·

We say that a function f E (!J(TC) is slowly growing with respect to a semigroup {Ud (leaving the cone C invariant) and a function p if it satisfies the estimate

(27)

for k > ko and some positive constants a, b, M (here A(y) is the distance from y to the boundary of the cone C).

Theorem 13. Let f E H (C) be the Laplace transform of a function 9 E [j>' (C*). The function 9 has a quasiasymptotic with respect to a semigroup {~} preserving the cone C* and a function p(k) if and only if the function f is slowly growing with respect to the semigroup {Ud, Uk = (~*rl and the function p and

(28)

as k -+ 00 uniformly on compacta in T C. Moreover, IX is the Laplace transform of fl·

The sufficient condition can be considerably weakened. Namely, it is sufficient to require that the function f(z) satisfy (27) only along one "imaginary" direction, i.e. for z = x + ibe where e is some fixed vector of C, 0 < b ::;; 1. For the limit (28) it is sufficient to require its existence for points z = XO + iy where XO is fixed and y belongs to an open subset of C. There are more detailed versions of Theorem 13 for some particular cases (especially, for functions from H+(TC )), cf. Drozhzhinov-Vladimirov-Zavialov (1984), Drozhzhinov (1982), Drozhzhinov-Zavialov (1979, 1985). Using this theorem, extensions of the Fatou and LindelofTheorems for functions in HCXJ(TC ) (cf. Chap. 2, Sect. 3.2) and in H+(TC ) (cf. Drozhzhinov (op. cit.)) have been proved.

4.4. Linear Passive Systems. The results given in Sect. 4.1-4.3 can be applied to the theory of linear passive systems, i.e. systems of linear convolution equations

Z*u =f where Z(~) is a real N x N-matrix function with components from .@'(lRm) which is passive with respect to a proper open convex cone r. The last assertion by definition means that

Re Lr <Z *CP, <p) dC2 0

for any vector-function <p E ['@(lRm)]N.

Theorem 14 (Vladimirov (1969d, 1972)). A matrix Z is passive with respect to a cone r if and only if its Laplace transform L[Z] is a nonnegatively-real holomorphic matrix function in T C with C = int r*, i.e. the function L[Z] is

holomorphic and has nonnegative real part in T C with L [Z] (z) = L [Z] ( -"2) for any Z ETc.

Using this theorem, we can apply the results formulated above to passive systems. We note that the equations of Dirac, Maxwell and many other equations of mathematical physics are passive systems.


The Cauchy-Bochner representation considered in the first section of this chapter was constructed by Bochner (1944) for tube cones and classical domains. For Sect. 1.1, 1.3, 1.5 and related results cf. Vladimirov (1979). The Cauchy-Bochner integral for classical domains was studied in Hua (1958). For a discussion of the properties of the Cauchy-Bochner integral in U spaces cf. Stein (1971). Uniform estimates for the Cauchy-Bochner representation in the future tube are obtained in loricke (1983), Sergeev (1986). The general properties of the Poisson integral given in Sect. 2.1 and at the beginning of Sect. 2.3 were


considered in Vladimirov (1979), Stein-Weiss (1971); the Poisson integral for classical domains was studied in Hua (1958). The second part of Sect. 2.3 which deals with admissible limits is of introductory character; a detailed exposition and review of earlier results is contained in Koraityi (1972). Section 2.4 is based on the results of Rudin (1969). For further results related to Sect. 2.5 cf. Berline-Vergne (1981), Lasalle ( 1984a, b). The derivation of the Cauchy-Bochner and 10st-Lehmann-Dyson integral representations from the general CauchyFantappie representation was proposed in Bros-Itzykson-Pham (1966), Lu (1965); we note also that basically the idea of the derivation of the CauchyBochner representation from the Cauchy-Fantappie representation is contained in formula (1) of Sect. 1.1. For a detailed exposition and proof of the estimates of Sect. 3.2-3.4 cf. Sergeev (1986). For a further development of the results of Sect. 4.1, 4.2 cf. Vladimirov (1979). The assertions of Sect. 4.3 obtained in Drozhzhinov-Vladimirov-Za vialov (1984), Drozhzhinov-Za vialov (1979, 1985) can also be extended and improved. For a detailed exposition of the properties oflinear passive systems (Sect. 4.4) cf. Vladimirov (1979, 1969d, 1972).

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Author Index

Aleksandrov, A.B. 120,161,167,168,172,211,

Cartan, E. 182 Cartan, H. 80 Cauchy, A. 120

234 Alexander, H.J. 94,98,155,172 Alexander, J. 4, 10 Amar, E. 154,155, 161, 172 Andreotti, A. 31

Bacharach 47 Baire, R. L. 169 Baum, P.F. 55 Bedford, E. 61,70, 71, 74, 83, 86 Bell, S. 212 Beltrami, E. 129 Bergman, S. 27, 115, 128,235 Berndtsson, B. 156 Bernstein, S.N. 62, 87, 94, 95, 103 Bertini, E. 55 Besov,O.V. 143 Beuriing, A. 218 Bezout, E. 37 Bishop, E. 26,44, 159, 160 Blaschke, W. 149-155 Bloch, A. 125 Bochner, S. 26, 199,201,214,224 Bogolyubov, RR 100, 182,215 Bolotov, V.A. 31,36 Bony,l-M. 223 Boole, G. 163 Borovikov, A.A. 15 Bott, R. 55 Bourgain, J. 158 Boyling, lB. 21 Brelot, M. 61 Bremermann, H.J. 61,69, 74, 89, 95 Bros, I 200, 222 Budan, F. 36 Burmann, H. 31 Burns, D., Jr. 159

Caccioppoli, R. 26 Caratbeodory, C. 87 Carieman, T. 120 Carleson, L. 135, 136, 159

Cayley, A. 33,47, 116, 187 Chaumat,J. 170,171 Chern Shiing-Shen 70 Chirka, E.M. 68,87, 103, 110, 133, 134, 183,

208 Chollet, A.-M. 170,171 Choquet, G. 75, 85, 86 Coif man, R.R. 138, 142 ColelT, N.R. 29 Cumenge, A. 161

Dautov, Sh.A. 151, 154 Davie, A.M. 159 De Rham, G. 3, 10 Descartes, R. 36 Dini, P. 146 Dirichlet, P.G.L. 70 Dissi, P. 146 Dolbeault, P. 3 Drozhzhinov, Yu. N. 183,208,244 Duchamp, Th. 161 Duren, P.L. 135, 142 Dyson, F.J. 182,218,238 Dzherbashyan, M.M. 149

Egorychev, G.P. 22

Fatou, P. 159,207 FelTerman, Ch. 227 Forelli, F. 99,128,162-165,213 Fornaess, J.E. 67,170,173 Fourier, S. 36 Freeman, M. 197 Froissart, M. 4, 6 Furstenberg, H. 235

Garnett, J.B. 136,137 Garsia, A. 143 Gleason, A.M. 147-148 Glicksberg,1. 157 Golubeva, V.A. 26 Gonchar, A.A. 62, 87, 89, 103, 208

256 Author Index

Gowda, M.S. 147, 174 Grauert, H. 201 Green, G. 94 Griffiths, Ph.A. 7,8,50,51 Grothendieck, A. 4, 14, 39,40

Hakim, M. 155, 168, 170, 172 Hankel, H. 96,144-146 Hardy, G.H. 39, 126 Harnack, A. 84 Harris,J. 51 Hartogs, F. 61,66,76,90,91, 100 Haustus, M.R. 22 Heisenberg, W. 116 Henrikson, B.S. 160, 170, 173 Herrera, M. 29 Hormander, L. 136, 222 Horowitz, Ch. 138 Hua Lo-keng 235

Iagolnitzer, D. 201, 222

Jac6bczak, P. 161 Jacobi, c.GJ. 46,92 Janson, S. 141 John, F. 141 Joricke, B. 160, 165, 228 J osefson, B. 71, 79 J ost, R. 182, 220, 239

Kashiwara Masaki 219 Kawai Takahiro 219 Kazaryan, M.V. 103 Keldysh, M.V. 62 Kellogg, O.D. 83 Khenkin, G.M. 110, 150, 151, 153, 154,157-

161, 183,212,214 Khrumov, Yu. V. 208 Khrushchev, S.V. 162 Kiselman, C.O. 86 Kobayashi Shoshichi 87 Kodaira Kunihiko 53 Konig, H. 157 Koppelman, W. 25, 26 Koranyi, A. 133, 136,210,212,232,233 Korenblum, B. 149 Krantz, S.G. 142 Kuprikov, A.V. 31 Kushnirenko, A.G. 38 Kytmanov, A.M. 36

Lagrange, J.L. 31 Landau, L.D. 18 Landkof, N.S. 61

Laplace, P.S. 65 Lasker, E. 53 Latter, R.H. 136, 138 Lavrentiev, M.A. 62 Lebesgue, H. 64 Lefschetz, S. 15, 18 Lehmann, H. 182, 220, 238 Leinartas, E.K. 8, 22 Leiterer, J. 110, 161 Lejbenzon, Z.L. 147, 148 Lelong, P. 29,61,69, 71, 79, 84 Lempert, L. 86 Leray, J. 4-7, 15,25,26, 56 Levi, E.E. 91 Levine, H.I. (Levin) 70 Lie, S. 190 Lindelof, E. 133, 134, 208 Lindenstrauss, J. 139 Lew, E.(Low) 120,167,173,211 Lumer, G. 165 Luzin, N.N. 80

Macaulay, F. 52 MacMahon, P.A. 23 Malliavin, P. 153 Marcinkiewicz, J. 131 Martineau, A. 216,221 Martinelli, E. 10, 26, 30, 31 Mather, J.N. 17, 55 Minkowski, H. 194 Mityagin, B.S. 158 Morimoto Mitsuo 218

Nagel, A. 158 Nehari, Z. 145 Nevanlinna, R. 103, 125, 149, 168 Newton, I. 36 Nirenberg, L. 70, 141 Nishino Toshio 91 Noether, M. 53 Norguet, F. 8,9,31,55

Odoni, R.W. 22 Oka Kiyoshi 61,69,91-93 0ksendal, B.K. 159 Oppenheim, A. 39 Orlicz, W. 137

Pade, H. 89 Pascal, B. 47 Pedan, Yu.V. 21 Pelczynski, A. 139, 158 Peller, V.V. 110 Penrose, R. 182, 195

Author Index 257

Perron, o. 74 Petrovskij, I.G. 15 Pflug, P. 218 Pham, F. 18 Picard, E. 15, 18 Pick, G. 168 Pinchuk, S.1. 110,160 Plessner, A.I. 133 Poincare, H. 4, 5, 11,30,56, 114 Poisson, S.D. 63 Poletskij, E.A. 87 P6lya, G. 91 Pontryagin, L.S. 4, 10 Power, S.c. 135 Privalov,l.I. 61

Riemann, B. 120 Riesz, F. 65,66,70, 157, 159 Riesz, M. 128,157,159 Rochberg, R. 139 Romberg, B.W. 142 Rouche, E. 27 Rudin, W. 110,128,158,168-170,173,213,

234 Ry\1,1. 120, 172

Sack, R.A. 33 Sadu\1aev, A. 160, 170 Safonov, K.V. 22 Sato Mikio 219 Schneider, R. 141, 147 Seever, G. 157 Sergeev, A.G. 197,235 Serre,1-P. 13,53 Shabat, B.V. 87 Shields, A.L. 142 Shilov, G.E. 88 Sibony, N. 155, 161, 168, 170, 172 Siciak,1. 87, 103 Skoda, H. 150,153 Slodkowski, Z. 91 Smirnov, V.1. 125, 149 Sorani, G. 26,31

Stein, E.M. 133, 134, 141,209,235 Stein, K. 67 Stokes, G.G. 72 Stout, E.L. 159-161, 173 Sylvester, 1.1. 33 Szego, G. 127

Tamm, M. 170 Taylor, BA 70,71,74,84, 86 Thorn, R. 16 Toeplitz, o. 144-147 Tsikh, A.K. 17,31,36-38 Tumanov, A.E. 159-161,212,214

Vagi, S. 136,210,212 Valiron, G. 103 Varchenko, A.N. 21 Varopoulos, N.Th. 152-154, 159, 172 Vinogradov, SA 110, 135, 162 Vladimirov, V.S. 61,217,219,242,245 Voronoy, G.F. 39

Wainger, S. 158 Walsh,1.L. 62,74,87,94,95, 103 Waring, E. 35 Weierstrass, K. 87 Weil, A. 27,96 Weiss, G. 138 Wermer,1. 94 Whitney, H. 17 Widom, H. 145 Wirtinger, W. 28 Woytaszczyk, P. 120, 158, 172

Yanashauskas, A.1. 22 Yuzhakov, A.P. 8, 11, 22, 27, 31, 32, 36, 37,

51

Zakharyuta, V.P. 86,87, 100, 103 Zariski, O. 18 Zavialov, B.!. 183,208,244 Zharinov, V.V. 183,221 Zygmund, A. 125

Subject Index

Automorphism 114

Boundary, distinguished 184 -, Shilov 88 Bounded in restricted sense 207

Capacity, condenser 77 -, projective 98 -, ,i!l'- 75 Class, Hardy 126 -, Nevanlinna 125 -, Smirnov 125 Class residue 6 Co boundary, Leray 4 Compactification, conformal, of Minkowski

space 194 Compact set, circled 97 -, pluriregular 76 Condition, Blaschke 152, 153 -, -, uniform 154 -, Noether local 53 Cone, dual 199 -, homogeneous 199 -, proper 199 -, regular 228 -, self-dual 199 -, tube 199 Cones, Yi;, i!l'/, i(J!/ 199 Conjecture, extended future tube 218 Converse of theorem on total sums of residues

47 C-properties 80 Current 68 -, residue 29 Curve, C-like 217 -, (P, 0- 134 -, (- 133 -, special (- 133

Degree of a zero 150 Dimension, Hausdorff 122 Disc, generalized unit 187, 189 Divisor 150 -, of a function 150 Domain, bounded symmetric tube type 200

-, Cartan classical 191,199,200 -, Dyson 218 -,ofpositivity 199 -, Siegel 199,200 Duality, Alexander-Pontryagin 10 -,De Rham 13 -,global 51 -, local 42

Equations, Hua 235 -, tangential Cauchy-Riemann 120 Estimate, conical 227 -, local 228 Expansion, Hefer 42, 236

Form, Levi 186 -, Martinelli 28 -, residue 5 -, semimeromorphic 6 -, simple differential 9 Formula, change of variables 43 -, Euler-Jacobi 45 -, Leray-Koppelman 25 -, Leray residue 6 -, logarithmic residue 24 -, Picard-Lefschetz 19 -, Poincare-Lelong 29 -, Poincar-Martinelli 30 Front, analytic wave 222 Function, automodal 244 -, barrier 236 -, -, Martinelli-Bochner 236 -, generalized Green 94 -, holomorphic 113 -, hypergeometric 118 -, inner 167,210 -, locally slowly growing 205 -, M-harmonic 130 -, maximal 131,132 -, outer 169 -, pluriharmonic 113, 233 -, plurisubharmonic 66 -, regularly varying 244 -, separately analytic 94 -, slowly growing 205

Subject Index 259

-, slowly growing with respect to a semigroup 244

-, subharmonic 62 -, T-invariant 119

Group, Heisenberg 116 -, of conformal transformations 186 -, Lorentz 186

Halfplane, generalized upper 187 Hull, C-convex 217 Hyperboloid, complex 218 Hyperfunction 204 -, multiplicative 223 Hypersurface, C-spacelike 206

Indicator, growth 242 Inequality, Bernstein-Walsh 95 -, Cauchy-Bunyakovskij 80 Infinity, space-time 194 -, light 195 Inverse of a holomorphic mapping 32 Integral, Cauchy-Bochner 227 -, Poisson 231 Intersection, essential 29

Kernel, Bergman 128 -,Cauchy 127,224 -, Cauchy-Fantappie 237 -, invariant Poisson 128 -, Poisson 129,229 -, Schwartz 228

Laplacian 65 -, invariant 129,234 Lemniscate 92 Limit, admissible 233 -, B 134 -, K 134 -, restricted admissible 232 -, semirestricted admissible 233

Manifold, generic 113 -, integral 114 -, interpolation 213 -, totally real 113 Mapping, inner 212 Matrix, i-unitary 114 Measure, associated 65 -, A- 156 -, analytic 156 -, CarIeson 134 -, -, of order t 135 -, Hausdorff 122

-, Khenkin 156 -, L- 156 -, &'- 75 ~, p- 161 -, representing 156 -, totally singular 156 Method for eliminating unknowns 35 Metric, Bergman 115 Minorant, best linear 242 Multiplicity of a zero of a function (see degree) -, of a zero of a mapping 24

Neighborhood, complex, of real domain 183 Non-straightening, holomorphic 196 Null set (see set) Number, Newton 38

Octant IR~ 199 Operator, Bergman 236 -, fractional differential 124 -, fractional integral 124 -, Hankel 144 -, Monge-Ampere 71 -, Toplitz 144

p-atomic function 136 --, holomorphic 138 Pluriregularity 76, 95 Point, Lebesgue 132 Polygon, Newton 38 Polyhedron, Newton 38 -, strictly pseudoconvex 197 -, Weil 96 Potential of a measure 65 Principal parts of a system at zero 38 Principle, maximum 62 -, Rouch€: 27 Problem, factorization 223 -, first Lelong 71, 79 -, Riemann-Hilbert 219 -, second Lelong 71,84 Product, Hadamard 21 -, Blaschke 211 Projection, Riesz 128

Quasiasymptotic of a generalized function 244 Quasimetric, nonisotropic 121

Ray, light (real) 184 -, -, complex 184 Relations, dispersion 227 Representation, Bergman 235 -, Bergman-Weil integral 42 -, Cauchy-Bochner 225

260 Subject Index

Representation (cont.) -, Cauchy-Fantappie 236 -, Herglotz-Nevanlinna 242 -, lost-lehmann-Dyson 239 -, Penrose 193 -, Poisson 229 -, Schwartz 228 Residue, global 51 -, Grothendieck 40 -, local 40 Resultant of a function with respect to a system

37

Sequence, Leray exact 4 Set, determining 209 -, J- 159 -, interpolation 159,212 -, Landau 18 -, local 170 -, M- 160 -, null 159,212 -, p- 159 -,-,peak 212 -,peak 170,159,212 -, piecewise-linear analytic 153 -, pluripolar 75 -, polar 64 -, uniqueness 160,209 -, z- 159 Space, Bloch 125 -, complex tangent 113, 185

A (T') 207 - .@' (£1) 183 - E(.@) 237 - H(C) 203 - H+(T') 241 -W(T') 202 - H(S) (C) 203

- Jf, 203 -- L~mp (IR") 183 - (I) (D) 183 Spectrum, singular 219 Sphere, Lie 191

Stratification 17 Support, microlocal singular 219,222 -, essential 222 System, passive 245

Theorem of Bogolyubov on the edge of the wedge 215

-, -, generalized 221 -, Bochner's tube 199,201 -, C-convex hull 217 -, decomposition 219 -, Fatou 207 -, finite covariance 218 -, Forelli 213 -, Froissard 6 -, Furstenberg 235 -, Grauert 201 -, Lindeliif 208 -, Macaulay 52 -, Martineau 216,221 -, Noether-Lasker 52 -, on total sums of residues 45 -, residue 11 -, -, for vector bundles 50 -, Rudin's "correction" 233 -, separate analyticity 215 -, Tauberian of Hardy and Littlewood

244 -, two constants 76 Trace 43 Transform, Hilbert 226 -, Cayley 116, 187 -, generalized Fourier 222 -, generalized Radon 223 Tube, future 184, 185 -, local 201 Tuboid 200 Twistor 193

Values, boundary, in the sense of hyperfunctions 204, 205

Width, zero 213

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Several Complex Variables II: Function Theory in Classical Domains Complex Potential Theory

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