(C.I.M.E.)

SOLVABILITY OF INVARIANT DIFFERENTIAL OPERATORS ON HOMONOGEOUS

MANIFOLDS

S. Helgason

Varenna. August 25 /September 2, 1975

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SOLVABILITY OF INVARIANT DIFFERENTIAL OPERATORS ON HOMOGENEOUS

MANIFOLDS

Sigurdur Helgason

MIT Cambridge Mass .

§1 . Introduction and summary .

Over a 100 years ago S . Lie [18] raised the

following question:

S. Helgason

Given a differential equation, how can knowledge

about its invariance group be utilized towards its solution?

( * )

Special case . Consider the differential equation

~ = Y(x,y) dx X(x,y)

in the plane, It is called stable under a one- parameter

group cj>t (t E ffi) of diffeomorphisms if each

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S. Helgason

of rotations around the origin permutes the integral curves,

i.e., leaves the equation, stable.

For a one-parameter group (~t) of transformations

in the plane let U denote the induced vector field

U = [·d C ~ t • P ) J a a = ~(p)~x + n(p)~Y p dt t=O o o

2 p ~ JR •

Theorem (Lie). Equation(*) is stable under (~t) if and

only if the vector field Z = x-l + y_l satisfies - ax ay

[U,Z] = t..Z (>.. a function)

In this case (Xn - Y~)-l is an integrating factor for the

equation Xdy - Ydx = 0.

Thus knowing a stability group for a differential

equation provides a way to solve it. For the example above

the equation is stable under the group

$t(x,y) = (x cos t - y sin t, x sin t + y cos t)

for which

a a u = - Yax + xay

and the theorem gives the solution

In spite of extensions of this theorem to higher

order partial differential equations these methods have had

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S. Helgason

a limited influence on the general theory of differential

equations. Nevertheless, Lie ' s question has been of enormous

importance in mathematics, in fact leading to the theory of

Lie groups.

Aiming for less generality we can pose a few

problems in Lie's spirit. Let G/H be a coset space, H

being a closed subgroup of ~ Lie group G and let D be a

differential operator on the manifold G/H which is

invariant under G in the sense that

D(f o T(g)) = (Df) o T(g) 00

f i C (G/H)

for all g c G, the mapping T(g) denoting the translation T(g):xH ~ gxH of G/H onto itself. Let o denote the

origin eH of G/H . We can then ask the following questions.

A. Does there exist a fundamental solution for D, i.e., a

dist~ibution T on G/H such that

DT = o,

o denoting the delta-distribution at o on G/H?

B. Is D locally solvable, i.e., is

for some neighborhood V of o?

C. Is D globally solvable, i.e., is

DC""(G/H) = c""(G/H) ?

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S. Helgason

Let D(G/H) denote the algebra of all G- invariant

differential operators o~ G/H. For example, if G is the

group of translations of Rn and H = {0} then D(G/H)

consists of all differential operators with constantv

coefficients; by results of Ehrenpreis and Malgrange,

questions A) E) and C) have positive answers in this

case . We shall see however that in general the answers

depend on the space G/H .

We shall now survey - in chronological order - some

of the principal results which have been obtained for the

questions above.

First if X = G/K is a symmetric space of the

noncompact type (i.e . , G noncompact, connected semisimple

with finite center and K a maximal compact subgroup) then

as proved in [11] each D E D(G/K) has a fundamental solu-

tion, and consequently, by convolution, D is locally

solvable . Since in solving the equation DT = o we may,

by the K-invariance of B and o, assume T to be K-invariant, the problem can be handled within Fourier

analysis of K- invariant functions on X. The global

solvability discussed later does not see~ to be accessible

by this method .

It was proved in Peetre [19] (cf . also Hormander

[15] , p. 170) that a differential operator of constant

strength is necessarily locally solvable . Without entering

into the precise definition, a differential operator of . constant strength is in a certain sense a bounded perturba-

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S. Helgason

tion of a constant coefficient operator. One might wonder

whether an invariant differential operator is not necessarily

of constant strength. This can not be so because then it

would be locally solvable and the operator

which is essentially the non-locally solvable Levi-operator

is (cf. [4]) a left invariant differential operator on the

Heisenberg group

x,y,zE,lR.

Thus we see that B has in general a negative

answer. In view of this, one can modify the problem and ask:

For what Lie groups L are all left invariant differential

operators (i.e., all members of D(L) = D(L/{e}) locally

solvable. Using Hormander's necessary condition for the

local solvability of a differential operator [15], p. 157,

he, and independently C~rezo-Rouvi~re [4], proved that in

order that each D ( D(L) should be locally solvable it is

necessary and sufficient that the Lie algebra of L be

either abelian or isomorphic to the algebra of n x n

matrices where each row except the first is 0 (and thus has

an abelian ideal of codimension 1).

The contrast between this negative result for L

and the positive result for the symmetric space G/K

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disappears when we view L in the context of symmetric

spaces. A coset space ~lc (B ~·Lie group, C a closed

subgroup) is called a symmetric coset space if there exists

an involutive automorphism cr of B with fixed point set

C. The spaces G/K considered above have this property.

Now we consider L as a homogeneous space under left and

right translations simultaneously, i.e., we let the product

group L x L act on L by

g E L.

The subgroup leaving e fixed is the diagonal L* - L so

we have the coset space representation

L = (L X L)/L*.

Then the algebra D(L x L/L*) is canonically isomorphic to

the algebra Z(L) of hi-invariant differential operators on

L and the natural problem becomes: Given DE Z(L), is it

locally (globally) solvable on L?

This problem was considered by Rals for simply

connected nilpotent Lie groups L. He proved that each

D E Z(L) has a fundamental solution and hence is locally

solvable . The method is based on harmonic analysis on L.

Shortly afterwards [13] I proved local solvability for each

D E Z(L) for the case when L is semisimple. The proof was

an easy consequence of a structure theorem of Harish~Chandra

for the operators in Z(L) but did not require any harmonic

analysis on L.

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S. Helgason

At about the same time I proved global solvability,

DC00

(G/K) = C00

(G/K) for each invariant differential operator

D on the symmetric space G/K . The proof [13] is based on

a characterization of the image of C~(G/K) under the Radon

transform on G/K .

What about global solvability on L? A remarkable

example of Cerezo-Rouviere [5] shows that for any complex

semisimple Lie group L there exists an element w € Z(L)

which is not globally solvable . This w can be taken as the

imaginary part of the complex Casimir operator on L . But

L viewed as a real Lie group has a Casimir operator C

which they showed, using harmonic analysis on L, is globally

solvable . This global solvability was more recently proved

by Rauch- Wigner [21], for all non compact real semisimple

Lie groups L, with finite center, using entirely different

methods . The main step in their proof is verifying that no

null bicharacteristic of C lies over a compact subset of

G.

The case of a solvable Lie group L has so far been

left out . Rais' method for the nilpotent case was extended

by Duflo - Rals [7] to solvable L and they proved that each

D € Z(L) i s locally solvable . An alternative proof was

found by Rouvi~re [22], modifying in an interesting manner

methods which Hormander had introduced for constant

coefficient operators . His method, being local, does not

give the supplementary result of [7] that if L is an

exponential solvable Lie group then D has a fundamental

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solution .

The results I have quoted on the Lie groups L and

on the symmetric space G/K suggest the following more

general question.

Let B/C be a symmetric coset space and D a

B- invariant differential operator on it . Is D necessarily

locally solvable?

§2 . Solvability results. Indications of proofs.

We shall now indicate the proofs of the principal

results already stated. We emphasize that the proofs of

Theorems 1 - 5 are quite independent and socan be read in any

order.

We have already mentioned the fact that a left-

invariant differential operator on the Heisenberg group is

not necessarily locally solvable . In order to describe the

proof of Rais ' positive result for nilpotent groups I recall

the main results from harmonic analysis on nilpotent Lie

groups [17] .

Let G be a simply connected nilpotent Lie group ,

~ its Lie algebra ; G acts on ~ via the adjoint repre-

sentati6~ and by duality on OJ-*, the dual vector space to ~· To each fE ~* is associated a unitary irreducible

representation rrf as follows : Let H be a closed

connected subgroup with the property that its Lie algebra

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S. Helgason

·"9 satisfies f( [~ ,~]) = 0 . Ass.ume H of maximum 'dime!"lsion with these properties. Then we can define a homomorphism

of H into the circle group T by

xf(exp X) = 2nlf(X) e ,

and let nf denote the representation of G induced by Xr ·

It is known that nf is independent of · the choice of H;

moreover if ·f 1 , f 2 E ·Of*, then equivalent if and only. if fl

and nf are unitarily 2

f 2 are in the same

orbit· of G actin.g on '}* . Thus the orbit space 0}*/G is "" identified with G, the· dual spac·e of G . The character Tf

of nf is a distribution on G (Dixmier [6]) which is given

by the formula

(1)

where denotes the Euciidean Fourier transform and ~ is

a G-invariant measure on the orbit G• f. Finally there is a

Plancherel formula: there exists a measure on tJ* IG

such that

( 2)

f 0 denoting the orbit G•f , Using (2) and (1) for

~ o exp =·ll' . we get for a Haar measure df on OJ*

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S. Helgason

(3) 1 .(f)df =I ri • of

U(,). Recall now that in terms of differential operators,

U(Uj) = D(a) and Z("J) = Z(a). The character' Tf is an eigend1str1but1on of each Z E Z(a)

(4)

and the eigenvalue Z(f) is just the polynomial s-1 (Z)

evaluated at f.

Let us also recall the following general fact

([2], [3], [20]): Let P > 0 be a positive polynomial on

En. Then if Re s > 0, P8 defines a distribution

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S. Helgason

This can be used to construct a fundamental solution for a

constant coefficient differential operator D on mn:

Taking Fourier transform f + f we have (Df)- = Pf where

P is a polynomial, which we may assume positive, replacing

D by DD 1 if necessary (*=adjoint). The tempered

distribution

then satisfies T1 * Ts = Ts+l which is holomorphic near

s = -1. Expanding Ts in a Laurent series near this point

we get Ts Iak(s+l)k where each ak is a distribution. -r

The relation above then gives T1 1 a 0 = o and up to a

factor, is a fundamental solution for D.

We can now prove the following result (cf. [20]).

Theorem l. Each bi-invariant differential operator Z on a

nilpotent Lie group G is locally solvable.

We may assume G simply connected and use the

notation above. Replacing Z by ZZ we may assume the A

polynomial P = Z(f) in (4) to be positive. Let ~ denote

the distribution ~ + (Z•~)(e) (• =adjoint). Then by (2) -

( 4)

( 6)

In analogy with Ts above we define ~s for Re s > 0 by

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By the result about P quoted, s ~ ~ extends to a s

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meromorphic d i stribution-valued function on C and by (5)

and ( 6),

Moreover , by (l) and (3),

( 7)

whence z~s = ~s+l' i.e .,

b 0 in the Laurent series

~l = ~ .

~ * ~s = ~s+l " Again the term 00 k

~s L bk(s+l) gives a - r

fundamental solution for Z . ..

Duflo and Rais [7] extepded this argument to

solvable groups giving

Theorem 2 . A hi-invariant differential operator D on a . solvable Lie group is locally solyable.

A completely different proof was found by Rouvi~re

[22] . We sketch his argument below .

Let

and let

x1 , . . . , Xn be a bas i s of any Lie algebra '1' denote the derived algebra [~,'}] . For any j 1

l < j < n let aj denote the endomorphism of U(~) given

by

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( 8)

Lemma 1 . Assume Xj ~ DOf·

a J z

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on an open set U C G, u E cJ(u) f E c""(u), P any

differential operator on U with adjoint P*, where

P' = [P,f], P" = [P' ,f] . Using this on P E z 1 and P € Z(~),

P $ Z(Dr~). Then there exists an operator

Q E Z(Dr~) 1\ Z(Dr-l~), Q + 0 and a neighborhood U of e in G such that

( 11 ) II Qu II 5., II Pu II

for all u E. C~(U) .

This is proved by induction on r. The case r = 1

comes from (10) and Lemma 1 . In fact, let

j = dim ~ - dim D~. Then j > 1 and we take for Q a nonzero derivative of p of maximal order in

Then Q E. z

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that DrGf = {O} so Z(Dr~) = c . Then Lemma 2 gives an inequality

Jl u II ~ C IIPu II

which as well known [14] implies local solvability, i . e.,

Theorem 2.

Next we consider the semisimple case . Here we have

by [10]'

Theorem 3. Each hi-invariant differential operator D on a

semisimple Lie group G is locally solvable .

For this let u, be a connected open neighborhood of 0 in Cf such that exp is a diffeomorphism of

onto an open neighborhood UG of e in G. In addition we

assume, as we can (cf . [9]) that u,. is completely invarian~ that is, for each compact subset c c UOJ-' the closure

Cl(Ad(G) • C) C U,. Let n denote the Jacobian of the

exponential mapping, i . e . ,

1 -ad X n ( x) = det ( - :d x· )

Then according to Harish-Chandra there exists a specific

Ad( G) - invariant differential operator Pn on 't with constant coefficients such that

(12) (Df) o exp = n- l/2p (n 112f o exp) D

whenever f is a C00

function on UG' locally invariant

under inner automorphisms . (This formula is proved in [9]

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S. Helgason

by computing the radial parts of the operators D and p0 ;

for an analogous global formula for the symmetric space

G/K (with G complex) see [11]).

Denoting by A the mapping

we have by (12),

( 13)

If G is compact, the Ad(G)-invariant distr!bu-

tions on form in a natural way the dual to the space of

Ad(G) - invariant functions in C~(U~) . The adjoint of A is

given by A*S (n112s)exp, the image of the distribution

n112s under exp, so by (13)

if S is an Ad(G) - invariant distribution on U~. Harish-

Chandra proved ([8], [9), p .· 477), that (14) is true even if

G is noncompact; since the process of averaging over G is

not available, entirely different methods are required. Now

(p0*)* has a fundamental solution S on ~ which we may

take Ad(G) - invariant (cf . the construction of a 0 before

Theorem 1) . Using (14) we get a fundamental solution for D

on UG' DT = o, and Theorem 3 follows by convolution. Global solvability does not in general hold in

Theorem 3 as shown by the following example of Cerezo -

Rouviere [5) . Let G be a complex semisimple Lie group,

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K a maximal compact subgroup and OJ, {l their respective Lie algebras. Then OJ = ~ + J-k. where J is the complex structure of~· Let (Ti) be a basis of~. orthonormal

forthe negative of the Killing form, and put

w = L (JTi)·Ti· i

Then if TE-i, [T,Ti] =}: cijTJ' where (cij) is skew j

so [T,w] = L [T,JTi].Ti + (JTi).[T,T1J i =

=

L cij(JTJ)·Ti- L cij(JTJ)•Ti = 0. i,j i,j

=

Similarly [JT,w] = 0 since w can also be written w = L Ti(JTi). Thus w is a hi-invariant differential

i

operator on G. Since it annihilates all C® functions on

G which are right invariant under K it is not globally

solvable.

Nevertheless we have by [5] for complex G and by

[21] for real G,

Theorem 4 . Let G be a connected noncompact semisimple Lie

group with finite center . Then the Casimir operator C is

globally solvable .

This follows from general results of Hormander [16]

once the following three facts have been established [21].

(I) c

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is one-to-one on c""(a). c

S. Helgason

(II) For any compact set r C G there exists

another compact set r' C G such that

r C Int f' and if u e E'(G) (distributions

on G with compact support) with supp(Cu)C r

then supp u c f'.

(III) No null characteristic of c lies over a

compact subset of G.

The Holmgren uniqueness theorem implies for the

analytic operator C that if a hyper-surface S in G is

non-characteristic at x0 then a solution u to Cu = 0 which vanishes on one side of S must vanish in a neighbor-

hood of in G. This implies (I) and .(II) without

difficulty because it is easy to determine plenty of non-

characteristic surfaces for C.

In order to describe the proof of (III) let us

recall a few facts concerning null bicharacteristics.

Let M be a manifold, T*M its cotangent bundle.

If (x1 , ... ,xn) is a local coordinate system on M then

is a local coordinate system (x1 , •.. ,xn'~l'''''~n) on

T*(M). The two-form

n S) = r d~ i " dxi

1

is called the canonical 2-form on T*M. It has the property

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S . Helgason

that if ~ is a diffeomorphism of M then the lifted

diffeomorphism of T*M preserves n (cf . e . g . [1], p . 97) .

Given a 1 - form w on T*M let X denote the vector field w

on T*M given by

(15) rl(• X)= w( • ) ' w •

If w = Eaidxi + Ebjd~j then

(16) a a 2Xw = Ebiaxi - Eaja~j

If w = df where f E C00

(T*M) then Xdf is called a

Hamiltonian vector field; the function f is easily seen to

be constant on the integral curves of Xdf ' In fact, if

(x1 (t), . .. ,~n(t)) is an integral curve of Xdf then by

(16),

so

1 af 2 n-' k

In particular, let c:T*G ~m be the principal

symbol of C, i . e ., if is the

metric tensor determined by the Killing form . The

b i characteristics of c are by definition the integral

curves in T*G - ( 0) to the Hamiltonian vector field xdc '

If the constant value of c on an integral curve of xdc

is 0 , this curve is called a null bicharacteristic .

By the invariance property of n stated above it

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S. Helgason

is clear that n, as a 2-form on . T*G, is hi-invariant.

Since C is hi-invariant on G, c is a hi-invariant

function on T*G so Xdc is a hi-invariant vector field on

T*G. Using in addition the fact that Xdc is Hamiltonian

the authors prove that its integral curves have the form

y(t) = (a exp tX, wa exp tX)

where and w a left invariant 1-form on G.

Now suppose y is a null bicharacteristic for C

lying over a compact subset of G. Then the same is true of

the curve

y0 (t) = (exp tX, w ) exp tX

so c(y0 (t) = 0 and exp tX lies in a maximal compact sub-group K of G. But then B(X,X) < 0 implies easily that

c is strictly negative on y 0 , which is a contradiction.

Finally, we have by [11] and [13],

Theorem 5. Let X = G/K be a symmetric space of the non-

compact type, D a G-invariqnt differential operator on X.

Then

(i) D has a fundamental solution.

(ii) D is g~obally solvable.

This theorem can be proved either by means of the

Fourier transform on X or by means of the Radon transform

on X. After (i) is proved, (ii) follows on the basis of

general functional analysis methods provided one can prove

for each ball B C X:

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(17) f( C~(X), supp(Df)~B -1> supp(f)C B.

Here risupp" stands for "support".

We now sketch how (i) and (17) can be proved by

means of the Radon transform on X. Let G = KAN be an Iwasawa decomposition of G where A is abelian and N

nilpotent. A horocycle in X is by definition an orbit of

a point in X under a subgroup of G conjugate to N. The

group G permutes the horocycles transitively; more precise-

ly, if ~O denotes the horocycle N•o (o = origin in X) then each horocycle ~ can be written ~ = ka·~o where a£ A is unique and k £ K is unique up to a right multi-

plication by an element m in the centralizer M of A in

K. Thus the space of all horocycles is naturally

identified with (K/M) x A. Each horocycle ~ is a sub-

manifold of X so inherits a volume element dcr from X;

the Radon transform of a function f on X is defined by

f(~) = Jr(x)dcr(x), ~

whenever this integral exists. There exists a G-invariant

differential operator D on the manifold such that

(18) (Dr)" = B? fEC~(X);

moreover fl has the form

(19) k E K,a € A),

where P is a translation-invariant differential operator on

the Euclidean space A (cf. [10], §4-5). Thus the Radon

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transform converts the operator D into a constant coeffici -

ent differential operator . This principle gives a fundament-

al solution for D by a method similar to the one used for

Theorem 3 (cf . [11]) .

In order to use this principle to deduce (17) the

following lemma is decisive .

Lemma 3. Let f E C~(X) and B C X a closed ball. Assume

that f( 0 = 0 whenever the horoc;Ycle ~ is disjoint from B. Then f vanishes identically outside B.

To verify (17) we may assume B centered at o.

Let R denote its radius. No~ supp(Df) ~ B implies by

(18) and (19) and d(o~ka·~ 0 ) ! d(o,a•o) that

d

Pa(f(ka · ~0 )) = 0 if d(o,a•o) > R,

denoting distance . But the function a -+ f(ka · ~ ) 0

has

compact support so by the Lions- Titchmarsh convexity theorem

we conclude that f(ka · ~ 0 ) = 0 if d(o,a•o) > R. But then Lemma 3 implies supp(f)

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Lemma 4. Let FE C00

(Rn) satisfy the conditions:

(i) For each integer k > 0, F(x)lxlk is

bounded.

(ii) There exists a constant A such that

F(~) = 0 for d(O,~) > A, d denoting

distance.

S. Helgason

Then F vanishes identically outside the ball

I xi < A.

Proof (cf. [12]). Suppose first F is a radial

function, i.e., F(x) = ~(jxj) where ~ E C00

(R) is even. A 00

Then there exists an even function ~ £ C (R) such that

$(d(0,0) = F(O. Now (20) takes the form

where A n-1 is the area of the unit sphere in n-1 'R • By a

simple change of variables (21) is transformed into Abel's

integral equation and is inverted by

where c is a constant. Now by (ii), ~(u-1 ) = 0 for -1\ 1

0 < u < A so· ~(s- ) 0 for 0 < s < A-1 , proving the

lemma for the case when F is radial.

Consider now the general case. Fix x ~ Rn and

consider the function

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= ( F(x+k •y)dk, J O(n)

n y € 1R ,

S. Helgason

where dk is the normalized Haar-measure on the orthogonal

group O{n); gx(y) is the average of F on the sphere with

center x and radius IYI. We have

gx(~) = J F(x+k·~)dk, O(n)

where x+k·~ is the hyperplane k·~ translated by x.

Clearly

d(O,x+k·~) ~ d(O,~) - lxl

so if d(O,~) > lxl + A we have gx(~) = 0. But

radial function so by the first part of the proof,

for IYI ~ lxl + A.

is a

This means that if S is a sphere with surface element dw

then

(22) J F ( s ) dw ( s ) = 0 s

if S encilioses the solid ball BA(O) (with center 0 and -,

radius A), Lemma 4 will therefore follow from another lemma.

Lemma 5. Let F E C~(Rn) such that for each k ~ 0,

F(x) lxlk is bounded. Assume (22) whenever S encloses the

solid ball BA(O) . Then F vanishes identically outside

BA ( 0).

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S. Helgason

The idea for proving this is to perturb S in

(22) a bit and ~1fferentiate with respect to the parameter

of perturbation, thereby obtaining additional relations. If

S = SR(x) and BR(x) the corresponding ball, then by (22),

so

that is

(23)

I F(y)dy = BR(x)

I F(y)dy Rn

a! J F(y)dy = o, i BR(x)

Using the divergence theorem on the vector field

V(y) = F(y)ei

we obtain from (23)

I F(x+s).:i dw(s) = 0. SR(O)

Combining this with (22) we deduce

I F(s)sidw(s) = 0. SR(x)

By iteration

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S. Helgason

J F(s)p(s)dw(s) = 0 SR(x)

if p(s) is an arbitrary polynomial, so the lemma follows .

Remark. The funct i on 1 F( z) = k z

(z E C) smoothed out

near the origin satisfies (by Cauchy's theorem) assumption

(ii) in Lemma 4 . Thus condition (i) cannot be weakened.

Similar remark applies to Lemma 5.

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S. Helgason

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