One-Parameter Families of Operators in C · One-Parameter Families of Operators in C 355 In [11],...

The Journal of Geometric AnalysisVolume 16, Number 2, 2006

One-Parameter Families of Operators in CBy Andrew S. Raich

ABSTRACT. We develop classes of one-parameter families (OPF) of operators on C!c (C) which char-

acterize the behavior of operators associated to the !-problem in the weighted space L2(C, e"2p) where p

is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a boundedoperator from Lq(C) to itself, 1 < q < !, with a bound that depends on q and the degree of p but not onthe parameter " or the coefficients of p. Last, we show that there is a one-to-one correspondence givenby the partial Fourier transform in " between OPF operators of order m # 2 and nonisotropic smoothing(NIS) operators of order m # 2 on polynomial models in C2.

1. Introduction

The goal of this article is to introduce classes of one-parameter families (OPF) of operatorson C which characterize the behavior of kernels associated to the weighted !-problem in C.The need for OPF operators stems from problems associated to the inhomogeneous !b-equationon polynomial models in C2 and the !-problem in weighted L2 spaces in C. A polynomialmodel M is the boundary of an unbounded weakly pseudoconvex domain of finite type of theform {(z1, z2) $ C2 : Im z2 > p(z1)} where p is a subharmonic, nonharmonic polynomial.M %= C&R and !b, defined on M , can be identified with the vector field L = !

! z " i !p! z!!t . Under

the partial Fourier transform in t , the vector field L becomes

Z"p = !

! z+ "

!p

! z, (1.1)

which we regard as a one-parameter family of differential operators acting on functions definedon C. OPF operators will be defined so that Z"p and Z"p = "Z'

"p = !!z " " !p!z are the natural

differential operators under whose action OPF operators behave well.

When " = 1, the differential operator Zp = !! z + !p

! z has been well studied [3, 2, 14, 15].Christ [3] and the author [14, 15] expressly cite the study of !b on polynomial models as motivationto study the !-problem on weighted L2 in C. In Section 1.1, we review the equivalence of the!-problem in L2(C, e"2p) with the Zp-problem, Zpu = f , in L2(C). When p is a subharmonicfunction satisfying mild hypotheses on (p, Christ [3] solves the equation Zpu = f on L2(C) viathe complex Green operator Gp for !p = "ZpZp where Zp = "Zp = !

!z " !p!z . Both Gp and

Math Subject Classifications. Primary 32W50, 32W30, 32T25.Key Words and Phrases. Finite type, NIS operator, one-parameter families, weakly pseudoconvex domain.

© 2006 The Journal of Geometric AnalysisISSN 1050-6926

354 Andrew S. Raich

the relative fundamental solution ZpGp are given as fractional integral operators. Also, Christshows that if Y # is a product of length 2 of operators of the form Y = Zp of Zp, then Y #Gp isbounded on Lq(C), 1 < q < !. When " = 1, Gp serves as a model for an order 2 OPF operator,while Y #Gp serves as model for an order 0 OPF operator. Christ and the author [14] find pointwiseestimates of the integral kernel of G"p and its derivatives (Christ in the case " = 1 and the authorfor " > 0), and the author [15] finds cancellation conditions for G"p and its derivatives when" > 0. Similarly to the ordinary Laplace operator, !p is a second-order, nonnegative ellipticoperator, and there is a strong analogy between Gp and the Newtonian potential N on C. Bothinvert “Laplace” operators, and if D2 is a second-order derivative, D2N is a Calderòn-Zygmundoperator and bounded on Lq , 1 < q < !. In Theorem 2.1, we will see that order 0 OPF operatoris bounded in Lq , 1 < q < !.

1.1. Connection of Z!p with "u = f on weighted L2

The interest in the !-problem on weighted L2 spaces in C started with Hörmander’s work [5]on solving the inhomogeneous Cauchy–Riemann equations on pseudoconvex domains in Cn.Hörmander’s methods, now classical in the subject [6], rely on proving that if diam($) # 1, thereis a solution to !u = f in L2($, e"2p) satisfying the estimate

!$ |u|2e"2p dz #

!$ |f |2e"2p dz.

Using the techniques of Hörmander, Fornæss, and Sibony [4] generalize the L2 estimate to an

Lq estimate, 1 < q # 2, and prove that !u = f has a solution satisfying:"!$ |u|qe"2p dz

# 1q #

Cp"1

"!$ |f |qe"2p dz

# 1q . They also show that the estimate fails if q > 2. Berndtsson [1] builds

on the work of Fornæss and Sibony and shows an Lq -L1 result. He shows that if diam $ < 1and 1 # q < 2, then !u = f has a solution so that )ue"p)Lq($) # Cq)f e"p)L1($). Berndtssonalso proves a weighted L!-Lq estimate when q > 2.

In [3], Christ recognizes that it is possible to study the !-problem in L2(C, e"2p) by workingwith a related operator in the unweighted space L2(C). If !u = f and both u = epu andf = epf are in L2(C, e"2p), then !u

! z = f *+ e"p !! z e

pu = f . However, e"p !! z e

pu = Zpu.Consequently, the !-problem on L2(C, e"2p) is equivalent to the Zp-problem, Zpu = f , onL2(C). Berndtsson [2] solves Zpu = f on smoothly bounded domains in C and views !p fromthe viewpoint of mathematical physics. He writes !p as a magnetic Schrödinger operator withan electric potential and his estimates follow from Kato’s inequality, a result from mathematicalphysics. The author [14] solves the heat equation associated to !p and uses techniques both frommathematical physics and the solution of the !b-heat equation on polynomial models in C2 [11].

1.2. The relationship between NIS and OPF operators

For computations involving !b on both polynomial models in C2 and the boundaries ofother weakly pseudoconvex domains of finite type in Cn, nonisotropic smoothing (NIS) operatorshave played a critical role in the analysis of the relative fundamental solutions of !b and relatedoperators. Nagel et al. [9] introduce NIS operators while analyzing the Szegö kernel on weaklypseudoconvex domains of finite type in C2. Nagel and Stein use properties of NIS operators intheir analysis of the heat kernel on polynomial models in C2 [11] and both the relative fundamentalsolution of !b and the Szegö kernel on product domains and decoupled domains in Cn [12, 10].A motivation for developing NIS operators is that the class of NIS operators have invariancesthat individual operators do not. NIS operators are invariant under translations and dilations,derivatives of NIS operators are again an NIS operators, and order 0 NIS operators have desirablemapping properties, namely Lp-boundedness [12].

One-Parameter Families of Operators in C 355

In [11], Nagel and Stein solve the !b-heat equation !u!s + !bu = 0 with initial condition

u(0,#) = f (#) where s $ (0,!) and # $ C & R. They write their solution using the heatsemigroup e"s!b and in turn express e"s!b [f ] as integration against a kernel called the heatkernel. NIS operators are one of the workhorses of their arguments because as a class of operators,NIS operators (1) commute with vector fields L and L', (2) remain invariant under translationsand scaling, and (3) change products of arbitrary compositions of L and L' to a composition of apower of !b with a well-controlled NIS operator. The analogy of NIS operators with Calderòn-Zygmund operators is strong. For example, (3) is analogous to writing an arbitrary derivative asthe composition of (k for some k with a Riesz transform.

A goal for OPF operators is to play the analogous role for objects associated to the operatorsZ"p and Z"p as NIS operators do to objects related to !b and !'

b on the boundaries of weaklypseudoconvex domains in C2. In [14, 15], the author solves the !"p-heat equation for " $ R,i.e., he solves the equation !u

!s + !"pu = 0 with initial condition u(0, z) = f (z). The solution iswritten as integration against a kernel, called the heat kernel which is shown to be smooth off ofthe diagonal {(s, z, w) : s = 0 and z = w}. Also, the author finds pointwise decay estimates forthe heat kernel and its derivatives. OPF operators play a fundamental role in these articles. Theyare an essential tool in the regularity arguments and the derivative estimates. Also, the abilityto scale an OPF operator and stay within the class of OPF operators is crucial in the time decayestimate of the heat kernel e"s!"p .

2. Main results

Theorem 2.1. If T" is an OPF operator of order 0, then T" , T '" are bounded operators from

Lq(C) to Lq(C), 1 < q < !, with a constant independent of " but depending on q.

Also, the classes of OPF operators fulfill the promise of being an analog to NIS operators.We can use results about OPF operators to study NIS operators and vice versa. We have thefollowing theorem.

Theorem 2.2. Given a subharmonic, nonharmonic polynomial p : C , R, there is a one-to-one correspondence between OPF operators of order m # 2 with respect to p and NIS operatorsof order m # 2 on the polynomial model Mp = {(z1, z2) $ C2 : Im z2 = p(z1)}. Thecorrespondence is given by a partial Fourier transform in Re z2.

3. Notation and definitions

3.1. Notation for operators on CFor the remainder of the article, let p be a subharmonic, nonharmonic polynomial. It will

be important for us to expand p around an arbitrary point z $ C, and we set:

azjk = 1

j !k!!j+kp

!zj! zk(z) . (3.1)

We need the following two “size” functions to write down the size and cancellation conditionsfor both OPF operators and NIS operators. Let

%(z, &) =$

j,k-1

%%azjk

%%&j+k (3.2)

356 Andrew S. Raich

and

µ(z, &) = infj,k-1

|&|1/j+k

%%azjk

%%1/j+k. (3.3)

It follows µ(z, &) is an approximate inverse to %(z, &). This means that if & > 0,

µ"z,%(z, &)

#% & and %

"z, µ(z, &)

#% & . (3.4)

We use the notation a ! b if a # Cb where C is a constant that may depend on the dimension 2and the degree of p. We say that a % b if a ! b and b ! a.

%(z, &) and µ(z, &) are geometric objects from the Carnot-Carathéodory geometry developedby Nagel et al. [13, 8]. The functions also arise in the analysis of magnetic Schrödinger operatorswith electric potentials [17, 18, 7, 14, 15].

Denote the “twist” at w, centered as z by

T (w, z) = "2 Im

&

'$

j-1

1j !!jp

!zj(z)(w " z)j

(

)

= i

&

'$

j-1

1j !!jp

!zj(z)(w " z)j "

$

j-1

1j !!jp

! zj(z)(w " z)

j

(

) . (3.5)

Also associated to a polynomial p and the parameter " $ R are the weighted differentialoperators

Z"p,z = !

! z+ "

!p

! z= e""p !p

! ze"p Z"p,z = !

!z" "

!p

!z= e"p

!p

!ze""p .

We need to establish notation for adjoints. If T is an operator (either bounded or closed anddensely defined) on a Hilbert space with inner product

"· , ·#, let T ' be the Hilbert space adjoint

of T . This means that if f $ Dom T and g $ Dom T ', then"Tf, g#

="f, T 'g

#. The Hilbert

spaces that arise in this article are L2(C) and L2(C & R). Since the L2-adjoints of Z"p and Z"pare different than their adjoints in the sense of distributions, for clarity we let W "p and W"p bethe negative of the distributional adjoints of Z"p and Z"p, respectively. Thus,

W "p,w = !

!w" "

!p

!w= e"p

!p

!we""p W"p,w = !

!w+ "

!p

!w= e""p !p

!we"p .

We think of " as fixed and the operators Z"p,z, Z"p,z, W "p,w, and W"p,w as acting on functionsdefined on C. Also, we will omit the variables z and w from subscripts when the application isunambiguous. Observe that (Z"p) = W "p and (Z"p) = W"p. Finally, let

M"p = ei"T (w,z) !

!"e"i"T (w,z) .

3.2. Definition of OPF operators

Let p be a subharmonic, nonharmonic polynomial. We say that T" is a one-parameter family(OPF) of operators of order m with respect to the polynomial p if the following conditions hold:


(a) There is a function K" $ C!""(C & C) \ {z = w}#& (R \ {0})

#so that for fixed " ,

K" is a distributional kernel, i.e., if ',( $ C!c (C) and supp' . supp( = /, then

T" ['] $ (C!c )0(C) and

1T" ['](·),(2C =**

C&CK" (z, w)'(w)((z) dw dz .

(b) There exists a family of functionsK",)(z, w) $ C!(C&C&R) so that if' $ C!c (C&R),

K",)[']C&R(z, " ) =*

C&R'(w, " )K",)(z, w) dw d"

and lim),0 K",)[']C&R(z) = K" [']C&R(z) in (C!c )0(C & R). All of the additional

conditions are assumed to apply to the kernels K",)(z, w) uniformly in ).

(c) Size Estimates. If YJ"p is a product of |J | operators of the form Y

j"p = Z"p,z, Z"p,z,

W"p,w, W "p,w, or M"p where |J | = * + n and n = #{j : Yj"p = M"p}, for any k - 0

there exists a constant C*,n,k so that

%%%YJ"pK",)(z, w)

%%%#C*,n,k|z " w|m"2"*

|" |n+k%(z, |w"z|)k if

+,

-

m < 2m = 2, k - 1m = 2, |w " z| > µ

"z, 1

"

#.

. (3.6)

Also, if m = 2 and |w " z| # µ(z, 1" ), then

%%%Mn"pK",)(z, w)

%%% # Cn

+.,

.-log

/2µ(z,

1" )

|w"z|

0

n = 0

|" |"n n - 1

. (3.7)

(d) Cancellation in w. If YJ"p is a product of |J | operators of the form Y

j"p = Z"p, Z"p,

or M"p where |J | = * + n and n = #{j : Yj"p = M"p}, for any k - 0 there exists a

constant C*,n,k and N* so that for ' $ C!c (D(z0, &)),

supz$C

%%%%

*

CYJ"pK",)(z, w)'(w) dw

%%%%

# C*,n,k

|" |n &+....,

....-

&21

log1

2µ"z, 1"

#

&

2)')L!(C)+

$

1#|I |#N0

&|I |33XI"p'(w)

33L!(C)

2& < µ"z, 1"

#and

m = 2, * = 0

&m"*

|" |k%(z, &)k

$

|I |#N*

&|I |33XI"p'33L!(C) otherwise

(3.8)

where XI"p is composed solely of Z"p and Z"p.

(e) Cancellation in " . If XJ"p is a product of |J | operators of the form X

j"p = Z"p,z, Z"p,z

or W"p,w, W "p,w and |J | = n, there exists a constant Cn so that*

RXJ"p

4ei" tK",)(z, w)

5d" # Cn

µ(z, t + T (w, z))m"n

µ(z, t + T (w, z))2|t + T (w, z)| . (3.9)

(f) Adjoint. Properties (a)–(e) also hold for the adjoint operator T '" whose distribution

kernel is given by K",)(w, z).

358 Andrew S. Raich

Note that for the " -cancellation condition (3.9), we do not need to consider the case Xj"p =

M"p since!

R!!"

"ei" (t+T (w,z))K",)(z, w)

#d" = 0.

In the size condition (c) and cancellation condition (d), the " k%(z, |z"w|)k and " k%(z, &)k

terms indicate rapid decay. If OPF operators are to be partial Fourier transforms of NIS operatorson polynomial models, rapid decay should not be surprising; it is consequence of being able tointegrate parts from the Fourier transform formula. This will be seen explicitly in Lemma 6.3.Ignoring the rapid decay terms, the size and cancellation conditions of OPF operators are familiar.An order 2 OPF operator should “invert” two derivatives, like the Newtonian potential. In R2,the Newtonian potential has a logarithmic blowup on the diagonal, just like an order 2 OPFoperator. For an order 0 OPF operator, the blowup on the diagonal is the same as a Calderòn-Zygmund kernel, and the decay of K" (0, z) is |z|"2, the same as a Calderòn-Zygmund kernel. Forthe cancellation conditions, if ' is “normalized” appropriately, the cancellation condition (3.8)simplifies to

33YJ"pT" [']

33L!(C)

! &j .

This is reminiscent of cancellation of a Calderòn-Zygmund operator or an NIS operator.

3.3. Notation for Carnot-Carathéodory geometry and vector fields on C ! RIn order to write down the definition of an NIS operator on a polynomial model in C2, we need

to establish notation for the Carnot-Carathéodory metric + and corresponding balls BNI

"(z, t), &

#.

If Mp is a polynomial model in C2 given by Mp = {(z1, z2) $ C2 : Im z2 = p(z1)}, thenMp %= C & R. Under the isomorphism, a representation of the Carnot-Carathéodory metricis the nonisotropic pseudodistance +

"(z, t), (w, s)

#= |z " w| + µ

"z, t " s + T (w, z)

#where

(z, t), (w, s) $ C & R. Since +"(z, t), (w, s)

#is a function of z, w, and t " s, we define a new

function

dNI (z, w, t) = |z " w| + µ"z, t + T (w, z)

#. (3.10)

We will see that dNI (z, w, t) is essentially symmetric in (z, w). The nonisotropic ball

BNI

"(z, t), &

#= {(w, s) : dNI (z, w, t " s) < &} .

We also define a volume function

VNI

"(z, t), (w, s)

#=%%BNI

"(z, t), dNI (z, w, t"s)

#%%% dNI (z, w, t " s)2%"z, dNI (z, w, t"s)

#.

That the volume function is comparable to dNI (z, w, t " s)2%"z, dNI (z, w, t " s)

#follows

from (3.4).

If " is the transform variable of t , observe that under the partial Fourier transform in t , underthe inverse partial Fourier transform, Z"p and Z"p map to the vector fields

Lz = !

! z" i

!p

! z

!

!tLz = !

!z+ i

!p

!z

!

!t

while W "p and W"p map to the vector fields

Lw = !

!w+ i

!p

!w

!

!tLw = !

!w" i

!p

!w

!

!t.


As we know from Section 1, !b (defined on M) becomes the operator Lz on C & R. It followsthat "Lz is the Hilbert space adjoint to Lz in L2(C & R). The translation invariance in t causesmany operators of interest to have a convolution structure in t . A consequence is that if we havea function f

"(z, t), (w, s)

#= f (z, w, t " s), we may study f (z, w, t). By the chain rule, Lw

and Lw are the versions of Lz and Lz in the w-variable. Finally, let

M = "i"t + T (w, z)

#.

3.4. NIS operators on polynomial models in C2

There are different notions of NIS operators (e.g., [9, 11]). We use the definition from [9].

Definition 3.1 (Nonisotropic Smoothing Operator of order m).

Let

T [f ](z, t) =*

C&RT"(z, t), (w, s)

#f (w, s) dw ds ,

where T"(z, t), (w, s)

#is a distribution which is C! away from the diagonal. We shall say that

T is a nonisotropic smoothing operator which is smoothing of order m if there exists a family

T)[f ](z, t) =*

C&RT)"(z, t), (w, s)

#f (w, s) dw ds ,

so that:

(a) T)[f ] , T [f ] in C!(C & R) as ) , 0 whenever f $ C!c (C & R);

(b) each T)"(z, t), (w, s)

#$ C!"(C & R) & (C & R)

#. The following two conditions hold

uniformly in ):(c) If X I = Xi1Xi2 · · ·XIk where Xij = Lz, Lw, Lz, or Lw, then

%%X I T)"(z, t), (w, s)

#%% # c|I |dNI (z, w, t " s)m"|I |

V"(z, t), (w, s)

# ; (3.11)

(d) for each * - 0, there exists an N = N* so that whenever ' is a smooth (bump) functionsupported in BNI

"(z, t), &

#,

%%X I T ['](z, t)%% # C*&

m"* supw,s

$

|J |#N*

&|J |%%X J ['](w, s)%% , (3.12)

|I | = *;(e) the same estimates hold for the adjoint operator T ', i.e., the operator with the kernel

T"(w, s), (z, t)

#.

4. Properties of T (w, z)

To prove Theorems 2.1 and 2.2, we need to understand the “twist” T (w, z) and how it behavesunder differentiation.

Proposition 4.1.

T (w, z) = "T (z, w) .

360 Andrew S. Raich

Proof. Since p(z) =6j,k1

j !k!!j+kp

!zj ! zk (w)(z " w)j (z " w)k, we have

!*p

!z*(z) =$

j-*k-0

j !(j " *)!

1j !k! .

!j+kp

!zj! zk(w)(z " w)j"*(z " w)

k.

Since p is R-valued, the twist [Equation (3.5)] T (w, z) = "2 Im"6

*-01*!!*p

!z*(z)(w " z)*

#, so

$

*-0

1*!!*p

!z*(z)(w " z)*

=$

*-0

1*!

&

7'$

j-*k-0

j !(j " *)!

1j !k! .

!j+kp

!zj! zk(w)(z " w)j"*(z " w)

k

(

8) (w " z)*

=$

j-0k-0

&

'j$

*=0

1j

*

2("1)*

(

) 1j !k!

!j+kp

!zj! zk(w)(z " w)j (z " w)

k

=$

k-0

1k!!kp

! zk(w)(z " w)

k =$

j-0

1j !!jp

!zj(w)(z " w)j .

The second to last line uses the identity6j*=0

"j*

#("1)* = &0(j). The result follows easily.

Corollary 4.2.

dNI (z, w, t) % dNI (w, z, t) .

Proof. This is a well-known fact [13, 8], but we are in a situation where the computations canbe explicit. We sketch a proof. If r = |t + T (w, z)|, it follows from Proposition 4.1 that it isenough to show that

|z " w| + µ(z, r) % |z " w| + µ(w, r) .

If µ(z, r) < |z " w| and µ(w, r) < |z " w|, there is nothing to prove, so (without loss ofgenerality) assume that µ(z, r) > |z " w|. By expanding p(z) around w and p(w) around z, itcan be shown that %(z, &) % %(w, &) if & > |w " z|. Thus, we see

%"w, µ(z, r)

#% %"z, µ(z, r)

#% r ,

and it follows that µ(z, r) % µ(w, r).

The next proposition contains two useful, though simple, computations.

Proposition 4.3.

!T

!z(w, z) = "i

!p

!z(z) " i

$

j-1

1j !!j+1p

!z! zj(z)(w " z)

j

and

!T

! z(w, z) = i

!p

! z(z) + i

$

j-1

1j !!j+1p

!zj! z(z)(w " z)j .


Proof. The proof is a short computation.

!T

!z(w, z) = i

&

'deg(p)"1$

j=1

1j !!j+1p

!zj+1 (z)(w " z)j "deg(p)$

j=1

1(j " 1)!

!jp

!zj(z)(w " z)j"1

"deg(p)"1$

j=1

1j !!j+1p

!z! zj(z)(w " z)

j

(

)

= "i!p

!z(z) " i

$

j-1

1j !!j+1p

!z! zj(z)(w " z)

j

since the first sum cancels all but the first term of the second sum. Since T is R-valued, !T! z (w, z) =!T!z (w, z) which gives the result for the second sum.

A useful consequence of these calculations is the following.

Proposition 4.4. Let YJ be a product of |J | operators of the form Yj = Lz, Lz, Lw, Lw.Then

%%YJ"t + T (w, z)

#%% # C|J |%(z, dNI (z, w, t))

dNI (z, w, t)|J | .

Before we prove the Proposition 4.4, we note that the result would be false if we replacedt + T (w, z) with t or T (w, z). Without both terms, there would be uncontrolled derivatives of p

remaining after applying Yj .

Proof. We have Lz

"t + T (w, z)

#= !T

!z (w, z) + i !p!z (z) = "i6

j-11j !!j+1p(z)

!z! zj (w " z)j.

Similarly, Lz

"t + T (w, z)

#= i6

j-11j !!j+1p(z)

!zj ! z(w " z)j . Analogous equalities (with z and w

interchanged and the sign switched) hold for Lw

"t + T (w, z)

#and Lw

"t + T (w, z)

#since

Lw

"t + T (w, z)

#=1!

!w" i

!p

!w

!

!t

2(t " T (z, w)) = "i

!p

!w(w) " !T

!w(z, w)

= "1

i!p

!w(w) + !T

!w(z, w)

2= "1!

!w+ i

!p

!w

!

!t

2(t + T (z, w)) = "Lw(t + T (z, w))

and Lw

"t + T (w, z)

#= "Lw(t + T (z, w)). But%%%%%%

$

j-1

1j !!j+1p(z)

!zj! z(w " z)j

%%%%%%# c1

%(z, dNI (z, w, t))

dNI (z, w, t).

Higher order derivatives are easier. As we just showed, the result of applying Y1 to t + T (w, z)

leaves a polynomial that is a sum of derivatives of (p (and hence well controlled). There are no t

terms remaining, so if j - 2, applying Yj is a matter of applying one of: !! z , !

!z , !!w , !

!w . Hence,the computation is simpler, and it can be done naively, i.e., there is no need to find any cancellingterms (which in general are absent).

5. Lq boundedness of order 0 operators

We are now ready to begin the proof Theorem 2.1. The idea is to show that e"i"T (w,z)K",)

satisfies the bounds of a Calderon-Zygmund kernel and the operator S" with kernel e"i"T (w,z)K",)

362 Andrew S. Raich

is restrictly bounded. These two facts, proven in Lemma 5.1 and Lemma 5.2, respectively, showS" satisfy the hypotheses of T(1) Theorem [19]. Consequently, S" is a bounded operator onLq(C). A result by Ricci and Stein [16] applies to pass from Lq(C) boundedness of S" to Lq(C)

boundedness of T" .

Lemma 5.1. Let T" be an OPF operator of order m # 2 with a family of kernel approximatingfunctions K",) . For k - 0, there exists Ck independent of " so that K",)(z, w) satisfies:

(a)%%%3z,w

4e"i"T (w,z)K",)(z, w)

5%%% # Ck|w " z|m"3

|" |k%(z, |w " z|)k . (5.1)

(b) If 2|w " w0| #| w " z|, then

%%%e"i"T (w,z)K",)(z, w) " e"i"T (w0,z)K",)"z, w0#%%% # Ck

%%w " w0%%

|w " z|3"m|" |k%(z, |w " z|)k . (5.2)

(c) If 2|z " z0| #| w " z|, then

%%%e"i"T (w,z)K",)(z, w) " e"i"T (w,z0)K",)"z0, w#%%% # Ck

%%z " z0%%

|w " z|3"m|" |k%(z, |w " z|)k . (5.3)

Also, the constants are uniform in ).

Proof. It is immediate from the Mean Value Theorem that (5.1) implies (5.2) and (5.3). Toprove (5.1), we use Proposition 4.3 and compute:

ei"T (w,z) !

!z

4e"i"T (w,z)K",)(z, w)

5= "i"

!T

!z(w, z)K",)(z, w) + !K",)

!z(z, w)

= !K",)

!z(z, w) " "

!p

!z(z)K",)(z, w) " "

$

j-1

1j !!j+1p

!z! zj(z)(w " z)

jK",)(z, w) .

Using the size estimate (3.6),%%%%!

!z

4e"i"T (w,z)K",)(z, w)

5%%%% # Z"pK",)(z, w) + "%(z, |w " z|)|w " z| K",)(z, w)

# Ck|w " z|m"3

|" |k%(z, |w " z|)k .

A virtually identical calculation shows

%%%%!

! z

4e"i"T (w,z)K",)(z, w)

5%%%% # Ck|w " z|m"3

|" |k%(z, |w " z|)k

which proves%% !! z

"e"i"T (w,z)K",)(z, w)

#%% satisfies the bound in (5.1). The bounds for the w

and w derivatives,%% !!w

"e"i"T (w,z)K",)(z, w)

#%% and%% !!w

"e"i"T (w,z)K",)(z, w)

#%%, use a repetitionof the calculations just performed and the identity e"i"T (w,z) = ei"T (z,w) (which follows fromProposition 4.1).

We now restrict ourselves to the case m = 0. Given an family T" of order 0, define arelated family of operators S" so that if K" (z, w) is the kernel of T" , the kernel of S" is given bye"i"T (w,z)K" (z, w). We have the following.


Lemma 5.2. S" and S'" are restrictly bounded uniformly in " , i.e., if ' $ C!

c (D(0, 1)),)')CN0

# 1 [where N0 is the constant from the cancellation condition (3.8)] and 'R,z0(z) ='( z"z0

R ), then33S""'R,z0#33

L2(C)# AR,

33(S" )'"'R,z0#33

L2(C)# AR (5.4)

with the constant A independent of " .

Proof. From the adjoint condition (f), it follows that we only have the prove the restrictedboundedness of S" .

33S",)"'R,z0#33

L2(C)=/*

C

%%%%

*

Ce"i"T (w,z)K",)(z, w)'

"w"z0R

#dw

%%%%2

dz

0 12

#/*

|z"z0|<2R

%%%%

*

CK",)(z, w)

4e"i"T (w,z)'

"w"z0R

#5dw

%%%%2

dz

0 12

+/*

|z"z0|-2R

%%%%

*

Ce"i"T (w,z)K",)(z, w)'

"w"z0R

#dw

%%%%2

dz

0 12

= I + II .

We estimate I first. By the cancellation condition (3.8)%%%%

*

CK",)(z, w)

4e"i"T (w,z)'

"w"z0R

#5dw

%%%%

# CN0

1max91, |" |N0%(z, R)N0

: supw$C

$

|I |#N0

R|I |%%%Y I"p

4e"i"T (w,z)'

"w"z0R

#5%%% .

We claim R|I |%%%Y I"p

"ei"T (z,w)'(w"z0

R )#%%% # C|I | max{1, |" ||I |%(z, R)|I |}. To see this, we first do

the case Y I"p = Z"p,w. It follows from Proposition 4.1 and Proposition 4.3 that

Z"p,w

4ei"T (z,w)'

"w"z0R

#5= ei"T (z,w)

R

!'

!w

"w"z0R

#

+ "ei"T (z,w)$

j-1

1j !!j+1p

!w!wj(w)(z " w)

j'"w"z0

R

#.

Hence,%%Z"p,w

"ei"T (z,w)'(w"z0

R )#%% # C

R (1 + "%(z, R)). Iterating this argument proves theclaim. Thus, for |z " z0| # 2R,

%%%%

*

CK",)(z, w)e"i"T (w,z)'

"w"z0R

#dw

%%%% # C ,

and

I # C

1*

|z"z0|<2Rdz

2 12

# AR .

When |z " z0| - 2R, |z " z0| %| z " w| for w $ supp'( ·"z0R ), so

II #C

/*

|z"z0|-2R

1|z"z0|4

1*

C

%%'"w"z0

R

#%% dw

22

dz

0 12

#CR21*

r>R

1r3 dr

2 12

# AR .

364 Andrew S. Raich

The final ingredient we need to prove Theorem 2.1 is a result by Ricci and Stein [16].

Theorem 5.3 (Ricci-Stein). In Rn & Rn, let K(· , ·) satisfy the following:

(a) K(· , ·) is a C1 function away from the diagonal {(x, y) $ Rn & Rn : x = y},(b) |3K(x, y)| # A|x " y|"n"1 for some A - 0,

(c) the operatorf 4,*

RnK(x, y)f (y) dy initially defined onC!

0 (Rn) extends to a bounded

operator on L2(Rn).

If P : Rn , Rn is a polynomial, then the operator T defined by

T [f ](x) =*

RneiP (x,y)K(x, y)f (y) dy

can be extended to a bounded operator from Lq(Rn) to itself, with 1 < q < !. The bound ofthis operator may depend on K , q, n, and the degree d of P but is otherwise independent of thecoefficients of P .

Proof of Theorem 2.1. The first step of the proof is to use the T(1) Theorem (p. 294 in [19])on S" . The T(1) Theorem says that if S is a continuous linear mapping from S to S 0 satisfying (5.2)and (5.3) (when k = 0) and S and S' are restrictly bounded in the sense of (5.4), then S extendsto a bounded linear operator from L2 to itself. In our case, this means S" extends to a boundedlinear operator. However, since all of the constants in Lemmas 5.1 and 5.2 are independent of " ,it follows that S" is a bounded linear operator from L2 to itself with constants independent in " .

Next, S" satisfies the hypotheses of Theorem 5.3, so T" is a bounded linear operator fromLq to itself for 1 < q < ! with a constant independent of " but possibly depending on the Lq

constant of S" and the degree of "T (which is # deg p), both of which are independent of " .

6. Equivalence with NIS operators

We now generate an OPF operator T" from an NIS operator T on a polynomial model Mp.Let k(p, q) be the kernel of an NIS operator T . On C & R, each kernel k can be associated witha kernel k by setting

k(z, w, t " s) = k((z, t), (w, s)) .

The convolution structure in t follows from the property that a polynomial model is translationinvariant in t = Re z2. Thus, we have (for appropriate '),

T ['](z, t) =*

C&Rk((z, t), (w, s))'(w, s) dw ds =

*

C&Rk(z, w, t " s)'(w, s) dw ds .

We set

K" (z, w) =*

Re"i" t k(z, w, t) dt (6.1)

and observe we also have

k(z, w, t) = 12,

*

Reit"K" (z, w) dt .


The integrals representing K" (z, w) and k(z, w, t) do not necessarily converge. For a tempereddistributionT and a Schwartz function', we know that ifF represents the partial Fourier transformin t , by definition, 1FT ,'2 =1 T , F'2. As an integral, this corresponds to:

1FT ,'2=*

C&Rk(z, w, t)

*

Re"it" '(w, " ) d" dw dt =

*

C&R

*

Rk(z, w, t)e"it" dt '(w, " ) dw dt . (6.2)

We make sense of (6.1) by the string of equalities in (6.2), and we say the integral!

R k(z, w, t)

e"it" dt is defined in the sense of Schwartz distributions. We similarly justify writing k(z, w, t) =1

2,

!R eit"K" (z, w) d" . If one of (or both of) the kernels is actually in L1(R) (in t or " ), then the

integral defined in the sense of Schwartz distributions agrees with the standard definition.

6.1. An NIS operator on C & R generates an OPF operator T! on C

Theorem 6.1. An NIS operator T of order m # 2 on a polynomial model Mp = {(z1, z2) $C2 : Im z2 = p(z1)} generates an OPF operator T" of order m with respect to the polynomial p.

Remark 6.2. The approximation conditions, (b) in the definition of OPF operators and (a) in thedefinition of NIS operators, imply one another since a partial Fourier transform is a continuousoperator on the space of Schwartz distributions. Also, the adjoint conditions (f) from OPFoperators and (e) from NIS operators, allow us to focus only k and K" as the computations willautomatically apply to k' and K'

" .

Theorem 6.1 is proved in a series of lemmas. We first show that if k is an NIS operator oforder m # 2, then K" is the kernel for a family T" of operators on C.

The proof that K",) satisfies the size conditions (3.6) and (3.7) is broken into two lemmas.We handle the m # 1 case and the m = 2 case.

Lemma 6.3. If m # 1, the kernel K",) satisfies the size condition (3.6).

Proof. It is enough to assume

YJ"p = Mn

"p = ei"T (w,z) !n

!"ne"i"T (w,z)

where |J | = n. Let - $ C!c (R) so that - 5 1 on ["1, 1], 0 # - # 1, and |-(n)| # cn. Also, let

-A(t) = -(t/A). We will estimate

!n

!"n

*

Re"i" (t+T (w,z))k)(z, w, t)-A(t) dt ,

and (3.6) will follow by sending A , !. The integral is compactly supported and the integrandis smooth, we can apply the derivatives inside of the integral. Integrating by parts (n + k) times

366 Andrew S. Raich

shows

cn

%%%%

*

Re"i" (t+T (w,z))

"t + T (w, z)

#nk)(z, w, t)-A

"t + T (w, z)

#dt

%%%%

= cn+k

|" |n+k

%%%%

*

Re"i" (t+T (w,z)) !

n+k

!tn+k

4"t + T (w, z))nk)(z, w, t)-A(t + T (w, z))

5dt

%%%%

= cn+k

|" |n+k

%%%%%%

*

Re"i" (t+T (w,z))

n+k$

j=0

cj!j

!tj

4"t+T (w, z))nk)(z, w, t)

5-

(n+k"j)A

"t+T (w, z)

#dt

%%%%%%

# cn+k

|" |n+k

n+k$

j=1

; *

|t+T (w,z)|#%(z,|w"z|)%(z, |w " z|)n"1"j |w " z|m"2 1

An+k"jdt

+*

%(z,|w"z|)#|t+T (w,z)|#2A|t + T (w, z)|n"1"jµ(z, |t + T (w, z)|)m"2 1

An+k"j

%%%-(n+k"j)" t+T (w,z)

A

#%%% dt

<.

If j = n + k, then

1|" |n+k

*

|t+T (w,z)|#%(z,|w"z|)%(z, |w " z|)n"1"(n+k)|w " z|m"2 dt

+ 1|" |n+k

*


An+k"j-" t+T (w,z)

A

#dt

#cn+k|w " z|m"2

|" |n+k%(z, |w " z|)k + |w " z|m"1

|" |n+k

*

%(z,|w"z|)#|t+T (w,z)||t + T (w, z)|"1"kµ(z, |t + T (w, z)|)"1 dt.

Using the substitution s = µ(z, |t + T (w, z)|)"1, | dsdt | % 1

µ(z,|t+T (w,z)|)|t+T (w,z)| , so

|w " z|m"1

|" |n+k

*

%(z,|w"z|)#|t+T (w,z)||t + T (w, z)|"1"kµ(z, |t + T (w, z)|)"1 dt

% |w " z|m"1

|" |n+k

*

|s|# 1|w"z|

1

%"z, 1

s

#k ds # cn+k|w " z|m"2

|" |n+k%(z, |w " z|)k .

If j < n + k, then using the support condition of -(j)A

"t + T (w, z)

#that |t + T (w, z)| % A, the

estimate simplifies to

1|" |n+k

*

|t+T (w,z)|#%(z,|w"z|)%(z, |w " z|)n"1"j |w " z|m"2 1

An+k"jdt

+ 1|" |n+k

*


An+k"j-(n+k"j)

" t+T (w,z)A

#dt

# cn+k%(z, |w " z|)n"j |w " z|m"2 1An+k"j

+cn+kAn"1"jµ(z, A)m"2 1

An+k"j+1A,!", 0 .

This complete the proof for m # 1.

Lemma 6.4. If m = 2, the kernel K",) satisfies the size conditions (3.6) and (3.7).

Proof. As is Lemma 6.4, we can assume that

YJ"p = Mn

"p = ei"T (w,z) !n

!"ne"i"T (w,z)


where |J | = n.

We first show the case µ(z, 1" ) - |w " z| and assume n = 0. From the definition of NIS

operators, |k)(z, w, t)| # c1%(z,|w"z|)+|t+T (w,z)| and | !k)!t (z, w, t)| # c2

%(z,|w"z|)2+|t+T (w,z)|2 . Sincek) is not integrable on R, we need to integrate by parts to obtain an estimate on K",) . However,since |w " z| is small, we need to be careful to integrate by parts as few times as possible andthen only for large t . Let A be a large number.

%%%%

*

|t+T (w,z)|# A|" |

e"i" t k)(z, w, t) dt

%%%% #%%%%

*

|t+T (w,z)|#%(z,|w"z|)e"i" t k)(z, w, t) dt

%%%%

+%%%%%

*

%(z,|w"z|)#|t+T (w,z)|# 1|" |

e"i" t k)(z, w, t) dt

%%%%%+%%%%%

*

1|" | #|t+T (w,z)|# A

|" |e"i" t k)(z, w, t) dt

%%%%%

! 1 +*

%(z,|w"z|)#|t+T (w,z)|# 1|" |

1|t + T (w, z)| dt

+ 1|" |

%%%%%

*

1|" | #|t+T (w,z)|# A

|" |e"i" t !k)

!t(z, w, t) dt

%%%%%+1

|" ||t + T (w, z)|

%%%%|t+T (z,w)|= A

"

|t+T (z,w)|= 1"

(6.3)

! 1 + log1

1/|" |%(z, |w " z|)

2+ 1

|" |

*

1|" | #|t+T (w,z)|# A

|" |

1"t + T (w, z)

#2 dt

! 1 + log1

1/|" |%(z, |w " z|)

2.

This is actually the estimate we are looking for since log" 1/|" |%(z,|w"z|)

#% log

"µ(z, 1" )

|w"z|#. Also, the

estimate is independent of A, so we can let A , !.

Now assume k - 1. Let - $ C!c (R), 0 # - # 1, supp -(· + T (w, z)) 6 ["2, 2],

-"t + T (w, z)

#= 1 if |t | # 1, and -(k)

"t + T (w, z)

## ck . We show the case |w " z| - µ(z, 1

" ).Let A $ R be large. Integration by parts n + k times shows:

%%%%!n

!"n

*

Re"i" (t+T (w,z))k)(z, w, t)-

4t+T (w,z)

A

5dt

%%%%

=

%%%%%%

n+k$

j=0

cj

"n+k

*

Re"i" (t+T (w,z)) !

j

!tj

""t + T (w, z)

#nk)(z, w, t)

# 1An+k"j

dn+k"j-

dtn+k"j

4t+T (w,z)

A

5dt

%%%%%%

# C

|" |n+k

&

'n+k"1$

j=0

AAn"1"jA"n"k+j +*

R

%%%%%!n+k

!tn+k

4"t + T (w, z)

#nk)(z, w, t)

5%%%%% dt

(

)

# C

|" |n+k

11

Ak+*

|t+T (w,z)|#%(z,|w"z|)%(z, |w " z|)"(k+1) dt +

*

|t+T (w,z)|-%(z,|w"z|)|t + T (w, z)|"(k+1) dt

2

# C

|" |n+k

11

Ak+ 1%(z, |w " z|)k

2.

Sending A , ! yields the desired estimate.

We have one estimate left to compute: The case |w " z| < µ(z, 1" ) and n - 1. Let A be a

large number. Let 0 # (1,(A2 # 1 so that 1 = (1 + (A

2 on ["A, A]. Let supp(1 6 ["2, 2]and supp(A

2 6 {t : |t | $[ 32 , 2A]}, and assume | !n

!tn(A2 | # cn

An if |t | - A2 and | !n(1

!tn |, | !n(A

2!tn | # cn

368 Andrew S. Raich

if |t | # 2. Since |z " w| # µ(z, 1" ), %(z, |z " w|) ! 1

" .%%%%!n

!"n

*

Re"i" (t+T (w,z))k)(z, w, t)

4(1"" (t + T (w, z))

#+ (2"" (t + T (w, z))

#5dt

%%%%

# cn

*

|t+T (w,z)|# 2|" |

|t + T (w, z)|n|k)(z, w, t)| dt

+n$

j=0

cj

%%%%%

*

R

"t + T (w, z)

#n !j(A2

"" (t + T (w, z))

#

!" jk)(z, w, t)e"i" (t+T (w,z)) dt

%%%%% .

Picking an arbitrary term and integrating by parts (n + 2) times, we have%%%%%

*

R

"t+T (w, z)

#n !j(A2 (" (t+T (w, z)))

!" jk)(z, w, t)e"i" (t+T (w,z)) dt

%%%%%

#cn+2

n+2$

k=0

*

R

%%%%%1

(t+T (w, z))n+2!k

!tk

4(t+T (w, z))nk)(z, w, t)

5!n+2+j"k(A2 (" (t+T (w, z)))

""n"2+k !" j !tn+2"k

%%%%% dt .

If n + 2 + j " k - 1, the term in the sum has support near 1|" | and A

|" | , so it is bounded by

*

R

%%%%%"n+2"k

"n+2

!k

!tk

4(t + T (w, z))nk)(z, w, t)

5!n+2+j"k(A2 (" (t + T (w, z)))

!" j!tn+2"k

%%%%% dt

# cn

|" |n+2

1|" |n"1"k

|" |n+2"k 1|" | + cn

|" |n+2

An"1"k

|" |n"1"k

|" |n+2"k

An+2"k+j

A

|" |A,!", cn

|" |n .

Finally, if n + 2 + j " k = 0, then j = 0 and k = n + 2 and we have the estimate

*

R

%%%%1

"n+2

!n+2

!tn+2

4(t + T (w, z))nk)(z, w, t)

5(A

2"" (t + T (w, z))

#%%%% dt

# cn

|" |n+2

*

|t+T (w,z)|- 12|" |

1|t + T (w, z)|3 dt = cn

|" |n .

Lemma 6.5. The operator T" has the w-cancellation condition (3.8).

Proof. Let YJ"p be a product of |J | operators of the form Y

j"p = Z"p, Z"p, M"p where

|J | = *+ n and n = #{j : Yj"p = M"p} and let ' $ C!(D(z0, &)). We have

K",)(z, w) =*

Re"i" t k)(z, w, t) dt ,

so that integration by parts yields

Z"pK",)(z, w) = Z"p

*

Re"i" t k)(z, w, t) dt

= !

!z

*

Re"i" t k)(z, w, t) dt "

*

R"!p

!z(z)e"i" t k)(z, w, t) dt

=*

Re"i" tLk)(z, w, t) dt .


Similarly, Z"p,zK",)(z, w) =!

R e"i" t Lzk)(z, w, t) dt . Also, recalling that Mf (z, w) = "i"t +

T (w, z)#f (z, w), we have M"pK",)(z, w) =

!R e"i" (t+T (w,z))Mk)(z, w, t) dt . Thus,

*

CYJ"pK",)(z, w)'(w) dw =

*

C

*

Re"i" tYJ k(z, w, t)'(w) dt dw ,

with the correspondence that if Yj"p = Z"p, Z"p, M"p, then Yj = L, L, M, respectively. Inte-

grating (n + k) times gives us:

*

CYJ"pK",)(z, w)'(w) dw =

**

C&R

"YJ k)#(z, w, t)e"i" t'(w) dt dw

= cn+k

"n+k

**

C&R

1!n+k

!tn+kYJ

2k)(z, w, t)e"i" t'(w)-(w, t) dt dw

+ cn+k

"n+k

**

C&R

1!n+k

!tn+kYJ

2k)(z, w, t)e"i" t'(w)(1 " -(w, t)) dt dw (6.4)

where - $ C!c (C & R) is a bump function on BNI ((z, 0), &). To estimate the integrals in (6.4),

the strategy is to expand"!n+k

!tn+k YJ#k)(z, w, t) and estimate an arbitrary term. It is important to

remember that in YJ , n of the terms are M and an L or L can hit either an M term or k)(z, w, t).

Expanding"!n+k

!tn+k YJ#k)(z, w, t), we see

!n+k

!tn+kYJ k)(z, w, t)

= !n+k

!tn+k

=

>$

|J0|+···+|Jn|=*

&

'c|J0|,... ,|Jn|X J0k)(z, w, t)

n?

j=1

("i)X Jj"t + T (w, z)

#(

)

@

A

=$

|J0 |+···+|Jn|=**0+···+*n=n+k

c|J0|,... ,|Jn|c*0,... ,*n

!*0

!t*0X J0k)(z, w, t)

n?

j=1

!*j

!t*jX Jj"t + T (w, z)

#, (6.5)

where X Jj is an operator composed only of X j = L and L. We pick an arbitrary term from thesum and show that it has the desired bound. Taking an arbitrary term from (6.5), we estimate theintegrals from (6.4) which reduces to the following two integrals:

I =

%%%%%%1

"n+k

**

C&R

!*0

!t*0X J0k)(z, w, t)

n?

j=1

!*j


#e"i" t'(w)-(w, t) dt dw

%%%%%%

and

II =

%%%%%%1

"n+k

**

C&R!*0

!t*0X J0k)(z, w, t)

n?

j=1

!*j


#e"i" t'(w)(1 " -(w, t)) dt dw

%%%%%%

where |J0|+· · ·+|J*| = * and *0 +· · ·+*n = n + k. Using Proposition 4.4 and the cancellation

370 Andrew S. Raich

condition (3.8), I has the estimate:

I # c|J0|,*0

|" |n+k

&m"|J0|

%(z, &)*0sup(w,t)

$

|I |#N|J0 |,*0

&|I |%%%%X

I

1e"i" t'(w)

n?

j=1

1!*j


#2-(w, t)

2%%%%

# c|J0|,*0

|" |n+k

&m"|J0|

%(z, &)*0sup(w,t)

$

|I |#N|J0 |,*0

&|I | $

|I0|+···+|In+1|=|I |cI0,... ,In+1

%%%%XI0"e"i" t'(w)

#

&n?

j=1

1X Ij

!*j


#2X In+1-(w, t)

%%%%

# cn,*,k

|" |n+k%(z, &)"k&m"* sup

(w,t)

$

|I0|#N|J0 |,*

&|I0|%%X I0(e"i" t'(w))%%

= cn,*,k

|" |n+k%(z, &)"k&m"* sup

w

$

|I0|#N|J0 |,*

&|I0|%%XI0" '(w)

%% .

To estimate II , we use size estimates and the support size of '.

II # cn,*)')L!

|" |n+k

*

|w"z0|#&

*

|t+T (w,z)|-%(z,&)

dNI (z, w, t)m"2"|J0|

%(z, dNI (z, w, t))1+*0

& %(z, dNI (z, w, t))n

dNI (z, w, t)|J1|+···|Jn|%(z, dNI (z, w, t))*1+···+*n dt dw

# cn,*

|" |n+k)')L!

*

|w"z0|#&

*

|t+T (w,z)|-%(z,&)µ(z, t+T (w, z))m"*"2 1

|t+T (w, z)|n+k"n+1 dt dw . (6.6)

If m # 2 or m = 2 and * - 1, then we use the substitution s = µ"z, t + T (w, z)

#"1, so| 1s

dsdt | %| t + T (w, z)|"1 and (6.6) becomes

II # cn,*

|" |n+k)')L!%(z, &)"k&2

*

|s|# 1&

s1"m+* ds # cn,*

|" |n+k)')L!%(z, &)"k&m"* .

If m = 2, * = 0, and k - 1, then a straightforward integration shows that II # cn,*

|" |n+k )')L!

%(z, &)"k&2. The integral in (6.6) diverges if m = 2 and * = k = 0, so we must estimate the tailterm more carefully in this case. With m = 2, * = 0, and k = 0, (6.6) simplifies to

II # 1|" |n%%%%

*

C

*

Re"it" !

*0

!t*0k)(z, w, t)

"t + T (w, z)

#n"(n"*0)'(w)"1 " -(w, t)

#dt dw

%%%% .

The key to this estimate is to recognize that !*0

!t*0k)(z, w, t)

"t + T (w, z)

#*0 satisfies the estimatesof an order 2 NIS operator. To integrate in t , we use the argument of (6.3) with & replacing |z"w|and see that

|II | # cn,0

|" |n*

C|'(w)| log

41

"%(z,&)

5dw ! cn,0

|" |n &2 log4

µ"z, 1"

#

&

5)')L!(C) .

Note that log( 1"%(z,&) ) % log(

µ(z, 1" )

& ). While this estimate is true for all " and &, the previous

estimate of II shows that we only have to consider the case when & # µ(z, 1" ) or equivalently,

"%(z, &) # 1.


Lemma 6.6. The kernel K",) satisfies the " -cancellation condition (3.9).

Proof. Since F"1F = I in the sense of Schwartz distributions,

%%X J k(z, w, t)%% # C|J |

µ(z, t + T (w, z))m"|J |

V (z, µ(z, t + T (w, z))))

implies 12,

!R XJ

"p

"ei" tK",)(z, w)

#d" = X J k(z, w, t) satisfies the same estimates.

The proof of Theorem 6.1 is complete.

6.2. An OPF operator T! on C generates an NIS operator k on C ! R

Theorem 6.7. A OPF operator T" of order m # 2 with respect to the subharmonic, non-harmonic polynomial p generates an NIS operator k of order m # 2 on the polynomial modelMp = {(z1, z2) $ C2 : Im z2 = p(z1)}.

We prove Theorem 6.7 in the same manner as Theorem 6.1. Remark 6.2 applies to Theo-rem 6.7 as well.

Lemma 6.8. The operator k satisfies the NIS cancellation conditions (3.12).

Proof. Let ' $ C!c

"B((z, t), &)

#. Also, let '(z, " ) =

!R e"i" t'(z, t) dt be the partial Fourier

transform in t of '(z, t). Let - $ C!c (R) with supp - 6 [" 2

%(z,&) ,2

%(z,&) ] and -(" ) = 1 when

" $ [" 1%(z,&) ,

1%(z,&) ]. Let X J be a product of |J | operators of the form of X j = Lz and Lz.

Then

X J**

C&Rk)(z, w, t " s)'(w, s) dw ds = 1

2,

*

C

*

R

*

RX J "ei" (t"s)K",)(z, w)

#'(w, s) d" ds dw

= 12,

*

C

*

Reit"XJ

"pK",)(z, w)'(w, " ) dw d"= 12,

*

Reit"XJ

"p

*

CK",)(z, w)'(w, " ) dw -(" ) d"

+ 12,

*

Reit"XJ

"p

*

CK",)(z, w)'(w, " ) dw (1 " -(" )) d" = I + II .

We estimate I and II separately. We first do the case m # 1 or m = 2 and |J | - 1. By (3.8),

|I | # c|J |&m"|J |*

Rsupw

$

|I |#N|J |

&|I |%%%XI

"p

4'(w, " )-(" )

5%%% d"

# c|J |&m"|J |*

R|-(" )| sup

w

$

|I |#N|J |

&|I |33X I'33

L1(t)d"

# c|J |&m"|J | 1%(z, &)

$

|I |#N|J |

&|I |33X I'33

L!(C&R)%(z, &) .

The last line follows from Hölder’s inequality and the size of supp'. The only difference betweenthe m = 2, J = 0 case and the previous estimate is the logarithm term in (3.8). The term toestimate is

%%%%

*

R-(" )&2 log

" 1"%(z,&)

#supw

%%'(w, " )%% d"%%%% . (6.7)

372 Andrew S. Raich

However, integration shows that! 1%(z,&)

0 log( 1"%(z,&) ) d" = 1

%(z,&) , so (6.7) simplifies to

&233'33

L!(C&R)

* 1%(z,&)

0log" 1"%(z,&)

#d" = &233'

33L!(C&R)

1%(z, &)

! &2)')L!(C&R) .

We estimate II in a similar fashion. We first cover the case when m # 1 or m = 2 and |J | - 1.

|II | = 12,

%%%%

*

Reit""1 " -(" )

# 1" 2

1XJ"p

*

C" 2K",)(z, w)'(w, " ) dw

2d"

%%%%

# c|J |*

|" |> 1%(z,&)

|" |"2&m"|J | $

|I |#N|J |

&|I |33" 2XI"p'(w, " )

33L!(w)

d" . (6.8)

The terms in the sum can be rewritten the more useful way:

33" 2XI '(w, " )33

L!(w)= sup

w

%%%%1

2,

*

R" 2XI

"pei" t'(w, t) dt

%%%%

= c supw

%%%%

*

Rei" t

1!2

!t2 X I'(w, t)

2dt

%%%% # c2%(z, t)

3333!2

!t2 X I'

3333L!(C&R)

. (6.9)

Using the estimate from (6.9) in (6.8),

|II | # c|J |&m"|J |*

|" |> 1%(z,&)

|" |"2$

|I |#N|J |

&|I |3333

1!2

!t2 X I

2'(w, t)

3333L!(C&R)

%(z, &) d"

# c|J |&m"|J | $

|I |#N|J |

&|I |%(z, &)23333

1!2

!t2 X I

2'(w, t)

3333L!(C&R)

(6.10)

# c|J |&m"|J | $

|I |#N 0|J |

&|I |333X I'(w, t)

333L!(C&R)

.

In the final estimate, we used the fact that %(z, &) !!t can be generated by commutators of &Xterms. As in I , the difference between the m = 2, J = 0 and the case already estimated is

the logarithm term in (3.8). However,!!%(z,&)"1

| log( 1"%(z,&) )|" 2 d" = %(z, &), so we can repeat the

estimate in (6.10) replacing |" |"2 with| log( 1

"%(z,&) )|" 2 and achieve the same conclusion.

Lemma 6.9. The operator k has the NIS size conditions (3.11).

Proof. It is enough to find the estimate on |k)(z, w, t)|. We handle the m = 2 separately. Firstassume m # 1. If dNI (z, w, t) = |z " w|, then we break the integral in two pieces and estimateeach piece separately.*

Rei" tK",)(z, w) d" = 1

2,

*

|" |# 1%(z,|w"z|)

ei" tK",)(z, w) d" + 12,

*

|" |- 1%(z,|w"z|)

ei" tK",)(z, w) d" .

Estimating the first integral gives us:%%%%%

*

|" |# 1%(z,|w"z|)

ei" tK",)(z, w) d"

%%%%% # c0|w " z|m

|w " z|2%(z, |w " z|) = c0dNI (z, w, t)m

V (z, dNI (z, w, t)).


The tail term is no harder: By (3.6) with * = n = 0 and k = 2,%%%%%

*

|" |- 1%(z,|w"z|)

ei" tK",)(z, w) d"

%%%%% # c2|w " z|m

|w " z|2%(z, |w " z|)2

*

|" |- 1%(z,|w"z|)

1" 2 d"

# c2|w " z|m

|w " z|2%(z, |w " z|) .

The case dNI (z, w, t) = µ(z, t + T (w, z)) is the " -cancellation condition (3.9).

Now assume m = 2. The estimate to prove is

|k)(z, w, t)| # CdNI (z, w, t)2

V (z, dNI (z, w, t))= C

1%(z, dNI (z, w, t))

.

Let - $ C!c (R) where supp - 6 ["2, 2], -(" ) = 1 if |" | # 1, 0 # - # 1, and

%% !k-

!" k (" )%% # Ck .

Let % = %(z, dNI (z, w, t)). We have

k)(z, w, t) =*

Rei" tK",)(z, w)-("%) d" +

*

Rei" tK",)(z, w)(1 " -("%)) d" = I + II .

Before we estimate I , observe!!&

log s

sk ds = "klog s

sk+1 + kk+1

1sk . Also, with the change of variables

s = 2µ(z,1" )

|w"z| , | !s!" | % µ(z, 1" )

|w"z|1|" | and %(s, |w " z|) % 1

|" | , so

I !*

|" |# 2%

log

/2µ"z,

1"

#

|w"z|

0

d" %*

|s|- µ(z,%)|w"z|

log s

s%(z, |w " z|s) ds

%* !

µ(z,%)|w"z|

infj,k-1

1%%azjk

%%|w " z|j+k

log s

sj+k+1 ds

! infj,k-1

1%%azjk

%%|w " z|j+k

&

7'log4

µ(z,%)|w"z|5

4µ(z,%)|w"z|5j+k+1 + 1

4µ(z,%)|w"z|5j+k

(

8)

! infj,k-1

1%%azjk

%%|w " z|j+k

|w " z|j+k

µ(z,%)j+k% 1%(z, µ(z,%))

= 1%

.

To estimate II , we need to separate the cases % = %(z, |w " z|) and % = |t + T (w, z)|. Wefirst do the case % = %(z, |w " z|). By (3.6) with k = 2 and * = n = 0,

II !* !

1%

1" 2%2 d" % 1

%.

Now assume % = |t + T (w, z)|. Then

II ! 1"t + T (w, z)

#2

%%%%%

*

|" |- 1|t+T (w,z)|

ei" (t+T (w,z)) !2

!"2

4e"i"T (w,z)K",)(z, w)(1 " -(" |t + T (w, z)|))

5d"

%%%%% .

If both " -derivatives are applied to K",) ,

1"t + T (w, z)

#2

*

|" |- 1|t+T (w,z)|

%%%%!2

!" 2

4e"i"T (w,z)K",)(z, w)

5%%%%"1 " -(" |t + T (w, z)|)

#d"

% 1"t + T (w, z)

#2

*

|" |- 1|t+T (w,z)|

1" 2 d" % 1"

t + T (w, z)# .

374 Andrew S. Raich

Next, if one " -derivative is applied to K",) and one to -, then

1"t + T (w, z)

#*

|" |- 1|t+T (w,z)|

%%%%!

!"

4e"i"T (w,z)K",)(z, w)

5%%%% -0(" |t + T (w, z)|)) d"

% 1"t + T (w, z)

#*

|" |% 1|t+T (w,z)|

1"

d" % 1"t + T (w, z)

# .

Finally, if - receives both " -derivatives,!|" |- 1

|t+T (w,z)|

%%K",)(z, w)%%-00(" |t + T (w, z)|)) d" %

1"t+T (w,z)

# .

Proving Theorems 6.1 and 6.7 proves Theorem 2.2.

References[1] Berndtsson, B. Weighted estimates for ! in domains in C, Duke Math. J. 66(2), 239–255, (1992).

[2] Berndtsson, B. ! and Schrödinger operators, Math. Z. 221, 401–413, (1996).

[3] Christ, M. On the ! equation in weighted L2 norms in C1, J. Geom. Anal. 1(3), 193–230, (1991).

[4] Fornæss, J. E. and Sibony, N. On Lp estimates for ! , in Several Complex Variables and Complex Geometry, Part 3,(Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., 52, Part 3, 129–163, Providence, RI, American MathematicalSociety, (1991).

[5] Hörmander, L. L2 estimates and existence theorems for the ! operator, Acta Math. 113, 89–152, (1965).

[6] Hörmander, L. An Introduction to Complex Analysis in Several Variables, North-Holland Mathematical Library,7, North-Holland Publishing Co., Amsterdam, 3rd ed., (1990).

[7] Kurata, K. An estimate on the heat kernel of magnetic Schrödinger operators and uniformly elliptic operators withnonnegative potentials, J. London Math. Soc. 62(3), 885–903, (2000).

[8] Nagel, A. Vector fields and nonisotropic metrics, in Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud.,241–306. Princeton University Press, (1986).

[9] Nagel, A., Rosay, J.-P., Stein, E. M., and Wainger, S. Estimates for the Bergman and Szegö kernels in C2, Ann. ofMath. (2) 129, 113–149, (1989).

[10] Nagel, A. and Stein, E. M. The !b-complex on decoupled domains in Cn, n - 3, Ann. of Math. (2), to appear.

[11] Nagel, A. and Stein, E. M. The !b-heat equation on pseudoconvex manifolds of finite type in C2, Math. Z. 238,37–88, (2001).

[12] Nagel, A. and Stein, E. M. On the product theory of singular integrals, Rev. Mat. Iberoamericana 20, 531–561,(2004).

[13] Nagel, A., Stein, E. M., and Wainger, S. Balls and metrics defined by vector fields I: Basic properties, Acta Math.155, 103–147, (1985).

[14] Raich, A. Heat equations in R & C, submitted, [arXiv:math.CV/0508571], (2005).

[15] Raich, A. Pointwise estimates for relative fundamental solutions of heat equations in R & C, preprint, (2006).

[16] Ricci, F. and Stein, E. M. Harmonic analysis on nilpotent groups and singular integrals, I, oscillatory integrals,J. Funct. Anal. 73, 179–194, (1987).

[17] Shen, Z. Estimates in Lp for magnetic Schrödinger operators, Indiana Univ. Math. J. 45(3), 817–841, (1996).

[18] Shen, Z. On fundamental solutions of generalized Schrödinger operators, J. Funct. Anal. 167(2), 521–564, (1999).

[19] Stein, E. M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, PrincetonMathematical Series, 43, Princeton University Press, Princeton, NJ, (1993).

Received May 16, 2005

Texas A&M Universitye-mail: [email protected]

Communicated by Steven Bell

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