Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability. (MN-37) [Course Book ed.] 9781400860739

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Analytic PseudodifFerential Operators for the Heisenberg Group and Local Solvability

Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability by

Daryl Geller

Mathematical Notes 37

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1990

Copyright © 1990 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, John Milnor, and Elias M. Stein

Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources

Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

Library of Congress Cataloging-in-Publication D a t a Geller, Daryl, 1950Analytic pseudodifferential operators for the Heisenberg group and local solvability / by Daryl Geller. p. cm. (Mathematical notes ; 37) Includes bibliographical references. 1. Pseudodifferential operators. 2. Functions of several complex variables. 3. Solvable groups. I. Title. II. Title: Heisenberg group and local solvability. III. Series: Mathematical notes (Princeton University Press) ; 37. QA329.7.G45 1990 515'.7242 89-24111 ISBN 0-691-08564-1

Dedicated to my mother, Libby, and to the memory of my father, Samuel.

CONTENTS Page Introduction

3

Chapter 1.

Homogeneous Distributions

69

Chapter 2.

The Space Z q

Chapter 3.

Homogeneous Partial Differential Equations..146

Chapter 4.

Homogeneous Partial Differential Operators

105

on the Heisenberg Group Chapter 5.

Chapter 6. Chapter 7.

168

Homogeneous Singular Integral Operators on the Heisenberg Group

202

An Analytic Weyl Calculus

256

Analytic Pseudodifferential Operators on H

: Basic Properties

Section One - Dependence of Constants in Theorem 2.11 on Degree of

Section Two

Homogeneity

285

- Generalized Convolution

308

Section Three - Analytic Pseudodifferential Operators on E

, Products

and Adjoints

343

Chapter 8.

Analytic Parametrices

376

Chapter 9.

Applying the Calculus

423

Chapter 10. Analytic Pseudolocality of the Szego Projection and Local Solvability References

453 489

Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability

Introduction Section One; Main Results The main purpose of this book is to develop a calculus of pseudodifferential operators for the Heisenberg group E T , in the (real) analytic setting, and to apply this calculus to the study of certain operators arising in several complex variables.

Our main new application is the following theorem

(Theorem 10.2 and Corollary 10.3): 1.

Suppose M is a smooth, compact CR manifold of dimension 2n + 1.

Suppose U C M is open and is a real analytic

strictly pseudoconvex CR manifold.

Further suppose:

(i) There is a smooth, bounded pseudoconvex domain D e l with boundary M.

(D may be weakly pseudoconvex.)

Let S denote the Szeg6 projection (onto ker 3, in L 2 (M)). (a)

If f

e

Then:

L 2 (M), V c (J, V open, and f is analytic on V,

then Sf is analytic on V. (b)

Say f e L 2 (M), ρ ε U.

Then there exists ω ε (E') 0 ^ (M)

with 3ίω = f near ρ if and only if Sf is analytic near p. (b) ' Say f ε L 2 (M), ρ ε U. Then there exists ω ε (E') ' 1 C M ) with 3*ω = f near ρ if and only if Sf can be extended to a holomorphic function on a neighborhood U of ρ in !C

4

INTRODUCTION

(c)

Let V be an open subset of U.

2 — If fε L (M) and 3,f is

analytic on V, then (I-S)f is analytic on V.

•

Even if D is strictly pseudoconvex everywhere, the result is new for η = 1.

(If η > 1, D strictly pseudoconvex, it follows

from the results in [77], [81].)

The theorem we shall present is actually more general than what we have stated.

For example, instead of (i), for (a), (b),

(c) one need only assume: (i)' The range of 3, : C°(M) + Λ 0 , 1 ( Μ ) is closed in the d° topology. (This is weaker than (i) by the results of Kohn [58].) Further, (a), (b) and (c) hold for f ε E' (U) + L 2 (M) (not 2 just f ε L .) As we shall discuss, in many situations, (a), (b) and (c) hold for f ε E ' (M) . The importance of knowing that S preserves analyticity was first recognized by Greiner-Kohn-Stein [35]. They proved the analogue of our result for M =IH . In the case of IH , 3* is the unsolvable operator of Lewy and (b) gives necessary and sufficient conditions on f for local solvability of the Lewy operator. Greiner, Kohn and Stein showed that (b) follows readily from (a).

In fact, from (i)', one sees that one can always globally

solve 3"To)1 =

(I-S)f; so we need only understand when we can solve

5

INTRODUCTION

3Γω~ = Sf near p.

If Sf is analytic near p, this can clearly

be done by Cauchy-Kowalewski.

If, on the other hand, 9*ω„ = Sf

near p, and ω ? ε E' (U), then Sf = S(Sf-3*a>2) is analytic near p, by (a) for E' (U) + L 2 (M). We prove (a) by use of the work of Henkin [43]. On strictly pseudoconvex domains, Kerzman-Stein [52] found a simple relation­ ship between S and the Henkin projector H (which also projects onto ker 8.).

S is the product of H and the inverse of a singular

integral operator.

In order to prove that, in appropriate situa­

tions, inverses of singular integral operators also preserve analyticity, we need a calculus of analytic pseudodifferential operators. Boutet de Monvel and Kree [9] developed an analytic calculus which is suitable for dealing with elliptic operators on E ; we need a calculus which is suitable for dealing with analo­ gous operators on H . The simplest operators that our calculus is intended to deal w i t h — t h e analogues of the Laplacian on R — Stein operators L (αεί).

are the FoIland-

These second-order differential operators

onffi are intimately connected to the Kohn Laplacian •,.

Folland

and Stein [20] showed that if α,-α^{η,η+2,...,}, then L φ = δ α α where φ

(0.1)

is homogeneous with respect to the parabolic dilations

which are automorphisms o f H n , and is (real) analytic away from

6

INTRODUCTION

η

O ε ΠΗ .

They used this fact to study Q, on nondegenerate CR

manifolds.

They speculated that there ought to be a calculus

of pseudodifferential operators modelled on the L

and parabolical-

Iy homogeneous distributions on IH , just as the usual calculus is modelled on Δ and homogeneous distributions on IR . Such a calculus would then be appropriate for the intrinsic (non-isotropic) Sobolev and Lipschitz spaces on a nondegenerate CR manifold. We present such a calculus here, and we do so in the analytic setting.

Our calculus is analogous to the C°° calculus of Taylor

[79], but our outlook is quite different from his, and our proofs of necessity —

are much more elaborate, since we are working in

the analytic setting. Besides application #1 above (to the Szego projection) we obtain a number of other new results in this book. Here is a summary of our main results: 2.

A very precise form of an analytic parametrix for Q, on

any nondegenerate analytic CR manifold

(Theorem 9.6) . From it,

one can read off simultaneously the analytic regularity and the (nonisotopic) Sobolev and Lipschitz regularity for [], ; 3.

An analytic calculus onIH , natural for dealing with •, and

operators like it.

Simple, explicit formulae for products and

adjoints (Theorem 7.11).

Simple and natural representation-

theoretic conditions, analogous to ellipticity, for determining if operators in the calculus are analytic hypoelliptic, having

—

7

INTRODUCTION

parametrices in the calculus (Theorem 8.1).

The calculus may

be transplanted to provide a very natural calculus on nondegenerate analytic CR manifolds, and more generally, on analytic contact manifolds. 4.

Generalization of the theory of operators like the L

beyond the study of differential operators, analogous to the way in which the usual theory of pseudodifferential operators introduces one to the notion of elliptic operators which are not differential operators. In this way we find a large new, natural class of analytic hypoelliptic operators (Theorem 8.1). Those which are not differential operators were not previously known to be analytic hypoelliptic. Our calculus is the first analytic calculus modelled on parabolic homogeneity, instead of the usual isotropic notion of homogeneity. 5.

A characterization of the Fourier transform of the space

{Κ ε -S' JR ) : K is homogeneous (with respect to a given dilation structure) and analytic away from 0} (Theorem 1.3). a

(The dila-

a

l n tions are to be of the type D χ = (r χ,,...,r χ ) for χ ε TR , r > 0, where a,,...,a are positive rationals). Of course, K is homogeneous. In the isotropic case (a =...=a =1) it is known that K must be analytic away from 0. This does not hold in general for other dilation weights. In the parabolic case, for instance, where a 1 = 2, a 2 =...,= a

= ι with

INTRODUCTION

8

(t,X) € lR

x

lR s = mn , and with dual coordinates (>"J:J, we have

the well-known formula

where H is the characteristic function of (0,00). familiar kernel of the heat operator

~~

J

o

is the

+ ala>.., and is not

analytic at >.. = O. Our characterization of understood in the case a

1

{i :

K as above} is most easily

= p, a = ...• =a = 1, P € (0, p > 1 2 n

(Theorem 2.11). In addition, we lay the groundwork for the following further studies: 6.

Generalization of our calculus to a wide class of nilpotent

Lie groups ("graded homogeneous groups"-see e.g. [12] for the definition).

We see no difficulty in doing this, but it would

then usually make more sense to work in the

~

setting, since

analytic hypoelliptic operators on most other groups are rare. Generalization to the study of operators like the La' but

7.

for a € ±{n,n+2, ... }.

In this case, one has not (0.1), but

L ¢

a a

= o-pa

(0.3)

with ¢a' Po. hanogeneous and analytiC away from 0, and with f

-+

f*Pa (f€L2) being the projection in L2 (lBn) onto [LaS (IHn) J!".

9

INTRODUCTION

is called a relative fundamental solution for L • a When a = n, La is the same as Db on functions on En, P a n is the Szego projection forIH , and (0.3) is one of the results ¢

a

of Greiner, Kohn and Stein alluded to above. application #1 above, for f near u

€

As in (b) of

E', L g = f is locally solvable

€

En if and only if f*P

a

a is analytic near u.

Peter Heller and this author have obtained a generalization of (0.3) to left-invariant differential operators on En which are "transversally elliptic." presented in a future paper.

This generalization will be

Historical references and further

discussions are in Section Two of this Introduction. Local Solvability and Analytic Pseudolocality of the Szego Projection We now explain in more detail our results on the Szego projection.

Again suppose that M is a smooth compact CR rnani-

fold of dimension 2n + 1; suppose U C M is open, real analytic, and strictly pseudoconvex, and that the range of db AO,I(M) is closed in the Coo topology.

jection on M. S : Coo(M)

~

Coo(M)

~

Let S be the Szego pro-

Our hypotheses are too weak to imply that

Coo(M), so we cannot necessarily extend S : E' (M)+

E' (M). What we shall show is that S is analytic pseudolocal on U when restricted to its "natural domain."

shall show S: Coo(M) so that S : COO(M)

+

+

Coo(U) continuously.

COO(V) continuously.

That is:

first we

Fix V

U, V open,

We define

C

V(S),

the

10

INTRODUCTION

domain of S, to be E' (1/) + L (M) . We shall show that if f ε p(s), W

c

U, W open, f analytic on W, then Sf is analytic

(By the results of Kohn [57], we may take V = M if

on W.

M = 3D, D a smooth bounded pseudoconvex domain in (C , provided D is of finite type if η = 2, or D is of finite ideal type if η > 2.) If η > 1, and M is in addition globally strictly pseudoconvex, analytic pseudolocality of S follows from the work of Treves [81] and Tartakoff [77]. Indeed, Q, on (0,1)-forms has a good "Hodge theory."

Let N invert [J, on (0,1)-forms on the

orthocomplement of its kernel.

Kohn's formula

I - S = 3*N9U b b

(0.4)

shows that the analytic pseudolocality of S may be deduced from that of N, which in turn follows from the results of [81], [77]. When η = 1, there is no good Hodge theory for Q, on 1-forms, so this method cannot be used in this case, even if M is globally strictly pseudoconvex. Our proof of analytic pseudolocality of S begins with a reduction to a local analogue, and it is in this reduction that we use the "closed range" hypothesis for 3, . The local analogue is this: V

= c

Say ρ ε U .

We shall show that ρ has a neighborhood

U, and there exist operators

INTRODUCTION

11

analytic pseudolocal on V

(0-5) on V mod (0.6)

analytic regularizing errors. Precisely:

if

then are all analytic on V .

(0.7)

If V were M and the equations in (0.6) were exact, then would have to equal S. say

In general, though, we can show this:

open,

Then we claim:

p has an open neighborhood

so that if

is analytic on

(0.8)

This would then clearly show analytic pseudolocality of S on U. To illustrate the method of proof, let us examine the analogue in the

setting (replace "analytic" by "smooth" in

(0.5), (0.7), (0.8); call the new statements (0.8)').

In (0.8)', we may as well assume

small as we like, by the pseudolocality of

supp t are as We may also

assume holds for

Indeed, to achieve these

relations we need only shrink V slightly and multiply the Schwartz kernels of

by an appropriate smooth bump function

12

INTRODUCTION

supported near the diagonal of

If supp

sufficiently small, the value of we change

, are is unaltered if

in this manner, so (0.8)' is unaffected.

establish ( 0 . 8 ) w e just have to look at hand, by the closed range property of for sane u

On the one we have

thus, on where

is smooth on

is smooth on smooth on V .

at once.

To

while

(0.8)' follows

(This part of the argument is similar to certain

reasoning in [10].) In the analytic setting, the bump function

cannot be

chosen to be analytic, of course; so we instead carry out this procedure with a sequence of special bump functions, due to Ehrenpreis.

These bump functions, and the errors

which result in the above argument, satisfy the conditions for analyticity (i.e. conditions like than or equal to a number N .

less

We may then let N -»• °° to establish

(0.8) . To establish (0.6), since

is analytic and strictly pseudo-

convex, we may in fact assume bounded strictly convex dcmian. If n

a smooth, If

we are done by (0.4) .

we use the work of Henkin and a method of Kerzman-

Stein. Henkin

shows that there are operators R,H on

so that

13

INTRODUCTION

(0.9) H is the Henkin projection onto the kernel of

These

relations apparently give us part of (0.6). But H , the generalization from I of the Cauchy projection, need not be orthogonal.

Thus we need not have

near p modulo an

analytic regularizing error. Kerzman and Stein, however, observed [52] that if the

projection on

then

is

(since

so that (0.10)

is invertible on Since

since

is skew-adjoint.)

, we would like to obtain (0.6) from

(0.9) by putting analytic pseudolocality on

The problem, then, is to obtain for

frcm (0.10); and for this

we need to develop an analytic calculus on the Heisenberg group. Overview Here now is an overview of our calculus.

Let A be a

classical pseudodifferential operator of order j on:

so

that (0.11) Here let us say the symbol a each a™ is smooth in u and homogeneous of degree

where

INTRODUCTION

14

We may then also formally write

where

(AfXu) = (Ku*f)(u)

(0.12)

K (w) = (2π)~η/β~1ν*ξ3(υ,ξ)άξ,

(0.13)

the inverse Fourier transform of a in the ξ variable. Let us call K (w) = K(u,w) the core of the operator A; the kernel of A is then K(u,u-v).

We have

K (w) ~ ^ ( w ) where Λ

near w = 0

(0.14)

is smooth in u and homogeneous of degree k + m (k=-n-j),

at least if k + m ft Z Z + = {0,1,2,...}.

(If k + m ε 2Z+, K 1 1 V)

may in addition contain a "log term" of the form ρ (w)log|w|, where ρ

is a hcmogeneous polynomial in w of degree k + m.)

A is a differential operator with analytic coefficients if all the κ

are supported at 0. A is elliptic if K (ξ) φ 0 for

u ε U, ζ / 0; then it has a parametrix of the same type on any relatively compact open subset V of U. We intend to develop a calculus on IH which is analogous to (0.12), (0.14).

In (0.12), * will be replaced by group con­

volution onlH , and in (0.14), the κ (w) will now be homogeneous in w (or homogeneous plus a "log term") in the parabolic sense. The condition analogous to ellipticity is that π (K ) be injective on Schwartz vectors for all non-trivial irreducible unitary representations IT of H .

INTRODUCTION

15

The idea of working with operators of type (0.12) in the C° category and in the nilpotent group situation is due to FoIland-Stein [20]. Such operators, and generalizations thereof, were also studied by Rothschild-Stein [70] and Nagel-Stein [66].

Taylor [19], in the C°° setting on JH , defines "pseudo-

differential operator" by (0.12), and then chooses to pass directly from these operators to define a new kind of symbol. We shall stay instead on the core level, although the Fourier transform will still play an integral role in our work. Also, we will work in the analytic setting.

For 3-step nilpotent

groups, in the C^ setting, a calculus based on cores was also recently constructed by Cummins [14]. Of course, the concept of "core" is far from new. We have introduced the new name since we intend to place absolute emphasis on the core, as opposed to the kernel or the symbol. It is worth recalling some of the basic reasons why one usually prefers, in the standard situation on H , to use definition (0.11) instead of (0.12), when (0.12) has a much simpler appearance.

Two elementary reasons are:

(a) The Fourier transform converts convolution to multiplication, which is easier to handle. (b) The Fourier transform converts the finding of convolution inverses (such as fundamental solutions for constant coefficient differential operators) to division.

16

INTRODUCTION

Reason (a) loses much of its significance on a nilpotent group, where the Euclidean Fourier transform on the underlying manifold (R ) converts group convolution to a ccmplicated analogue of multiplication, while the ccmplicated group Fourier transform converts group convolution to multiplication. One could, then, write down formulas analogous to (0.11), but their complexity makes them undesirable to manipulate compared to (0.12).

We shall see that, using convolution, we can obtain

an analogue of the Kohn-Nirenberg product rule forIH

(and more

generally, for graded homogeneous groups). As for reason (b), even when one inverts, it is possible in many circumstances, to avoid use of the Fourier transforms. The problem can be reduced to inverting the principal part of the operator (in (0.12), (0.14), the operator with core equal to K (w) near w = 0 ) . The rest can be handled with a Neumann series, after a product rule is established. Thus, the initial problem to be solved is: Given K, homogeneous and analytic (resp. smooth) away from 0, can we find K~ homogeneous and analytic (resp. smooth) away from 0, with Κ ? *Κ, = (c) appears to be sharper than in Gelfand-Silov (who required all the elements of U to precede all the elements of V), and the equivalence with (e) appears to be new. an analogue of the well-known fact that i f g g

€

€

(e) is

2 L (Rs), then

S if and only if its Hermite coefficients decay rapidly. (a) ==>(b) of this theorem states that the decay of f,

s which is initially assumed only onIR , must persist into a sector.

This is a consequence of Phragrnen-Lindelof.

fact hOlds for zq '. q,]

A related

Thus, suppose a function f satisfies all

the conditions for membership in zq " except that (0.2) is q,] s only known for i:; = t:t:R , 1t:1 > 1, rather than for i:; in a complex sector.

Then f can be shawn to be in zq '. q,]

This

relatively deep fact is shown in Theorem 2.6. Whenever one is dealing with a space defined by asymptoties, like zq " it is cammon to ask whether, given a formal q,]

series, there is necessarily an element of the space which is asymptotic to it. finition.

Precisely, let us make the following de-

52

INTRODUCT ION

Definition.

We say that zq . is ~ if the following condi-

tion holds.

Given any set of functions {gf(~)} as in the

q,J

definition of zq . (i.e. go is hOlamorphic in a sector q, J

-L

{~I Inl < cl ~I}, homogeneous of degree j - pi and satisfies

Igf(~) I

< CRff!p-l for

f E zq . such that f ~ q, J

~

E

s, I~I = 1)

then there exists

Lgo. -L

We have proved the following: Theorem (a) (Theorem 2.12).

If q is an even integer and j E

~,

then zq . is ample. g,] (b)

If q

E

~, q > 1 and j EIR then zq . is ample. q,]

(b) will not be proved or used here. of (b) works equally well if j E

~,

(Probably our proof

but we have not checked

the details.) Homogeneous Partial Differential Equations Chapter 3 contains a result about a rather general class of partial differential operators.

As before, we assume

(al, ... ,a s +l ) = (p,l, ... ,l) and we use coordinates (t,X)E with dual coordinates

(A,~).

lRxIR

We study differential operators

L satisfying these conditions: L is homogeneous, with homogeneous degree k; the degree of L is also k; the coefficients of L are polynomials in the xf's; and we can write

(0 . 48)

s

53

INTRODUCTION

where

is a constant-

coefficient differential operator in the only, and where

is elliptic.

An example (with

is

On the other hand,

does not satisfy (0.48);

its degree is 6. Note that

must be homogeneous with homogeneous degree

k, and degree k . thesis that

Given all the other assumptions, the hypo-

is elliptic is evidently necessary for L to be

analytic hypoelliptic, as one sees easily by considering functions of x alone. This class of operators includes: (I)

constant coefficient homogeneous differential operators in (t,x);

(II)

transversally elliptic homogeneous left invariant differential operators on the Heisenberg group

(III)

Grusin operators in (t,x) (for these, L satisfies (0.48) and is also elliptic away frcm

If the Fourier transform of L .

We

then have the following result. Thom-CTn t ?

Suppose L satisfies (0.48) and

Suppose

INTRODUCTION

54

that Z~,_(O+k+Q) is ample.

Then of the following conditions,

(a) tmplies (b), and (b) is equivalent to (c): (a)

L is analytic hypoelliptic and L and Lt are hypoe 11 iptic .

(b)

For any Kl

AKo there exists K E: AKo+k with

E:

LK ::: K . 1 (c)

For all A, if F that ~ G

s zq q (R ) , there exists G E: zq q such

E:

= F.

In particular, (c) is necessary for (a) to hOld.

As

we

discuss in Chapters 3 and 4, condition (c) is often satisfied for operators of type (II) and (III).

As

is evident, it can

never be satisfied for operators of type (I).

We hope that

our methods will ultimately be the starting point for the construction of analytic parametrices of the "right type" for operators of type (III). The most difficult part of the proof, and the only place where the ampleness assumption is used, is (c) ==>(b).

The key

fact used is the following: Theorem 3.5.

Suppose j

satisfies (0.48). K'

E:

Say K

AK-(j+Q) and K2

such that

J

E:

1

E:

a: and that

zq . is ample. Suppose L q,J AK-(j+k+Q) Then there exist

E:

AK-(j+k+Q) such that LK'

2A (f,;) = K2 (A,f,;)

E:

Z~(JRs) for all A of

= Kl - K2 and O.

55

INTRODUCTION

Once this is known,

follows easily. One has

only to construct

with

for then L

By Theorem 2.11, it suffices to construct with

this can be done by (c).

(This argument must be modified slightly if We illustrate the method of proof of Theorem 3.5 with an example.

Say s = 1, p = 3/2, L

Thus

By Theorem 2.11 it suffices to con-

struct 1

such that

for then

we can choose K with

It is easy to see that

1 has a formal asymptotic solution of the form Estimating the a^ and using the ampleness condition, one finds that there does exist F such that

as desired.

In the above example, L is a proved by

operator, and was

[39] to be analytic hypoelliptic.

(However,

one still does not know how to construct an analytic parametrix of the "right type" for L.) for this L .

In particular, Theorem 3.2(c) holds

As we show in Proposition 3.7, this may also be

shown simply and directly by use of Theorem 2.3(a) (c), the "basic estimate" for of a method of

operators, and an extension frcm [62].

The Group Fourier Transform on The main result of Chapter 4 is Theorem 4.1, which we

INTRODUCTION

56

restate now. Theorem 4.1. Suppose L is a left invariant differential opera­ tor on H

which is homogeneous of degree k and transversalIy

elliptic. Then there exist K e A K ~ 2 n - 2 + k and P ε Α Κ ~ 2 η ~ 2 such that LK = 6 -P, and such that the map f -*• f *P is the projection in L 2 (H n ) onto [LSiIH11)]4-.

If f ε L 2 or E' and if q ε lHn, then

there exists a distribution u with Lu = f near q if and only if f*P is real analytic near q. To prove Theorem 4.1, we begin by observing that L satis­ fies the condition (0.48).

Thus Theorem 3.5 holds for L; we

are interested in the case K =

δ of Theorem 3.5. The question

is how far one can go in eliminating K~. be using the group Fourier transform.

For this, we shall

It is convenient, then,

to change notation; from now on, in this introduction, F means Euclidean Fourier transform while " means group Fourier trans­ form. Let us use coordinates (t,x,y) on Ή , where ζ = x+iy. Then, if f ε L 1 ( E n ) , (Ff)(X,p,q) = / H

exp[i(Xt+x.p+y-q)]f(t,x,y)dtdxdy

(0.49)

where we are using (X,p,q) as dual coordinates to (t,x,y). The group Fourier transform is given by a rather similar-looking formula.

If f ε L (H n ), f is a family of bounded operators

57

INTRODUCTION

here

acts on a separable Hilbert space

ranges over

The formula for

and

is (0.50)

where

are certain

n-tuples of unbounded operators on alized as

is commonly re-

In this realization, the operators

and

take this form: (0.51) (We are using coordinates The significance of definition(0.50) is that the map

is an irreducible unitary representation of

for

further all infinite-dimensional irreducible unitary representations arise in this way up to equivalence. as

(If

is realized

this is called the Schrodinger representation).

consequence is the rule Plancherel and inversion formulas for Let us write to us is the map

A

also, one has . Of great interest

which makes the following diagram commute:

INTRODUCTION

58

funct ion on JR2n Figure 1

W is called the Weyl correspondence; let us explain some

A of its properties.

It is often extremely useful to consider the matrix (RaS) of an operator on HA with respect to a certain distinguished orthonormal basis {E ,} • a,/\. aE (lIZ +) n sentation, {(-i)

la IEa ,I/4}

(In the Schrodinger repre-

are the Hermite functions.)

us put

S(H ) = {REB(H ) for A A

all

I (REa,A,ES,A) I Z~(HA)

N there exists
0,0 < r < 1, we have

Let

INTRODUCTION

59

It is well known that

We also have:

ProDosit ion 4.2. (Again, strictions to

is the space of functions which are reof

functions.)

Returning to the situation of Theorem 4.1, we invoke Theorem 3.5 to find

as in Theorem 3.5 with

Lei

If we attempt

to eliminate

that is, if we seek

reduced to producing

for

with

with

is what we previously called

geneity we need only do this when equation.

Set

with

where

By homo-

We apply

to the

We seek to solve

Here

ed operator on

we are

is a certain unbound-

in fact one could write

is the Dirac distribution on

This is the same as

solving (0.52)

INTRODUCTION

60

We seek to produce the matrix of Ht one column at a time.

Let

HA = {V=Lva EaA €H __ > 0 such that .>.. Ifor all N>O there exists C-N OO

~ = {v€HAlfor sane e

> 0, 0 < r < 1,

IVai < erial

for all a}. (In the SchrOdinger representat ion, with HA = L2 l;Rn), one has 00

HA

= S CR n ) and

w H,).

=

2

n

Z2 OR ), by Theorem 2.3 (a) 2.

If Re k > -Q, KI ELI. Thus Re IKIII < C( sup IK(x) I) f Ixl kdx < C sup I K(x) I. Ixl=l Ixl an and for fixed t;' :=:

J(~'

,t: n ).

=

~

1 and it follows readily :=:

O.

Even more, suppose

(t:l, ... ,t: n- l ) with t:l ::f 0, set

Then F is better than real analytic--it is

the restriction to:R on an entire function on a:. {t: l

= a},

(1.10)

= 1.

that J is real analytic away fran t: 1

F(~n)

77

Along

J may well be worse than real analytic.

This agrees

with the statements in the introduction about the case S (p,l, ... ,l).

=

We study that case in detail after Theorem 1.3.

With this motivation in mind, we turn to the details. (b) below is a restatement of (1.10), while (c) and (d) are variants. Theorem 1.3.

For J

(a)

J E A) j

(b)

For sane C,R, It::e I

E

lalla .fl aQJ(t:) I

for all .f,a,I~1 (c)

= 1.

a s/a:e s a la:e Forsanec,R'/~:e/ m la!J(t:)j < CR s.1m

for all s (d)

Jj, the following are equivalient:

E

7l,

all :e,m (l~~n), and It: I

= 1.

n1 s n s (n +..• +n ) For sane Co,Ro ' It:l I ... j~nl n la~J(t:)I -Q.

Then, for any RI > 0 there exist

C,R,Co,Ro > 0 so that the inequalities of (b), (e) and (d) hold whenever J = for a K € AKk which satisfies

K

for all y

€

+n •

(7G)

Then for any C,R,C,R > 0 there exist o 0 CI,R > 0 so that if the inequalities of (b), (c) or (d) hold I and i f J = K then su~ !aYK(x)! < cIJlyl/y! for all y c(7i)n. (2)

Suppose Re j > -Q.

1;;;;/xl;;;;2

Proof.

We prove (a)

~ (b)

evident, i f one takes aa

=> (c)

=

~

(d) => (a).

a~ in (b).

(b) => (c)

Also (d) =>

evident, i f one takes n.1 = (b)

(a).

These we have motivated above; we

present the tiresome detailed proof now.

is

As we said, we

would rather have (e) => (d); this is shown as follows. Assuming (e), write nt

is

The reader

HOMOGENEOUS

DISTRIBUTIONS

79

may skip this proof without loss of continuity.

The uniformity

assertion will follow frcm the proofs of (a) =s> (b) and of (b) =s> (These, and future uniformity assertions, will be used in Chapters 7 and 8.) Suppose As we have motivated, we wish to examine distributions that look like

and apply Proposition 1.1(b).

evident problems—that we might not have Re First, fix choose

The

might not be an integer, and that are easily dealt with as follows.

with

we could

We need prove (b) only for those a with denotes greatest integer function.)

If where

Thus and in particular, Re

It suffices then to show that (1.11)

for

and (1.12)

For by Proposition

we would then have is less than or equal to the right side

CHAPTER I

80

Since I~,el

of (1.11).

lal/a,e

laaJ(~) I is less than or equal to

the former quantity for I~I = 1, (b) would then follow. For (1.11), it suffices to show that (1.13)

a,e for 1 ;;; Ixl ;;;2,lyl ;;; N. Indeed, [Ial/a,el! ;;;!al!;;; aa II lI aa a a n a (alaI) 1. .. (anan )! ;;; n 0, by (1.8), (1.7), (1.8), Thus

(1.6).

(1.11) is a consequence of (1.13).

(1.13) would surely be true if K were replaced by a [ lal/a.[l-p x 0,

(Cnl for all .[, /z,e-x.[1 < r}.

B (x). 1;;;2 r

Now, for scme r

>

X

E:

n

R , let

Also let

0, K/

B nRn

has an extension

r

to a bounded analytic function on B , which we also denote K. r

For

Z

E:

B,

r

/Zo.{. I

< C

r for all .f, for some Cr > O.

Accordingly

we have an estimate (1.14)

for

Z

E:

B .

r

Thus, for all S, (1.15)

by (1.6).

Thus i f Iyl ;;; N, and R2 = c!r, we have

HOMOGENEOUS

DISTRIBUTIONS

81

But

for some and we are done.

by (1.6). Thus (1.13) is proved,

The first uniformity assertion, also, follows

from the proof just given. For each m , fix any p For

put

with Then In particular

so that

where Re

Our plan will be to show that for

seme (1.16)

for all m , if

1 and

Once this is known, we shall suppose 1.1(b), we shall have that for any

By Proposition there exists B > 0

such that (1.17) for

The real analyticity of K would follow

at once if we had these estimates for instead of deal with the desired variant below.

ve shall

82

CHAPTER 1

We begin by proving (1.16). m = 1.

=

~

Write

(~l'~')'

=

13

For ease in notation, suppose

Sl- i

qq.

= L (.)Sl···(Sl-l+l)~l

diJIQ(O

i==O

iJ

Note that

(13 1 ,13').

1

q

Now 13 1 ", (Sl- i +l) ~ si, while iIo to show that for aIlS with lsi

(i)

~

, 13' [ISj/all-p+q-i

(~)

dl

q

== 2

J(~).

then it will suffice

j

alP, all i,q with

1 , and all ~ with lsi = 1, we have B -i ,[IBI/a l-p+q-i II I ISII (s')S dl 1 J(s) I < CR 131\31. With Y == (Sl-i,S'),

i

~

q-p

q

~

[N/all and i

= P,

~ 13

[Iyl/all+p this is the same as I~Ydl J(s) I

aJlyll+i(Yl+i)! (y')! since IBI/al-i = Iyl/a . l show this for all P with -p

~

P

~

[N/all - p

i ~ N, and for all y with [Iyl/all + P ~ O.


0 such that for all P with -p ~ P ~ M and for all y with [hi/all + P ~ 0 we have

rlyl/all+p

I~Ydl Iyj.

J(OI
y the right

, with a different C and R. Bu and (c>2 follows.

holds, then

similarly for some

and (c)^ follows from Cauchy-Schwarz.

holds, the analogue of (c) holds with thus (d) holds. In particular (c). The uniformity in the uniformity in

If (c)^

replacing so by symmetry,

follows from this proof and The uniformities in

also follow frctn this proof and the uniformity in

and

THE

SPACE

117

be the Hermite operator. As is well known, (2.2)

By

and the simple inequality

have that for sane (2.3) for any

Thus if for seme

and

may be

M written as a sum of (2s) expressions of the form Since the that For

are orthonormal, if ||aj| = N we have in particular Thus By Lemma 2.2(c)

(a) with

find f(N) < Ae "" for sane A,B > 0, as desired.

be the annihilation and creation operators. As is well known, if

(1 in the £th spot), then

implication(zero will follow readily frcm the fact that for seme

118

CHAPTER

1

Let us first shew (2.4). Since, for fixed functions

the

are orthogonal, we have

For seme

re have

for all N;

thus for seme ' , yielding (2.4) at once. (2.3) quickly follows, for seme so that

. Indeed, each

:an be written as a sum of

expressions of the font with

this gives

Remark.

Thus (2.3) follows, since if

does have a generalization for

eralization is not for

but for

but the gen-

Details are left to the

interested reader. For a related result, see Hille [44]. (e) suggests a rough analogy.

is the unit circle,

consists of those functions with Fourier expansions whose coefficients decay rapidly. on

analytic functions

consists of those functions with Fourier expansions whose

coefficients decay exponentially (by Laurent's theorem). consists of those functions with Hermite expansions whose coefficients decay rapidly, while

consists of those functions

with Hermite expansions whose coefficients decay exponentially.

THE

Thus we can think of

SPACE

119

its variant,

"a natural

Schwartz space in the real analytic category." We now show how Lemma 2.4 can also be utilized to show the fact claimed towards the end of the first section, that

Further, we have this uniformity. Suppose there exist

then

so that whenevei

then

Proof. As we remarked after the definition

we have

For the converse, we shall use the following lenma: Suppose such that of degree

be a set of functions

s holcmorphic in

and homogeneous

there, and such that Then there exist

for and a function

f which is defined and continuous on the closure of the sector holcmorphic in S', and satisfies: (2.5)

Further, we have this uniformity. Suppose there exist

3 so that whenever

then is a set of

120

CHAPTER

1

functions as described in the hypothesis, there then exists f defined and continuous on the closure of I holcmorphic in S 1 , and which satisfies (2.5). Suppose and

As we remarked before the definition

g^ can be extended to functions

the

which satisfy the hypo-

thesis of Lemma 2.7 for sane

Select

:

as in

the conclusion of Lemma 2.7. Then we surely have [

in S 1 , for seme 1

Further, for

we have By Lemma

_ then there exist

sane

_ i^ia so that exist

there D that if

then

Thus for sane I for by Lemma

so that

for sane If we combine this with

(2.5), we find that for sane as desired. The uniformity assertion follows from the arguments and the uniformity assertions of Lemmas 2.2, 2.4, and 2.7. We claim that we may assume Re j < 0. Indeed, for any fixed N e IN, put

•

THE

The

SPACE

121

are then holomorphic in some sector

are hcmogeneous of degree

and for sane

they satisfy

If the

construction we seek is possible for Re

then if N is

large enough there exists a function F such that for seme is holomorphic ii

continuous

on the closure, and satisfies The function .s then as desired. The uniformity assertion for Re

would also follow fran these arguments

and the uniformity assertion for Re We therefore assume Re

and construct

We free

all the notation introduced in the last paragraph. For simplicity, let us first give the argument when s = 1. Write the function

as a sum of an even and an odd function. The

even part is just a multiple of multiple (

the odd part is a

Thus < , where

The even part is just

so that

has

a holonorphic extension to any sector

namely taken accord-

ing as Re functions). estimate

(principal branches of the power note that we have the simple for sane

since

122

CHAPTER

1

(We shall prove a more precise estimate in detail in Lemma

We next define functions

for

as follows:

This series converges absolutely for estimate for

as the

together with the estimates for the

A„ ,K show at once. L -C /simply can put

e

t

W

for C in any sector

e where

claim that we arctan

(We use the principal branch of the (q/2)-power function; note cannot be on the negative real axis.) Indeed, f is clearly holcmorphic in S' and continuous in the closure of S'. Fix where the ± sign is taken as Re

or Re

It suffices

to show that for some for

Clearly we may assume Re 5 > 0, so that In the expression for eT, write L

where

For

in the sector S', there exists a constant

such that

Thus we may estimate

Substituting this in the above, and estimating all the integrals by integrals from (

we find that for some

as desired. The uniformity assertion, when s = 1, follows also frcm the above arguments.

124

CHAPTER

When

1

we must be more precise in our estimates.

Again we free all notaiton introduced since the beginning of the proof.

be a basis for the spherical

harmonics which is orthonormal on the unit sphere. We define by letting

be the degree of P^; we may suppose

the P are so ordered that

is a nondecreasing function. We

adopt the convention that in any expression, equality or inequality in which both m and .. occur,

the spherical harmonic expansion of

means

We consider

say

(this, as we said, is shorthand for . We shall need the following lemma. (The proof of the forward direction was shown to us by E. M. Stein): Lemma 2.8. Suppose

then there exist

and

as follows. Let g be a real analytic function

homo-

geneous of degree 0, such that Let

be the spherical harmonic expansion of g,

for

Then Conversely, given (

such that

then

and constants is real analytic

in a neighborhood < Let us accept this lemma temporarily and proceed. It is an easy consequence of the lemma that for seme

THE

SPACE

125

we have

Indeed, let this function is homogeneous of degree

0. There exist

such that for each

has a

holomorphic extension to the sector

, to the

functior

and such that, if Accordingly, for sane We apply

the forward direction of the lemma to whose spherical harmonic expansion is for

The asserted estimate, (2.7)

Now, there exists a sector

such that This is a

simple consequence of the following lemma: There exists

as follows.

there exists

such that for all and all m we have

(b) There exists a number a > 0 such that for a 5 have

with

126

CHAPTER

1

Accepting this lemma temporarily also, we verify the claim at once, as follows. Select any : with

(r as in

Use (a) of the lemma to select

such that in Further, select

such that : have

we

Then

where

is a constant depending only on

space of all spherical harmonics of degree page 140), there exists

and k.

As

is the is known

such that dim

( [ 7 6 ] ,

thus

the series converges absolutely and uniformly on compact subsets of

and therefore coincides with We next define functions of the case

there.

in analogy with the functions

We need a precise estimate for

this is given in the following lemma. Suppose Re (a) There exist

such that for all = greatest integer function).

(b) There exist

(c) For any

such that for all

there exist

such that for

THE

SPACE

127

The lemma is quite simple to prove. However, we just accept it also for the time being. We apply Lemma 2.9(b) with and observe that in (b) we can then write We can thus form (2.8)

which, as in the case

converges absolutely for

because of (2.7) . We put

and wish

to put (2.9) for

in a sector

where

small that

To see that f is

holcmorphic in so that

is sufficiently

Choose c^ as in Lemma 2.10(a), . Note that elementary estimate , Observe that

for

for seme

, we have

, by L erana

and

Lemma 2.10(b). By Lemma 2.9(a) and the inequality dim , we have that the series for f converges absolutely

128

CHAPTER

1

and locally uniformly, and that for seme

and

u Evidently, then, f is holomorphic on on its closure.

For

and continuous

let

(2.10)

It suffices to show that there exist such that if

then for all m, (2.11)

Indeed, (2.5) is an immediate consequence of this and Lenma 2.9(a) (where the

of that lemma is simply chosen with

In the expression for >

write (2.12)

Observe that by Lemma 2.10(b) and (2.7), there exist such that

129

(2.14)

(2.15)

Select

. Select

such that

with then Re

Then We estimate the absolute value of the integrals in (2.15) by the integrals of the absolute values, use (2.13) and also use the last estimate. We make an observation analogous to (2.6) again, and estimate all resulting integrals by integrals from

]

I

We find that for sane 1

we have

130

CHAPTER

and 1

1

such that This is (2.11), precisely

what we wanted. This gives Lemma 2.7, since the uniformity assertion also follows from the arguments. We must still demonstrate the technical Lemmas 2.8, 2.9 we turn to this now.

and

Let sphere

denote

norm on

the unit

We begin with the identity that if P is a spherical

harmonic of degree H, (2.17) It suffices to show that if Q is a homogeneous polynomial of since we may then

degree apply this identity

. To see

the latter identity, observe that

geneous of degree

. On the other hand, since

grad,

by Stokes1 theorem, if v denotes unit normal, since j Euler's identity. The identities follow at once. From (2.17) we evidently have that if Proceeding inductively,

THE

we have that if

SPACE

131

is any nonanial with It is known, however,

that there exist

appendix C.4)

such that for any spherical harmonic

R of degree

denotes

norm

s-1 on S

. Accordingly,

where

Using this and a power series expansion

about a point

we find the following. Suppose Then, for seme 1

and dist

where The lemma is now immediate. Indeed, for (a), given r^ with

we may select

then

Next we may select

such that if such that if then dist (

Then for seme

then

as desired.

For (b), observe that there certainly exists d > 0 such that for all (b) follows with

with

we have dist

where The converse is immediate frcm Lemma 2.9.

Indeed, with data as in the hypotheses of the converse, select with

and

as in Lemma 2.9. Then

converges to a holomorphic function in

132

CHAPTER

1

For the forward direction, let

be the spherical

Laplacian. Since Ag is a second order differential operator on the sphere, it is easy to believe that if the statement, then there exist

are as in

depending only on

such that for all (2.18)

Let us accept (2.18) for the mement and proceed. As is known ([73], Section 3.14),

consequently

Thus for seme 1

Then if for seme C" > 0, where

(Of course, we are assuming then with q = 1, for seme

By Lemma This is

what we wanted, since as is immediately checked, C^ and B can be chosen to depend only on R^. (2.18) is not quite obvious, and we give a direct proof. Let (This is shown in [20], Section 1]. For a quick proof, one can easily compute that any

for

Since every polynomial is a linear combination

of polynomials of the form 1

Theorem IV.2.1),

for any polynomial P, and in particular for every

THE

SPACE

133

polynomial of degree less than or equal to 2. Thus the two differential operators must be equal, since both are second order.) Now

can be written as a sum of

terms

of the form W, where W is a product of 2L factors, each of the form

for seme j,k. it suffices, then, to show that for

seme

depending only on

we have

whenever W is a product of M of the then use this for as a sum of

(Indeed, we can

Such a W, however, can be written

terms of the form D, where D is a product of M

factors, each of the form to show that there exist that for every such

for seme

It suffices, then,

depending only on R^, such . This, however, is easy

from Lemma 2.5. Indeed, let with

and for any fixed £ . There exists

that g, restricted to function g on however,

j

such

can be extended to a holcmorphic

such that

for

for all j. lemma 2.5 applies directly

to show

as desired. (a) follows at once frcm the elementary

inequalities

applied to

. and 1

To prove (b) and (c), we observe several facts about the ganma function: (i)

This follows from

134

CHAPTER

(ii) Fix

and

2

0. Then there exists

so that

, To see this, observe that we can obviously select for

so that . Now

select r,N so that

we may and so that

x = r + N. Then

as claimed. (iii) (iv) : as the inequality

shows.

To prove (b) and (c), we may assume

By

(i) and (ii) we see that we may assume j is real. By (iii) and (iv), we may further assume assume j = 0.

and in fact we may

(iii) and (iv) then further show that it suffices

to prove (b) and (c) with

replaced by

(b) is now immediate frcm (a), since by (1.7), For (c), note that, because of (a), it suffices to show that if such that then to show that if for all

there exists It suffices

there exists R > 0 such that (We could then apply

THE SPACE zq .

135

q, J

= r~,

this with r

for then r-[u/Q]

that we can choose R

= r;Q[u/Q]~r;~

We claim

n m = (l-r)-l. Indeed, (m+n n ) r (l-r) ~

the binomial theorem.

This completes the proof.

This completes the proof of Theorem 2.6.

1

by

•

As we indicated

in Chapter One, Theorem 2.6 implies the following result, which is our main result so far. with~.

=

.-

(p,l, ... ,l).

We use the

We let zq . (IRs) q,]

notation of Chapter One

=

{functions f onJRslf

is the restriction to JRs of a zq . function on [S}. q,]

We state

only one of the uniformities here. Then there exists

Theorem 2.11.

i f and only if:

J

-

(-1)

~

gr

J+, J_ If

E

zq . (IRs), and if J+ ~ Lg~ then q, ]

this is the case,

g.e. W

=

(a~J) (O,~)/l!.

(2.19)

Further, we have this uniformity. Let k Re k > -Q. J+,J

-

E

=

-Q - ji suppose

Then for any Rl > 0 there exist B,C,R,c > 0 so that

zq .(B,C,R,c) whenever J q,]

= K for

a K E AKk which satisfies

sUf la\(u)1 < Jlyll y ! for all Y E (zz+)s+l.

l~ Iu ~2

Proof.

This is an immediate consequence of Theorems 1.7, 2.8,

and 2.6.

The uniformity follows from the uniformities in

Corollary 1. 5 and Theorems 1. 7 and 2.6, in the case -Q < Re k < O. If Re k

~

0, for the uniformity we argue as follows.

Select

136

CHAPT.ER

4

and, with K as above, consider For seme

depending only on

we have

for all

Thus if

we have ing only on

^(B,C,R,C) with B,C,R,c depend-

. But

and it is clear, either directly

frcm this or through (2.19), that if

then

for and frcm

Frcm these relations,

the uniformity in the case Re

follows

easily frcm that in the case As we said after the proof of Theorem 2.3, one can think of

as "a natural Schwartz space in the real analytic category." might then be thought of as "a natural asymptotic symbol

class in the real analytic category." simply be

Its

there exist

analogue would functions

homogeneous of degree j - p£ and constants i such that for all L,6, if

By Theorem 1.7(a) and Theorem 1.8(a), one can characterize in terms of these spaces.

It is an important, and easy, result

in the theory of pseudodifferential operators that given any collection of

functions

homogeneous of degree

such that , then there exists

is (in

In order to apply our methods to study partial differential

THE

SPACE

137

equations, it is important to have the analogue of this result for

To be precise, let us make the following definition.

Definition. We say

is ample if the following condition

holds: Given any set of functions morphic in a sector S =

which are holo, homogeneous of degree

and which satisfy there exists

such that

•

Clearly, if it were in general known that it would be an improvement on Lemma 2.7.

is ample,

In fact, if q is an

even integer, the proof of Lemma 2.7 gives this at once. if Re

we can simply choose f as in (2.9).

exp

Indeed,

In this case,

is entire. Thus the argument after (2.9)

which showed that f is holomorphic also shows that f is entire and that there exist

so that

Indeed, in that argument, we can replace the sector It is then evident that f Re j

by

If instead

0, we can proceed as in the first paragraph of the proof

of Lemma 2.7. This latter argument then shows that if q is such that

is ample for all j with Re

it is then

ample for all We state the result as a theorem. Theorem 2.12. for all

Say q is an even integer. Then

is ample

Further, we have the following uniformity.

CHAPTER 2

138

Suppose c,C,R > 0; then there exist B,C,,R,,c, > 0 as follows. If {g (ζ)} are holomorphic in S = {|η| < c|ξ]}, are homogeneous ο

of degree j - pi

|ζ| = 1, then there exists f ε Z

g

for ζ ε S,

.(B,C,,R,,c,) such that

q, ]

f~ h

η—1

and satisfy | g-,(ζ) | < CR l\y

1 1 1

v

The uniformity assertion follows from the proof of Lemma 2.7. If q is not an even integer, βχρ[-τ(ζ-, + .. .ζ ) longer entire.

] is no

However, it is possible to replace it by similar

suitable functions which are entire—in fact in z"—and

to prove

the following fact.

Theorem.

If q > 1 and q is rational, then z"

. is ample for

all j ε Κ . This will not be proved or used here, but will be shown in a

later paper.

The key point is the construction of the re­

placements of exp[-x(ζ,+...+ζ ) ^ ] . Our construction is lengthy. To see the difficulty, note that it is not even evident that Ζ™ φ {θ}. This was, however, proved in [22] in an argument attributed to B. mentary proof.)

IA. Levin.

(Silov has an earlier, less ele­

We, however, need an element of Z q with very

specific properties, and must use a very different construction. For a sketch of our construction, see [30]. Our construction probably works for all j ε C, but we have not

checked all the details.

Probably the construction could

THE SPACE Z

q

139

q» J

be carried out for q irrational as well.

(However, we are

only concerned with the case of q rational, as we said in Chapter One.) From Theorem 2.12, we now find the following fact and uniformity, which is a counterpart to Theorem 2.11.

Corollary 2.13. k = -Q - j .

Say q is an even integer, Re j > -Q; let

Say c,R > 0; then there exist C,,R. > 0 as follows.

Let Ig* (ζ) } be a set of functions such that g„ is holomorphic in S =

{|η| < cI ζ t} an^i

that \gAO

homogeneous of degree j - p£, and such

\ < R £ ! p - 1 for |ζ| = 1, ζ ε S.

K ε A K k which satisfies

Then there exists

sup |9 Y K(u) | < C R |

Y

V for all

1S|U|S2

γ ε (K+)s+1, Proof.

and such that K(I,ζ) ~ ^g (ξ) in Z q ..

This is an immediate consequence of Theorem 2.12, Theorem

1.8(b) and Corollary 1.5. We close this chapter with some generalizations that will be needed in Chapter 7 and 8. suppose k ε (C. K

Say Ω

(1) + (2)' follows form testing on appropriate f. If instead

for each

we still say

depends

holcmorphically on the parameter co if (1) and (2) hold. Again, (1) implies (2) if

is

equivalent to (1) + (2)'. If

respectively), and depends holomorphically

on oo, and if 1 S m S n, note that there exists K™ e

(or

respectively) depending holcmorphically on o>, with 9/du^K^f)] = . In fact, if K = G away from 0 where G is C°°. CO (JO 0) then

away frcm 0.

and if

If

then

is as in (2)', then

Say now

for sane n, W open,

for each w e W. We naturally say

, and we are given depends analytically

on the parameter w if there is a complexified neighborhood fi of W in £ n , and an extension of the family

. where

THE

SPACE

141

depends holcmorphically on

. We have a similar defini-

tion of analytic dependence for the

spaces.

Fran (1) and (2), and the proof of Proposition 1.1(a), it is easy to see this: Say

for each

Let

Then

depends holomorphically on the parameter w if and only if

(2.20)

does.

Say now

depends holcmorphically on the parameter

and that for each

there is a constant

so that

Let

Then the proof of Proposition 1.1(a) shows that

for each

there is a CJ, so that

In this case it is easy to see that, for any y, and

depend holcmorphically on co. Indeed,

we need only check the case where there is a single differentation. Property (2) for

is clear, while property (1) follows

from arguments involving difference quotients and uniform convergence on ccmpact subsets of

. Similarly for

In Chapters7 and 8, we will frequently examine the following situation:

142

CHAPTER

depends holomorphically on

4

; and

(2.21)

For seme (2.22)

for all In that case, we have the following result: Say ; then there exist as follows. Suppose we have (2.21) and (2.22), and in addition that Re

Suppose

(2.23)

for each is a holcmorphic function of w £ !!. Indeed, since the statement about

follows at once

frcm the uniformity in Theorem 2.11, we have only to check the last statement.

By (2.19), (2.20) and the remarks following

(2.20), we see that if is a holcmorphic function of to.

then Further, we know that

for sane

(2.23) now follows frcm an argument involving power series whose coefficients depend on to and which converge uniformly for to in a compact subset of

THE SPACE Z

q

. q, J

143

We shall also at times need a version of (2.23) which holds when k = -Q: (2.23) holds for k = -Q, provided that g

= 0

ω θ

(2.24)

for all ω ε Ω. To see this, note that by (2.19), J (Ο,ξ) = 0 for ξ φ 0. We may therefore define J' ε J p as follows: J'(λ,ξ) = J (λ,ξ)/(-ίλ) if λ φ 0; J/(Ο,ξ) = i(9 J )(0,ξ). LU

LU

UJ

Λ UJ

Q+P

Select K' ε K ~ with K' = J , and observe that TK' = K . ω ω ω ω ω Thus if (t,x) Φ 0,

K'(t,x) = / K (s,x)ds W

UJ

or -/00K (s,x)ds. 4- UJ

Z-

—oo

(Either integral gives K'(t,x) if χ φ 0. works for t < 0, the second for t > 0.) clearly that K' ε AiC

If χ = 0, the first These formulae show

, depends holomorphicalIy on ω, and

satisfies a condition analogous to (2.22). apply (2.23) to K'. J"

We may therefore

We then may use simple relations, comparing

and its asymptotics to J

and its asymptotics, to complete

the proof of (2.24). Having listed all these elementary properties of holomorphic dependence on a parameter, we now have a non-trivial assertion to make.

It is the generalization of Theorem 2.12 to the case

where the functions depend holomorphicalIy on a parameter.

144

CHAPTER

Corollary 2.14.

4

Say q is an even integer.

Suppose

then there exist

as follows. Say

seme n,

w

For each

holanorphic in

Dn

suppose

for is

is homogeneous of degree

and satisfies Suppose, further, that for each function of such that

is a holcmorphic

Then there exists and such that for each

is a

holcmorphic function of to c Q. Proof.

This is actually a Corollary of the proof of Theorem

2.12. We reduce to the case Re j < 0 as in that proof. One has then only to check that if fw (c.) is defined by the method evidenced in (2.9), the result will be holanorphic in ai. This follows easily frcm the estimates proving the convergence of the series in (2.9), and the fact that the parameterized versions of the A ^ in (2.8) will surely be holanorphic in to. Fran Corollary 2.14 we now extract the parameterized generalization of Corollary 2.13. Corollary 2.15.

Say q is an even integer. Re j > -Q: let

k = -Q-j. Say c,C,R > 0; then there exist Say

. For each

morphic in and satisfies

as follows.

, suppose 0, a function F

HOMOGENEOUS

supported in

PDE

ON

with

IH"

147

such that F is not

smooth at 0; for frcm F, the appropriate f could be found. Now

has the desired property. Were

F N smooth at also.

would be also, so

If '

would be

would also be smooth at 0, and would be also.

If

for any

then put

as in (1.2) . Then P(3), so that P O ) is

It is not; it equals

is a fundamental solution of

hypoelliptic.

It suffices to show

that K is real analytic away from 0 if and only if If

use Corollary 1.4.

If not, we show that J cannot

satisfy the conclusion of Theorem 1.3. Assume it did. Suppose Fix

with

and let us

examine the real analytic function H(^n) = an estimate

•

it is the inverse of a polynomial in Write

But

; consequently, it is

The polynomials

vanish if So

have

Since

H is the restriction tolR of an entire function in

constant.

We

; by continuity, they vanish identically. Thus

thesis and completes the proof.

; this contradicts our hypo•

To study analytic hypoellipticity for homogeneous partial differential operators, it is therefore necessary to let the

148

CHAPT.ER

4

coefficients be polynomials. We restrict attention to the case

We use coordinates with dual coordinates

, and we put

Q = p + s as in Chapter One. We restrict attention to those partial differential operators L with the following properties: L is hcmogeneous, with homogeneous degree k; the degree of L is also k; the coefficients of L are polynomials in the

's; and we can write L = L' +

(3.1)

, where L' is a constant-coefficient differential operator in the

's only, and where L'

is elliptic. Note that L' must be hcmogeneous, with hcmogeneous degree k, and degree k.

Given all the other assumptions, the hypo-

thesis that L' is elliptic is evidently necessary for L to be analytic hypoelliptic, as one sees easily considering functions of x alone. If L = the "Fourier transform of L". We shall often impose the following additional conditions on L: For all

r

there exists G (3.2)

such that

HOMOGENEOUS PDE

149

We shall see many examples later of cases in which (3.2) holds.

The main result of this section is the following.

Theorem 3.2. Suppose L satisfies (3.1) and σ ε ¢. that Z _/ .1^4^\ is ample.

Suppose

Then of the following conditions

(a) implies (b), and (b) is equivalent to (c): (a)

L is analytic hypoelliptic and L and L

(b)

For any K

(c)

(3.2) holds.

are hypoelliptic.

ε ΑΚσ, there exist K ε A K a + k such that LK = K . -

In particular, (3.2) is necessary for (a) to hold.

As

we said at the end of Chapter 2, Z q . is ample for all j ε IR, q,j — and we shall demonstrate this in a later paper. just include it as a hypothesis.

For now we

We will in fact not use this

hypothesis in proving (a) =* (b) (for σ = -Q) => (c). by Theorem 2.12, if q is an even integer then 7,

(Of course,

. is ample

for all j ε (E.) In (b), the special case where K, = δ and where L is left invariant on the Heisenberg group will be studied in detail in Chapter 4.

In this case, (b) implies LK = δ and L

is hypo­

elliptic and analytic hypoelliptic. For (a) =* (b), we can work in a more general setting. Lemma 3.3. Suppose D

is a differential operator with polynomial

coefficients which is homogeneous of degree k with respect to

150

CHAPTER

4

the dilations

Suppose

and

are hypoelliptic.

exists

such that

Chapter One.)

Proof.

Then for all

there are as in

In particular, in Theorem 3.2, (a) =>(b) if

By Corollary 1, page 540 of [80],

has a fundamental

kernel J(x,y) in seme neighborhood V of 0; the map dy takes

to

. Considering, then,

dy where

near 0, we see that there

exists a neighborhood U of 0 and U.

such that

in

It is now necessary to modify K to obtain

case

In

= 5 , it is a theorem of Folland and the author that

this can be done ([29], Theorem 3.)

In case

is general,

the procedure in [29] can still be followed, almost word for word, to obtain

•

In fact, by making minor modifications in the proof in [29], one could obtain, for any I and any with

could then be shown in general frcm

this. However, the general (a) =s> (b) will ccme out automatically later, so we do not explain the modifications needed in [29] here. Instead we turn at once to the heart of the matter, (b) (c). We shall work almost entirely on the Fourier transform side, so for later notational ease, we write -(j+k+Q) for a

HOMOGENEOUS

in (b), where

PDE

151

is appropriate.

From now on, if

is a multi-index, we write By Theorem 2.11, if with

then solving

is essentially the same as solving

with

for each

, where

for

Let us then begin by remarking the following proposition: Proposition 3.4. Suppose L satisfies (3.1) . Then for all

Proof.

is a sum of terms of the form

where

(We reiterate that

Since

for all j, it suffices to show that for all and (multiplication by The second fact is trivial; the first follows at once frcm the Cauchy estimates. We now prove (c)

•

(b) in Theorem 3.2. The key fact which

we use is the following. Theorem 3.5.

Suppose j e £E and that

satisfies (3.1) . Say and

is ample.

Suppose L

. Then there exist such that

and

such that Remark.

Once this is known, (c) ==>(b) follows at once.

it suffices, under the hypothesis of (c), to construct

Indeed,

152

CHAPT.ER

4

then

To do this,

observe that by (3.2) there exist By Theorem 2.11 we can select such that Thus

is supported at 0, so

polynomial p^. We may assume Then since

where

and that

for sane

otherwise we are done. , we must have

is homogeneous of degree a. Using the Cauchy-

Kowalewski theorem, we can find f, real analytic in a neighborhood U of 0, so that Lf =

.If

is the polynomial which

equals the sum of the terms in the Taylor expansion of f about 0 which are homogeneous of degree

, then evidently

so we can put Proof of Theorem 3.5. Let Let

Suppose

construct known.

~ £h.

It suffices to

so that

Say

For, say this is

Using the ampleness of

so that

. select

Using Theorem 2.11, select

such thai

Then

; also in

So ; by Theorem 2.11,

also, as desired. Write

where P is a homo-

geneous polynomial of degree k such that P(£) = 0 only when

HOMOGENEOUS

P D EONIH"

153

It suffices to construct an asymptotic series that

is holcmorphic in a sector S =

such

homogeneous

of degree j - pi and satisfying, for seme S, with the property that formally.

(3.3)

That is, the left side of (3.3) is a formal sum of terms homogeneous of degree j + k - pi, as the proof of proposition 3.4 shows. ness of

If this sum formally equals select

we can, by the ample-

with

of Proposition 3.4 shows that

and then the proof as desired.

Define D: Put

We need to construct formally.

as before (3.3), with

To do this, we shall make sense of

is holcmorphic in a sector hcmogeneous of degree j - pi, and satisfies for

We may assume P(£)

that

0 for £ c S^. Note

reduces the homogeneity of a function

by

, since |a| - |B| - k = -pm as in Proposition 3.4.

Accordingly, for any N,L, we can write

where

is hcmogeneous of degree j - pi and the sum is finite. Let and M ^

We check (3.3). Observe t h a t i f

N a I

Thus, for any I, we can write (3.4)

154

CHAPT.ER

4

Therefore, where the last sum is finite and j -

New, if

is homogeneous of degree

is formally computed, the term which

is homogeneous of degree j - pi is the same as the term which is homogeneous of degree j - pi in the expansion of This is just Now

so (3.3) has been established.

is holcmorphic in

and homogeneous

of degree j - pt. Let

To finish

the proof it suffices to show that for seme for

We have written D = Now write i for the tuple (a,g,m) ,

and write with

where I is an index set.

If

, write length Now we can write

where

is homogeneous of degree j - pi and the sum has only one nonzero term.

Further

The number of

terms in this sum is no more than

. Thus it

suffices to show that for sane for

, whenever 0 i v i I and length(I) i I. we can write, with r = length(I) ,

By Lemma 2.5, then, there exist

such that

If

HOMOGENEOUS

Here

denotes

PDE

155

norm on

Noting

v,r £ 1 and that for some N, we have we find that for seme then

if . It suffices then

to show that (3.6) for then . the fact

(We have used (1.7), (1.8), and

and we have let [ ] denote greatest integer

function.) To demonstrate (3.6), first note that by (3.5) we define Then as a term in

occurs

We have

Now, for the

first time, we use another of the hypotheses of (3.1), that the degree of L is k. As a consequence, that

so

(3.6) follows and the

proof is complete.

•

The following lerrena will complete the proof of Theorem 3.2.

156

CHAPTER

4

Lemma 3.6. Suppose L satisfies (3.1). Further suppose that there exists

as follows:

such that

for all

there exists

Then (3.2) follows.

Remark. Once this is known, Theorem 3.2 follows at once. Indeed (c)

(b) by Theorem 3.5; (b) =>(c) by Lemma 3.6; and

(a) -> (c) by combining Lemmas 3.3 and 3.6. Proof. We prove (3.2) for and the case of general siderations.

The case \ = -1 is similar,

follows from simple scaling con-

Suppose

By Theorem 2.11, we can select

such that

By hypothesis,

we can select

such that

then

We have only to

show that G must be in

Suppose it is not.

Suppose

sleect M such that Write

as in the proof of

Theorem 3.5.

In the sum for

Thus, in

where In particular,

but contradiction.

By the way, the proofs we have given show a

• version

of Theorem 3.2. Namely, suppose L satisfies (3.1) . Then of the following conditions, (a) implies (b) and (b) is equivalent to (c): (a) L and

are hypoelliptic; (b) for any

• if

HOMOGENEOUS PDE

K

ε f , there exists K ε K °

+k

157

such that LK = K 1 ; (c) for all

n

n

λ, if F ε S!lR ) then there exists G ε 5 0R ) such that L G = F. λ

One can even drop the hypothesis in (3.1) that the degree of L is k. Three groups of interesting operators which satisfy the hypotheses (3.1) are: (I)

constant coefficient homogeneous hypoelliptic operators in (t,x);

(II)

transversally elliptic homogeneous left invariant differential operators on the Heisenberg group H ;

(III)

Grusin operators in (t,x).

In case (I), it is evident that (3.2) could never hold. This gives an alternate proof of Proposition 3.1 in the case a = (p,1,...,1).

(The ampleness hypothesis is not used in

proving (a) => (b) (for σ = -Q) * (c) .) Case (II) will be discussed in Chapter Four. We close this section with a discussion of case (III); this will not be used in the rest of the book. The operators L in class (III) can be described, with minimal hypotheses, as follows:

CHAPT.ER

158

4

L is homogeneous, with hcmogeneous degree k; the degree of L is less than or equal to k; (3.7) the coefficients of L are polynomials in the and L is elliptic for x ^ 0. (3.1) follows frcm (3.7). Indeed, write where

is a differential operator in the

Since L' is

hcmogeneous with hcmogeneous degree k and degree less than or equal to k, it follows at once that and degree k.

has constant coefficients

In particular, L must have degree k.

since L is elliptic for

Further,

must be elliptic.

For these operators, Theorem 3.2 takes a very clean form. Proposition 3.7. Suppose L satisfies (3.7) and that (a)

is ample.

(c) F

Then the following are equivalent:

is analytic hypoelliptic;

(b)

there exists o

r

i

s

(d) For (a)'

(c)1 F (d)' For

surjective; is injective;

is hypoelliptic;

(b)1 if

there exists o

r

i

Suppose

s

surjective; is injective.

HOMOGENEOUS

PDEONIH"

159

Proof. We shall prove the implication

below in

Lemma 3.8; we assume it for now. We now show that the remaining implications follow easily frcm this, Theorem 3.2, and, especially, the work of Grusin.

Indeed, Grusin demonstrated

We shall show .

This will be enough, since

(a) as we have said. (b) => (c) follows frcm Theorem 3.2 (c) => (d) since, given (c) and

we may write

Since since, as we have said,

Thus (b) =£> (c) =£> (d) => (a); similarly .

Thus we need only show (a) => (b) and

First note (a) =t> (d) . In fact, we can prove more generally that if L is any homogeneous analytic hypoelliptic differential operator in (t,x) with coefficients which are polynomials in the

must be injective on

Indeed, suppose

By Theorem 1.8 (b) (with all there exists Clearly LK = 0.

so that If L is analytic hypoelliptic, K is analytic

at 0, hence K = 0, so F = 0. Thus (a) = (d); analogously (a)' = (In fact the analogous argument, for observed by Taylor [79].)

, was

160

CHAPTER

4

New assume (a). We have (d), which implies that is also injective.

Since (d)

(a) and (a)',

is hypoelliptic and analytic hypoelliptic. , there exists

with

(b). Similarly

(a)

By Theorem 3.2, Thus

this completes the proof,

except for the demonstration of

, which we turn to

now. Metivier proved (d) =>(d)1 in[62] in the case p = 2; Helffer [40] adapted Metivier's proof to the case where The proof which we present here for general p is an extension of Metivier's method. For

, define the Hilbert space

Write

Note that if

then

(where the

derivatives are taken in the sense of distributions) and multiplication by

In fact, for all j and

Further, for some V

It is traditional, when dealing with Grusin operators, to study not

but its inverse Fourier transform in

keep with this tradition.

Thus, if

and we

HOMOGENEOUS

PDE

161

(3.8) we examine (3.9) instead of In the expression for

note that for each term, the

relations Frcm this it is easily seen that for any

we have

Now, under hypothesis

(3.7), Grusin showed that (for seme For all u where

(3.10) Frcm this we shall prove the following lemma,

which clearly (by Theorem 2.3

implies Proposition

3.7 Lemma 3.8.

Suppose

degree

IN and L is a differential operator of

Df the form

term

where in each Suppose (3.10) holds, and that

(3.10) also holds if L is replaced by Lg

Then:

If

S and

then

Proof. As in Theorem 2.3(c), define

We use the notation X for a product of

162

CHAPTER 3

the fonnxl ..• x + , where {XI, ..• ,xm+n } c:U uV. mn

If

#{x.lx. E U} = m and #{x.lx. E V} = n, we define w(X), the 1.1.

1.1.

weight of X, to be m/(p-l) + n. p - 1 = (a-b)/b.

Recall p = alb, so that Define 7l"+/A = {n/Aln EZZ+}.

Put A = a-b.

Then for any X, w(X) E

For f E S, j E ZZ"+ /A, put

'lJ,+ /A.

[fl. =rrax{II(l+lxI2)r(p-I)/2xElI2 : r E?l+ /A, J

O~nmax(l,q-l) ,r+w(X)~ j}.

L

I + (Note q - I = (p-l) E ?l /A.)

We claim:

f E z~ if and only if there exist CO,R j E ?l+/A, [flj < CoRg(Aj) !l/Aq.

O

suppose f E S.

Then

> 0 such that for all

For, suppose f E

z~.

Using

Theorem 2.3 (a)(c)2' select C,R > 0 so that if {XI, •.. ,x + } c: mn U u V, #{x.lx. E U} = m and #{x.lx. E V} = n, then 1

1.

1

1.

m n Ilx I' . . X + II 2 < CR + m.,lipn.,l/q . mn L o~ r~

max(l,q-l), r+w(X)

Say JElL< . ""+/A , r

~ j.

E

""+/A, ""

PickNE1NwithN~

Then for some C ,C ,R > 0, 11(1+lxI2)r(p-I)/2Xfll 2 2 I I

(p-l)/2.

~

L

II (1+1 x1 2 )NXf ll 2 ~ C Rm+n (m+2N)! l/Pn! l/q ~ C2R~+n[m!A/ (p-l) n!All/lq~ I L

C Rm+n(A') ,l/Aq 2 I J.

Say {XI""'Xm+n} c: U let X = Xl" 'Xm+n'

U

V, #{x.lx. E U} = m, #{X.IX.E V} = ni 1 1 1. 1.

Then

II xfll

2 L

~ [fl n+m/ (p-l)
0, 0 < r < 1 so o 0

that whenever Ilvll :£ 1, w

= Rv, we have IWal


0 and 0 < r 1 ,r2 < 1 so that I (REa '13)I < clrlal+lsl and I(S\'13)I < c2r~al+ISI for all a,S. For (a), note I (RSEa '13) I

= I (SEa,R*E S) I

c c rlalrlsl I(r r ) Iyl 121 2 1 2 y

RS

€

z~ (H).

For

(b),

:£ I I (SEa,E y ) II (Ey,R*ES) I < y

= c rlalrlSI

for some C > 0 so 3'

312

say Ilvll < 1, w

Rv.

=

Then Iwal :£

i

IlvQ11 (E ,RE )1 :£ Ilvll(II (E ,RE ) 12)1/2 < c r a \ (Ii\8\)1/2 1 S ~ aSS a S c4rlal for some C4 > 0, as desired.

=

Without further ado, we pass to the proof of Theorem 4.1. Proof of Theorem 4.1.

We write

commuting polynomial.

Also we write L

L =

p(Z,Z) where p is a nont

= Pt

-

(Z,Z),

HOMOGENEOUS POE ON ~n

193

t

+

Let R = P (W ~*

= -p(~+ ,-~).

,-YJ) ,

(We just take this as the definition of

~*,

even though in fact ~ is indeed the adjoint of the unbounded operator ~.

(v,R~W).). R*

,!(f, (L

f)

Note, hONever, that if v, w € Then i f f

rr;:,

then (R v ,w) A S, (Lf)A = ~, (L*f)~ = fE*, (LRf)A

€

A

A

= R* f. k-2n-2

and K2 € APV such 2 2n and such that F2A = F?2 € Z2 nR ) for all

By Lemma 3.5, there exist KI s AK

By Proposition 4.6, J 2 + '-

= W+IF2 +

€

'-

-

Next, using Proposition 4.8, select ~l homogeneous . 2M 2M+k + of degree -k, wlth J I + : H ->- H for all M Ell, such that (fIK I )

= cn(fIJ I )

whenever

f

E

Q. Applying Proposition 4.8(a)

repeatedly we see that whenever f (f IL*KI ) = cn(f I~l~* ).

€

Q,

cn(fII-~2)

It follows that for all a,A,

A

But R(A) * : Hk and J I (A) : Ho

->-

= (fI8-K 2)

->-

H0 boundedly,

k boundedly; hence

H

J (A)R(A) * v I

=

(I-J 2 (A) ) v

for all v

Again using Lerrrna 3.5, we select K3

E

such that LR*K3 = 8 - K4 , and such that K4

AK

E

Hk •

k-2n-2

= ~4'

(4.11)

and K4

J4 +

,-

€

€

APV

Z~(H+I)'

We also select ~3 homogeneous of degree -k, with J 3,+ ~ ..,. H2M+k for all M € 7l + , such that (f IK ) = c (f I:h) whenever 3 n A

f

€

Q. Applying Proposition 4.8(a) repeatedly we see that

194

CHAPTER 4

whenever Since

the uniqueness assertion of Proposition 4.7

shows

Thus, for all for all

3y Proposition

(4.12)

and

compact. Also,

are

boundedly. By Atkinson's

theorem ([16]), then,

is Fredholm; that is, it

has closed range and finite dimensional kernel and cokernel. Indeed, the kernel and cokernel are contained in H!j] . For if then

so

by Proposition 4.9(b). Further if then or that

It suffices to show for all

from (4.12). Also note

. But this follows indeed, since v e

for all a. Thus, in fact,

Now let Q(A) denote the projeciton in the cokernel of

onto

Since the cokernel is contained in

by examining the matrix of Q(A) we see that Indeed, if

is an orthonormal basis for

, we have

an estimate for seme

Note Q is homogeneous of degree 0,

so by Proposition 4.6, there exists P e APV such that P = Q.

HOMOGENEOUS

PDE ON H "

195

Note so

(4.13)

To see the significance of all this, let us first verify that if

is given by

jection in

then B is the pro-

onto a subspace of

B is idempotent,

since

by Proposition

4.3. B is self-adjoint, since if f,

then

Thus B is a projeciton. Further, that if

This is the same as saying

then

To see this, observe that

also

for all

and since

Since

boundedly. Thus

(Note, as a consequence, that

as claimed. since

for

We see, thus, that B projects onto a subspace of [LS]-1-. In fact, B projects precisely onto

this is

easy to believe, but will be verified only at the end of the proof. Let us speak heuristically for a moment and motivate the rest of the proof. In order to prove Theorem 4.1 we are

196

CHAPTER 4

seeking K so that

whenever

we expect that

or that

We are trying to find

and in light of (4.13) it

seems reasonable that the theorem would follow if we knew the following: There exist

such that

Returning now to the formal proof, we shall establish (4.14) in case k is even, and show how the theorem follows from it. (4.14) is indeed the main point. Assume, then, that k is even. We construct only the construction of H_ is similar. We construct H by constructing its matrix columns

where

We abbreviate

denote the columns of S by We would like

so that

Since

and since

has closed range, we can and do select

so that

and so that for seme need only show that there exists : Let

We

so that

; it suffices to show that there exist so that

Proposition 4.9(a). Let so that exists

for all

so that

Now,

by

there exist As a consequence there Now, by (4.15)

HOMOGENEOUS

New

PDE ON H "

197

and k is even, so by Proposition 4.7, Further, there exists

so that

, In addition. Also

position

so by Pro-

there exist

so that Thus, indeed, by (4.15),

for seme

we have

claimed, and (4.14) follows if k is even. Still assume k is even, and define if

Then H is homogeneous

of degree

and, by (4.14),

Select

so that then it is elementary that

for all

2.10 there exists whenever

Let

By Theorem

so that

Then

since for appropriate just as in the proof of

Proposition 4.6(a). Applying Proposition 4.8(a), we see that Now

whenever

so by the uniqueness assertion of Proposition As we argued before (4.12), however,

and

Now consider the

given by

198

CHAPTER 4

-1

R

f (u) = f(u ); extend ~ to 5'. Further, if K by f "* f *K

Then if g ε 5', (L *g)~ = Lg. 2 ε PV, the adjoint of the operator on L given

is the operator given by f -> f *K . Since B is 2

self-adjoint (recall Bf = f*P for f ε L ) , P = P . AK

k-2n

2

~ .

K ε AK

k_2n

Then LK = 6 - P. ~

2

Let K = K" ε

Thus, if k is even, there exists

with LK = δ - P.

If η = 1, k may be odd.

In this case, however, we can

still find K" ε A K 2 k - 2 n ~ 2 with LL*K" = δ - P ' , where if P' = Q \ then Q' (λ) is the projection in H

onto {ν ε Η ω | R(A)*R(A ) ν = 0} .

This latter space equals {ν ε tfwJR(A)v = 0 } , so Q' (λ) = Q(A) and P' = P.

If we set K = L*K" then again K ε AKk~2n~2

and

LK = δ - P. Using the facts that LK = δ - P and L^> = 0, one can now verify the local solvability assertion of Theorem 4.1. See [26], page 549, for the argument, which is the same as that which Greiner-Kohn-Stein [35] gave for the particular case L = /..

(On page 549 of [26], our P is called P,. Lemma 2 4.6(c) of [26] is used to show that if f ε L + E' and if

f = 0 on an open set U =JH , then f*P is real analytic on U. This, however, is easily seen without Lemma 4.6(c). On pages 549-551, it is also shown that, if P φ 0, there exists f ε C°° η L HH ) such that f is not real analytic near 0, but f = f*P; thus, if P φ 0, Lu = f is not solvable near 0.)

HOMOGENEOUS PDE ON ~n

199

To complete the proof, then, it suffices to show that 2

I - B projects L

n

--

onto LS.

(H )

and we have already sho.vn LS LS is dense in (I-B)L 2 (lIn).

c

(Recall Bf == f*P for f (I-B)L 2 (II n )

Let

Q==

€

2

L ,

We must show

.)

{fl fE: Q}.

It suffices

to show: v

(I-B)Q is dense in (I-B)L 2

(I)

v

(I-B) Q c LS.

(II)

(I) is evident, since

Qis

dense in L2 and (I-B) : L2 +L2 v

continuously.

For (II), say h

€

(I-B)Q so that h == f* (o-P)

v

for some f

€

Q.

It suffices to show that f*K

h == f*(o-P) == L(f*K) follows. each N

LS.

€

To show f*K

€

€

S, for then

S, we proceed as

As we observed in the proof of Proposition 4.7, for + N 7l , f == LofN for some fN with fN € Q. Thus f*K == A

€

fN*K_, where K_ == RNK N

-N

0

E

Kk.-2N-2n-2.

Now ajax. == (x.+2f)j2, ]

]]

ajay. == (Y.+~)j2; we thus see that if D is any differential ]

]

]

monomial in the a/ax., a/ay. and T, and if k. - 2N - 2n - 2 == J

]

-M < 0, then Dg(u) == O(l+jul)-M.

Since M is arbitrary, Dg

€

this completes the proof. The following corollary was known (see [81], [77], [78], [62] for the equivalence of (a), (b), (d) and (e), and [29], Theorem 3 for the equivalence with (c).) Corollary 4.10.

Suppose L is a left- invariant homogeneous

differential operator onHn which is homogeneous of degree k.

S;

200

CHAPTER 4

Define R by (Lf)" = ig, for f ε S .

Then the following are

equivalent: (a)

L* is hypoelliptic

(b)

L* is analytic hypoelliptic

(c)

There exists K ε A K k _ 2 n ~ 2 so that LK = 6

(d)

I L is elliptic; R ( D v = 0, ν ε H => ν = 0; R(-l)v = 0, OO

ν εΗ =>v = 0. (e)

I L is elliptic; R ( D v = 0, ν ef/° =>v = 0; R(-l)v = 0, ν ε Η ω = > ν = 0.

Proof.

It is evident that (c) =>(a) and (b). Next, if (a)

or (b) holds, observe that I L is elliptic (examine functions of ζ alone.)

Thus we can select K,P as in Theorem 4.1. Since 2 f -v f*P is the projection in L onto (LS)-1-, L*P = 0, so P is smooth, so P = 0.

Thus (a) =>(c) and (b) => (c) , so (a)

(b) (c) . If any of these hold, then I L is elliptic and P=O,

so (d) and (e) follow.

Finally, if we have (d) or (e),

we can again choose K,P as in Theorem 4.1, and by hypothesis P=O,

so (c) follows.

Remarks.

1.

The main point of the proof of Theorem 4.1,

besides Lemma 3.5, was (4.14) . The proof of (4.14) shows that if ν e (1-Q+)H η Η ω , then there exists w ε Ηω so that R(+D* w = v .

This was shown in [26], and also in [62]; in

HOMOGENEOUS PDE ON Η

Π

201

the latter, however, it was assumed ν = 0. equivalence of the definitions of π the present book.

(By the way, the

was not realized until

[26] and [62] used different definitions.)

The method in [62] is that of Lemma 3.8.

Indeed, in the

Schrodinger representation, R(+l)* is a differential operator which satisfies the hypotheses of Proposition 3.7 with ρ = 2. Note also that the estimate (3.10) for L = R ( + 1 ) * is an immediate consequence of our methods, specifically (4.11). 2.

If L satisfies the hypotheses of Theorem 4.1, then it

also satisfies the conclusion of Proposition 3.7.

Indeed

(a) =>(b), since if (a) holds, LL* is hypoelliptic and analytic hypoelliptic by Corollary 4.10, so w e may use the argument previously used for Proposition 3.7(a) => (b) .

(b) => (c) = >

(d) exactly as in the proof of Proposition 3.7. Next, for (d) => (a), assume (a) fails.

Then by Corollary 4.10, if K,P

are as in Theorem 4.1, we have P φ 0.

Λ 2 Since Ρ(λ) ε Z2(H^)

for all λ , Ρ χ ρ ε Z^SR 2 ") for λ φ 0 by Proposition 4.6. But L*P = 0, so L*f\P = 0. fails.

Since F, P φ 0 for λ = 1 or - 1 , (d)

Thus (a) => (b) => (c) => (d) => (a) ; similarly (a)' =>

(b) ' => (c) ' => (d) ' => (a) '; finally (a) (a) ' by Corollary 4.10.

5. Homogeneous Singular Integral Operators on the Heisenberg Group Corollary

gives a necessary and sufficient condition

for a homogeneous left-invariant differential operator

on

to be analytic hypoelliptic. What is the same thing, if L has degree k, it gives a necessary and sufficient condition for there to exist

such that

then

Let

and we have the precise conditions under

which there exists

with

(since

In this chapter we answer the more general question:

, when does there exist with

(The meaning of this convolu-

tion will be discussed presently.) We shall also show that the existence of such a

is the same as the analytic hypo-

el lipticity of the operator

(reca

by

and is extended to a map from Corollary 4.10 answers this question precisely when supported at

is

since it is easy to see that any homogeneous

distribution supported at 0 has the form L6 for seme homogeneous left-invariant L. answered;

(The analogous question on!

exists if and only if

is easily

does not vanish away from

if and only if convolution with

is an elliptic pseudo-

differential operator. In fact, by Corollary 1.4, it is necessary and sufficient that

. All of this

SINGULAR

INTEGRALS

ON

203

follows from techniques we shall see soon.) The theorem on inverting singular integral operators, to be proved in this section, will be a crucial ingredient in the theory of analytic pseudodifferential operators presented in Chapters 7 and 8. We state the main theorem now. Not all the terms have been defined precisely yet, but we shall do this immediately after stating the theorem. Theorem 5.1. Suppose

for sane

Define in the

sense. Then the following are equivalent: (a)

is hypoelliptic.

(b)

is analytic hypoelliptic.

(c) There exists

so that

(d) and (e) •

The equivalence of (a), (c) and (d) in the C°° situation will also follow frcm our methods; see also [12] for this. We proceed to explain our terms in detail. First we explain what is meant by a convolution

204

CHAPTER 5

Proposition 5.2. Suppose (a) Write

where

the distributions

of compact support, and

Then for f exists and is independent of

the choice of (b) There exists

(independent of f) with

We write (c) Similarly, we can form

this then equals

(d) If R is right invariant. invariant, then (e) Suppose

If L is left so that

For ". Then if

(f) (g) Suppose also Then (h) Select

with

Define Still suppose

but do not assume Re

necessarily. Let be homo-

geneous differential operators with polynomial coefficients, of homogeneity degrees

respectively, and

SINGULAR

I N T E G R A L S ON

205

suppose Re

Then, in the sense

of distributions, m

Most of these facts were proved in [26], Theorem 4.4, or in [12], Lemma 9.5; we take the latter as our reference since it is easier to read. The basic idea is to also split where

and to put (5.1)

This is easily seen to exist and satisfy ments for

The argu-

in [12] were carried out only for

Rhom^,

real, in which case it is shown that The proofs in the case

are the

same. In the more general case

the proofs

of (a)-(e) go through as before, but they only show is C00 away from 0. However, it is very easy to see from (5.1) that if

then Now suppose

Re

Using (4.10), write Then each we can write

where and

, Continuing in this way we see that as a sum of terms of the form Re a, Re

where Thus

206

CHAPTER 5

as claimed. place of

(a)-(e) also hold for

and 2n in place of

same or easier.

in

the proofs are the

(g) and (h) were not proved in [12]; we

include the proofs in an appendix to this chapter (in part A.1). We also prove there that if and Re j, Re k. Re

Proposition 5.2(f) is a simple consequence of the following lemma which will also be used later. Lemma 5.3. Suppose

for some

and

is open.

Suppose that one of (a), (b) or (c) holds: (a) f is a distribution of compact support which vanishes (b) on f U; (c)

f vanishes on U, and for sane a n d f is real analytic o n U .

Then

and

are real analytic on U.

we define for exists for

this clearly under the hypotheses on f.)

Remark. Proposition since if

(In case (b),

is open and

follows at once from Leirma does not contain 0, then we can write

as a sum of 3 distributions f each of type (a), (b) or (c).

SINGULAR

Proof of Lemma 5.3.

207

INTEGRALS ON J)"

Say ρ ε U; we show f*K is analytic near

p; by replacing f by a translate we can and do assume ρ = 0 and U is a neighborhood of 0.

Define J,τ

: C°° -> C°° by

(JF) (u) = F ( u - 1 ) , (τ F) (u) = F(uv); extend J, τ put g = Jf.

Pick U, open with 0 ε U,, U, c

U.

In (a) or (b),

we can and do regard f as an element of V (U.). g(x K) for u εIH

close to 0.

: V -> V ;

Note (f*K)(u) =

It is then easy to use this

formula to extend f*K(u) to a holcmorphic function of u ε (C

,

u close to 0; this proves the theorem for f*K, in cases (a) and (b) . (c) is more subtle.

For f*K, the case -2n - 2 < Re j
0 so that for all 5 e S with (5.7) we have It follows that all

and all

have holcmorphic homo-

geneous extensions to S. We still use the notation for the holcmorphic extensions. Select and fix c' with and let

Then the main point

is to show the following: There exist

so that for all

with

If this were known, Lemma 5.6 would follow at once. Indeed, by Theorem 2.12, we could then select

so that

By Theorem 2.11 we could select

n

SINGULAR INTEGRALS ON H

225

K3 E: APV so that F +(!;,) = (FK ) (I,!:;), F _(!;,) = (FK ) (-I,!;,). 3 3 3 3

= K3*Kl = K3*(o-K).

Let 0 - K4

~30(~-~)

that .

In

z2

2,0.

=

~,

By Lemma 5.7 and the fact ~

it follows at once that (FK 4 ) (I,!;,)

0

But K4 E: APV, so by Theorem 2.11, we have also

Thus we need only show (***). an explicit formula for

g3~(C,).

For this, we shall examine

To derive this, first put

D = (a/aC,1, ••• ,a/aC,2n)' 5 = (a/aC,n+l,··.,a/aC,2n' -a/ac,l, ..• ,-a/a gem) =

~

I

Sn ) on

a+b+lal=~

~

2n

.

Observe that in S we have

[(2i) iaila!]IP g Da g(m-l) = I g . Da g(m-l) a b a+b+lal=~ a,a b

where we have set gaia (c,) = (2i) lalnaga (r)/a! for c,

r

c,

E: S.

This

gives at once, by induction, the formula

(5.8)

in S, where the sum is taken over all (a , ••. ,a _ ) E:N and O m1 1 1 m all 2n-tuples a ,···,a of nonnegative integers such that 1 1 m a + ..• +a _ + la 1+ ••• +la - 1 =~. We estimate gim) (c,) for o m1 C, E: S'.

The number of terms in the summation in (5.8) is no

more than the number of ways of writing

~

as a sum of

m + (m-l) (2n) nonnegative integers, which is (~+m+(m£1)2n-l) ~ 2!+(m-l) (2n+l) < 2(2n+2)~ i f m ~~. 0.)

(Recall that i f m >~,

Let us then estimate the size of a typical

term in the summation.

Select c" with c' < c" < c.

By the

ZZ6

CHAPTER 5

Cauchy estimates, (5.7) and the definition of the gaia we see that there exist C ,R > 0 so that for all a,a and 2 2 all

S" with 1/2

I:; €

~

Select 0 > 0 so that if centered at {(, €

o

I:;

o

2, we have Ig «(,)I < aia S',

€

/s 0 /

c2R~+lala!.

= 1, then the polydisc

with polyradii (0, ... ,0) is contained in

S"ll/2 < 11:;1 < 2}.

and (,


0,

.t = a + ••. +a - + ml o

Iglm) (c;) I

i f m ~.t.

(Indeed, since

I I) a o ··• .am- l ·INI. (d).

Say Kl is as in Theorem 5.1(d).

in Lemma 5.6. J i + = Ji

(1)

Say K , = J, in the Q sense (i=1,3,4). 1

(i=1,3,4)

i

~l

then J 4+

E

By Proposition 5.5(b), for all v If v

€

Hoo, Jl+v = 0, then v

By (el, v (d) => (cl.

Select K ,K as 3 4

2

Z2(H) E

= J 4+v

= O. Similarly if v

Define

by Proposition 4.6(b).

Hoo, J +J l +v = (I-J 4+)v. 3 E

E ~,

W

H by Proposition 4.9(bl. Jl_V

= 0, then v = O.

This is the only remaining implication, and is

the main point.

For this we shall adapt the proof of Theorem 4.1.

SINGULAR

INTEGRALS

ON

227

Because of Lemma 5.6, only minor changes are needed. For the time being, we use only the hypotheses that the hypotheses that kernel in

have zero

will be used only at the end of the proof.

Suppose

in the

sense. Define

by

and extend " to a map frcm S' to S'. Say K^ = J^ in the Q. sense. We claim that for all then

if

To see this, suppose select f with

Now

also,

so that

This is true for all for all

so

Since

extensions to bounded operators from

have

and

by Proposition 5.5(a), as claimed. Now, using Lemma 5.6, let us select and

APV so that

and Using (5.1) and the widely-

applicable rule

we see that

Also

in the Q. sense

then

By Proposition 5.5(b), if then (5.9)

CHAPTER 5

228

(S.10) By Proposition S.S(a) and (c), J (A) can be extended to a 1 -k

0

bounded operator fram H

to H ; J (A) and JS(A) can be ex3 o -k tended to bounded operators fram H to H ; J 4 (A) can be

o 0 extended to a bounded operator fram H to H , and J 6 (A) can be extended to a bounded oeprator fram H-

k

to H- k •

From now

on, when we write J. (A) (i=1,3,4,S or 6) we mean the aforel.

mentioned extensions and not the operators with domain H. Then (S.9) holds for all v V E

E

H- k and (S.10) holds for all

~.

By Proposition S.S(c), J (A) and J (A) are compact. 6 4

Thus,

by the analogue of the simple argument given after equation (4.12), J (A) has closed range, and it also has finite-dimen1 sional kernel and cokernel which are contained in~. fact,

[J 1 (A)H-kl~

= {v

E

~IJI (A)V = a}; let Q(A) denote the

projection in HA onto this space. 4.1, Q(A) E exists P

Z~(HA)'

E APV

9 is "

In

As in the proof of Theorem

homogeneous of degree 0, and there

with P = Q.

Of course, if we have the full

hypotheses of (d) (that J 1+,J _ have no kernel in~) then 1 P = O.

In general, without these hypotheses, we shall show

that there exists K2

E

AK -k-2n-2 so that K2*Kl = 0 - P. Clearly

this will establish (d) =>(c). following (4.13), f

+

Note that, by the argument

f*P is a projection in L2 aHn ); we post-

pone discussing what its range is.

SINGULAR INTEGRALS ON ~n

We let J i + = J i (±1) (i=1,3,4,5,6),

229

Jl +

J l (±1).

=

As in

(4.13), we have (I-Q+) (I-J 4+) = I - J 4+ and

J l +J 3+ - -

= (I-Q+) - (I-Q+)J 4+ -

(5.11)

The theorem will follow at once if we can show (5.12) Indeed, say (5.12) were known. K7

E

AK

-k-2n-2

J 7 (±1).

~

so that if K7 =

By Proposition 5.5(d), select ~7

in the Q sense then H+ =

Put K2 = K3 + K7 ; by (5.11), (5.12) and homogeneity,

(K l *K 2 ) , taken in the Q sense, equals I - Q.

PVOHn ),

(K l *K 2 )

Kl*K2 = I - P.

Since Kl*K2 E

= I - Q in the sense of Proposition 4.3. Now

P=

Thus

P since the operator of convolution with

P is self-adjoint on L2; thus K2*Kl = I - P as desired.

Thus we

have only to show (5.12). To prove (5.12), we construct only H = H+, since slinilar.

H_

is

This is done exactly as in the proof of Theorem 4.1,

so we will be brief. Let S = (I-Q+)J 4+, Sa = SEa; we can a -k -. a a select H E H so that J l +H = S and so that for some Co > 0, a IIHall_k < collsallo for all a. Let H~ = (EB ,H ); i t suffices to show IH~I < c r}a l +IBI for some C > 0, l l z2 < r l < 1, for then there exists H E 0 2 so that a H , and H will be as desired. There exist HEa C 2 ,C

3

>

0,

o
0, M,N

Then there exist Say f,g

e,R

E

zt.

> 0 as follows:

E' (lHn), f of order M,g of order N.

E

Say supp f

SI' sing supPaf c BI , supp g c S2' sing sUPPag c B . 2

elRl~I~!

laYf(u)

I

for u

E

P and all y.

Say k

E

0::,


0 .

of the form

< 1 to be

Then

for some

Now e need only be chosen so that v

V,

this is easy to do, so (7.37) is established.

Thus v

V,

Finally, then for |u|
0.

Then there exist C,R 3 > 0

320

CHAPTER

Suppose

7

for some

Suppose

supp

is analytic

and that

for all Then K*f is analytic on U. Further

Proof. This is immediate frcm Lemma

and

(7.34) . Lemma 7.8.

Suppose

Select s > 0 so that Select s-^

with

denote the character-

istic function of Suppose

and suppose

there Suppose, exist i

Then

with the following properties. for seme Then:

(a)

Suppose

for all If Re

for

and

then

is analytic

ANALYTIC

CALCULUS

ON I'll

holomorphically on a parameter oj, then so does (c) Suppose

for

is analytic for

Then Further, (7.38)

If

depend holcmorphically on a parameter

a), then Proof.

is holcmorphic in u> for i Clearly we may assume, without loss of generality,

that (a) is inmediate from Lemma 7.6(b), Case 2. As for (b), note first that there are only finitely many values of for which the hypothesis Re

could hold. This fact,

together with the usual decomposition it clear that there exist only on

so that

makes both depending whenever

supp

whenever

This is what we need to know for the condition (7.21). For away frcm

(7.20)

note that for

f/e may write

Because of (a), it is now clear that in order to prove (b) we need only shew (c), with would then obtain relations like

We

CHAPTER 7

322 not for I S

|u| S 2, but the information is easily carried

over to 1 < |u| < 2 by use of the homogeneity of K *K,. However (7.38), the main assertion of (c), is immediate from Lemma 7.6(a), (7.31) and (7.34).

The statement about

holcmorphic dependence on a parameter in (b) is easily proved through use of (5.1) and the map Q of Proposition 5.2(a).

Similarly, breaking up χ,K. and X 3 K 3 into two parts,

one supported very close to 0, enables one to prove the state­ ment about holcmorphic dependence on a parameter in (c).

Remark.

Note that Fact 5.10 does indeed follow from Lemma

7.8(b), as we claimed when we presented that Fact.

Indeed,

note that (in the notation of Fact 5.10), on any set Ω 1 of the type described there, there must exist Cl,CI,R',Ri > 0 k so that K ε AK V (C ,R ) for all ω ε Ω", ν = 1,2. νω ν ν Proof of Theorem 7.5.

Our first task is to define K = K *K,.

If Re κ < 0, we set K *K, = K*K,. Let us motivate our procedure.

Say then Re κ i 0. We do not use the de­

finitions (7.18), (7.19); rather we give a different definition which we later prove to be equivalent to (7.18), (7.19). strategy will be to first specify what K

Our

= 9'Κ/γ! is for

|γ| = [Re κ] + 1 or [Re κ] + 2 and then to invoke Corollary 7.2 to produce the needed K. tion, we could define γ!K

by

If * were Euclidean convolu­

ANALYTIC

C A L C U L U SONI'll

(7.40)

with

chosen so that

since Re

; this makes sense

Re

We wish to avoid differentiating kernels the real parts of whose homogeneity degrees are less than or equal to -2n - 2, if at all possible.

For this reason, we would distinguish

three cases: (7.41) (7.42) (7.43) If we have (7.41), we would be sure to choose Yv(v=l,2) in so that Re

and

it is easy to see that this is always possible, at least if Re

(For this, Then

It is then easy to write least if

where

If we have (7.42) we would choose Y^ so that at least Re

=

(This

follows frcm Re

Similarly,

in case (7.43) we would choose

In this way

we would be able to take full advantage of as Lemma 7.8(b) (to estimate

for

as well and Corollary

324

CHAPTER

7.2. The trouble is that

7

is not Euclidean convolution,

so we need a replacement for the identity this unfortunately forces us to contend with more technicalities.

There will also be a slight additional

technicality in the case We are writing (1= j fin). We intend to avail ourselves of these widelyapplicable identities:

The third identity, for instance, follows frcm and New let U denote the set of finite sets of ordered pairs of distribution on

thus

if for seme We abuse notation

by writing such a set of ordered pairs as a formal sum (The formal sum of two ordered pairs is another ordered pair.)

If

ANALYTIC

we define elements

CALCULUS

Pmu,Qmu

P1u=

ON

3Z5

U as follows:

(f, Tg)

Q 2 u = (Tf,g) and if

(7.44)

Then f,g

u = (f,g) and 2n + 1, we have (7.45)

If now u

let us put, for 1

for v = 1,2.

Now, suppose (Re

+ (Re

for v = 1 or 2.

> 4, so that Re

For u =

2n + 1, let us define if Re

2n + 1,

Suppose that + 2n + 2 > 2

of this form, and as follows:

+ 2n + 2 > 2

= Qju if Re k 1 + 2n + 2 < 2 (in which case Re Finally, let

2n + 2 > 2) .

326

CHAPTER

7

for al1

where

is independent of and if Then

Thus,

we define and

and

we can form

and Now, returning to the notation of the statement of the theorem, let us write

If

then

Let us assume

for now, then; we explain the necessary

modifications to handle the case

later. Then we can

form say. Note 2n; then vhere Re and we can form

Thus for all

ANALYTIC

CALCULUS

ON

I'll

(7.47)

(Because of (7.45), we have every heuristic right to think of this as a replacement for define

We shall now

in such a way that

for all y.

We shall be using Corollary 7.2, so we shall need to show that, if

then

For this, note that write for sctne for

or m.

where each

In the notation of (7.47) we may (7.48)

where D ^ is a homogeneous differential operator. Now choose (S e C"(Hn) with

Thus for

and let

Thus, byPut

Put

By (7.47) and Proposition then

for and note

then,

328

CHAPTER

7

(7.49)

where the limit is taken in the sense of distributions, is an immediate consequence of (7.49). We may therefore put and we say

as in Corollary 7.2,

It is easy to see, from the above

construction, as well as the definition of the map F (see the proof of Corollary 7.2) that ± is bilinear from

Let us now prove (a) and (b), still assuming then indicate the necessary modifications for is immediate if Re If Re

(a)

by Lemma 5.7 and Proposition 5.5(b).

let L and R be as in (7.22). Write

where the Pp are hcmogeneous polynomials. A homogeneity check reveals

for all

p. We may therefore write or RLK

hence

for all where

By the definition of K and

where the limits are in the distribution sense; we used Proposition

ANALYTIC

CALCULUS

This proves (7.22).

ON I'll

In particular (7.50)

for

Since

as in Corollary

follows at once frcm the construction of Corollary 7.2 that if

away from 0 then

By

(7.50), however, G is given by (7.18), so that K is given by (7.19). The remaining statements in (a) follow at once frcm

Lemma 5.7 and Proposition 5.5(b). Let us now prove (b), still assuming

suppose

' for seme

that (7.25), (7.26) hold. We put

and

and we put

prove (b), we must estimate the

of

To in the sense

made clear by (7.10) of Corollary 7.2, with constants depending only on

and

Writing (7.51)

in the notation of (7.47) and (7.48), and noting by (7.44) that

we see that it suffices

to consider each term separately. We need to use another property of the map With notation as directly after (7.46), say where

and

tfhere

330

CHAPTER

7

Suppose that for seme

we have (7.52)

Then it must be the case that Re

(7.53)

and for seme

we have (7.54)

This is easily seen by induction on and (7.46).

and by use of (7.44)

(Loosely speaking, here is why this holds:

the process of applying

In

repeatedly, we never differen-

tiate a distribution if the real part of its homogeneity degree is

at worst we multiply it by

Further, in the process, a distribution the real part of whose homogeneity degree is

will never be differentiated

to the point where the real part of the new homogeneity degree is and

This, then, is why we used both the

to define

as we said at the outset of the proof,

we wished to avoid differentiating any distribution if the real part of its homogeneity degree is Now let us return to the notation of the theorem and of (7.51).

In (7.51), we may write any

where each

in the form

ANALYTIC

or

CALCULUS

ON I'll

Letting

it is

clear from repeated applications of (7.44) that Further,

It now

follows easily from Lemma 2.5 and (7.25) that

1

for certain

depending only on the

Say that

for all

Note, by the dis-

cussion surrounding (7.52)-(7.54), that if seme Re

then Re

Then and

we have

and all

there exists so that for such

and

depending only on we

have

whenever

depending

332

CHAPTER

Since

7

for all

Lemma 7.8(b) and

(7.9) together imply that for seme only on the kV and for all y.

> 0, depending

we have

By (*) and Corollary 7.2, for seme A,R depending only on the kv and

This proves (b) for n > 1. If n = 1, u = then if if

= [Re x] + 1 = [Re

+2

1 we have h(u)

2 we have h(u)

2 +

3 +

: while In either

case we may either write (1)

where i,j = 1,2 or 3 and where h(u) be able to apply

say.

We may not

to u , but we can at least form

Then we define in Case (1) ;

in Case 2. We new define (*), put K =

as in (7.47), again verify (7.49) and and verify (a). The slight problem

arises in (b), since by using the

we may have now

differentiated distributions the real parts of whose hemogeneities were

-4. However, say that

ANALYTIC

CALCULUS

ON I'll

The arguments for (b) in the case i for

M there exist

a n d d e p e n d i n g I

show clearly that with

only on t h e a n d s o

for all

that

Note also

Re

Since we need now to apply the

P's at most twice to get the that for

it is now easy to see

N there exists

with

, depending only on the

that for all

and the

so

now

follows at once as in the case We now tackle (c), one of the main points of the entire discussion. With

as in (7.44), let us write, for (7.56)

Here (dropping y superscripts)

is a homogeneous differ-

ential operator of homogeneous degree geneous polynomial of degree

is a homo-

and (7.57)

If at least one of f,g has ccmpact support, Thus, if

Proposition 5.2(h) and (7.57), then

3 l i

CHAPTER

Writing

7

in this expression, we find,

for (7.58) where

For seme

depending only on

and

and for some

0, depending only on

and

we

have

This follows at once from Lemma 7.6(a) and (7.31) and (7.34). Further, for sane : , depending only on

and (7.60)

and

(7.61)

The number of terms in the summation for F

Y

does not exceed

Further, we may write where each Now, from (7.44) , Frcm Lemma 2.5,

ANALYTIC

for all

335

depend only on

Frcm Lemma 7.8(a) and (7.61), we see

is analytic for

where

ON Hn

where and

that

CALCULUS

depend only on

and

and

In particular, (7.62)

Now, for

sufficiently small, let us form

By (7.62) this is analytic near 0. We have

Here and

are constants depending only on is chosen with

and

We claim that, if

then

Indeed, it suffices to observe that

(sum over

so This is evident

336

CHAPTER

7

from (7.58). With

we have from

(7.58), (7.59), (7.60) and (7.63):

Here

depend only on

and r. This is almost

all that we need, except that (7.64) tells us only that (7.66) where p is a polyncmial all of whose terms have homogeneity degree less than

To complete the proof, it

suffices to show that

where

and

depend only on

and r. How-

ever, by (7.66), Lemma 7.8(c), part (b) of this Theorem and (7.65), we clearly have (7.67) for polyncmial of degree at most Theorem on p about any point

But p is a using Taylor's

with

we see by

enlarging Ag and Rg if necessary, (7.67) holds for Setting

we have completed the proof of (c).

As for (d), the fact that

depends holcmorphically

on the parameter is immediate frcm Lemma 7.8(b) if Re

A N A L Y T I C C A L C U L U S ON ΙΗ

Π

337

If Re k + m i 0, we define G as in (7.18) . Note (by (2.20)) that G(λ,ξ) is a holomorphic function of the parameter for 1/0.

N

Since, in (7.18), F ( t ( K ^ A 1 ) ) (Ο,ξ) is clearly

a non-zero multiple of G(O,ζ) for ζ φ 0, G(O,ξ) is holo­ morphic in the parameter also for ξ φ 0.

Hence Γ

depends

holomorphically on the parameter, and therefore K^iK-, does also.

Possibly decreasing r, we now see from Lemma 7.8(c)

that Q(u) = (χ~Κ *χ,Κ,)(u) - K(u) is holomorphic in the parameter for 0 < |u| < r.

Since (7.29) is satisfied for

all ω ε Ω, there is a point U n with 0 < IuJ < r so that Q(O) may be obtained by means of a power series expansion about U n for any ω ε Ω.

An argument involving uniform con­

vergence of holomorphic functions on compact subsets of Ω now shows Q(O) is holomorphic in ω as well.

This completes

the proof of Theorem 7.5. Although our arguments have been lengthy and careful, in one important way they have been crude. Theorem, say r, and r_ are small.

In (c) of the

If m is large, then Q

should be small on {u : |u| < r}, with size decreasing geometrically in m, since X~K„*X..K, and K are in some sense small there.

The point is, we have not kept careful track

of the dependence of R n in (c) o n r , and r_.

Fortunately,

dilations provide us an extremely easy way to recover this information, and to prove the following lemma, which is

338

CHAPTER

7

the backbone of all that follows. (Localization Lemma). and

Suppose

suppose Then there exist

with these properties. Suppose, for (7.68) for seme For be the characteristic function of

Then there exists

analytic on

such that (7.69)

and

(7.70)

Further if

depend holomorphically on a parameter

and satisfy (7.68) for all

then

is also a holcmorphic

function of to for Proof.

and K are homogeneous, this is a simple con-

sequence of Theorem 7.5(c), in the case use of dilations.

together with

In the general case, let us write, for with

ANALYTIC

the

CALCULUS

ON I'll

homogeneous, and with the p y f p polynomials. then of course,

recall

and write

funct ion. For Now choose

With

for f a

we have as in Theorem

sufficiently small that if f supp

Now for

We may assume r is , supp

then we have on

We obtain the needed estimate for each term separately. The term involving

is estimated by (7.29); the term involving

p log c is handled through (7.27) and (7.33) . The next term is handled through Lemma 7.6(a), (7.31), (7.34) and (7.33). The last two terms are handled through (7.33) and Proposition 7.7. This establishes (7.69) and (7.70).

340

CHAPTER

7

For the statement about holcmorphic dependence, note that since

depend holomorphically on

and p(u) will also be holcmorphic functions of to for any u.

Indeed, this is easily verified if one first examines in the case

then in the case

and then uses homogeneity.

The case

is trivial

since the polynomials are homogeneous, so that they are 0 at the origin if they are not constant. holcmorphic dependence of
0,

we let Ck(U;r,s,R,R') denote the space of all cores of the form 00

K (w) u

=

X (w) [ L ~(w) + Q(u,w) 1 m=O u

where X is the characteristic function of B r

(7.73)

= {w

Iwl


V = O.

0 ~lu

analytic

Then

parametrices

377

is analytic hypoelliptic on

in the follow-

ing sense: Suppose

open, and that

Suppose

and that

is analytic on V .

Then

is analytic on V . Further, say

Suppose

compact. with

Then there exists an open set

and there exists

(where and open sets

with

so that if

we have: I

f

t

h

e

n (8.1)

and

is analytic on

The core of

(8.2)

has the form

with

istic function of

and

for appropriate

but for

Further, suppose on a parameter

the character-

as in (7.75), (7.78), (not U).

in addition depends holomorphically Suppose E is compact and

Then there exists an open set

with

as follows: We may choose and

as above independent of

depending holcmorphically on

so

CHAPTER 8

378

that (8.1) and (8.2) hold for all ω ε Ω'.

The core of Q.

has the form χ (w)Q (u,w) with χ the characteristic function of {u : I u| < r}, and Q

as in (7.75), (7.78), for appro­

priate r,s,R independent of ω ε Ω', for u ε Tj'; further, 0 (u,w) depends holomorphically on ω ε Ω' for u ε U", |w|< r. Theorem 8.1 is an immediate consequence of Theorem 7.11(d) and the following assertion: Lemma 8.2.

Suppose K. is as in Theorem 8.1.

U' 0, i f J

!•/"(!,ζ)!

But by

= FFv ,

< C(m)eB^'

.

(8.22)

(7.21), J 1 1 1 U , ζ ) = (FK 1 1 1 Hl, ζ) = [ ( F K ) ( I , ζ ) ] 1 1 1 .

(8.23)

In general, (FK)(Ι,ζ) (the entire extension of (FK)(Ι,ξ) to I

) will grow like a Gaussian on (C

, say on the order of

e A | C ' . By (8.23), J111CL, ζ) should grow like e 1 " ^ ' , which flagrantly contradicts (8.22). Probably, then, the operator f •+• ( O.

(8.98)

v

If we replaced each Sv by J v + in

(B.9S), we would then have an estimate

This is rather like (B.87), since if we expanded out all \' 2/ 4, there would be 2r P's, and a! 1/2 r! A = LP. 1

~

(a+2r)! 1/2 .

Of course, this does not prove (B.87), but does help to motivate what we are about to do, which is to reduce Lemma

B.S to Sub 1emma B.7. Before we do this, however, let us now explain briefly why we could not work directly with the derivations (8.70), and had to bring in the derivation D. briefly, the trouble is this:

d~

in

Stated very

D is the commutator with A,

so victory for the D's is measured in terms of factorials. d.1+ is essentially the camnutator with a

~,' v

so victory

416

CHAPTER

8

for the d's is measured in terns of square roots of factorials.

But the crucial facts (8.90), (8.91), involve

factorials, not square roots of factorials. That, in a very few words, is why we have had to pursue the roundabout route of proving Sublenma 8.7 and seeking to derive Lemma 8.5 from it. We shall do so by means of a device, which is analytical rather than combinatorial in nature. The idea behind it is easy to explain. What is needed are analogues of the Riesz transforms. Suppose for the moment that we knew that there exists so that Convolution on the right with

would be like applying

Lemma 8.5, that all

Suppose also, in

so that

Say, for instance,

Then we could rewrite (8.99)

Now,

define

where we are writing The

are analogues of the Riesz transforms. Note Most importantly, it is clear frcm the

results of Theorem 7.5 that for seme satisfy estimates akin to (8.86), with some new R;

ANALYTIC

PARAMETRICES

417

C and the new R depend only on the old R.

Thus if we define

would satisfy estimates akin to (8.98), with a new

and

R' depending only on R. However, frcm (8.99) above.

Applying the same reasoning to each group of terms for I odd, we see that if L is even we have only to estimate

for certain Lemma.

much like the

of the statement of the

Such an expression can be estimated in the manner

that we indicated after (8.98), and the estimate needed on (8.87) would be obtained, since the number of A's is under our assumpt ions. There are several minor problems with this plan might not be L, L might not be even - and one major problem we don't know that there exists There is such a $ in we believe this

however - see

is, in fact,

effort to prove it.

with

, we have made no

Instead we opt to use seme exact

formulae of [28] to prove replacements for $. We define

Although

as

follows:

CHAPTER 8

418

1 (t,z)

(8.100)

Then by Proposition 7.1 of [28], we have: (8.102) where the operator families B

1AEa ,A

while

~2

=

~1'~2

are defined as follows:

(2IAI)-1/2[f(lal+n/2)/f(lal+(n+l)/2)]Ea'A if A 0 (8.103) (21 AI) -1/2 [r ( Ia 1+ (n+l) /2)[ ( Ia I+ (n/2) +l)]E if A< 0 a'A

does just the opposite: (21 AI) -1/2 [r (Ial + (n+l) /2)[ (Ial +(n/2) +1)] E ,if A> 0 a,/\ (8.104) (2IAI)-1/2[f(lal+n/2)/T(lal+(n+l)/2)]E , if A < 0 a,/\

(In the notation of Proposition 7.1 of [28], these are the cases k of A

-1

~1' ~2

•

= n+l/2,

j

= 1/2,

Y = n/2 or (n+l)/2.)

show that they are

rr

The forms

approximate rr square roots of

Moreover, (8.105)

ANALYTIC

Using

PARAMETRICES

in place of

419

we can now give a proof of

Lemma 8.5 which solidifies all of our heuristics. We define

by (8.100), (8.101),

(8.103), (8.104), and note (8.102) and (8.105). For

for seme k, we define

where we are writing The

are analogues of the Riesz transforms. We shall

also needdo to define,if for (N would convolution were We also write, if canmutative.)

Clearly

/

The

are analogues of multipliers, and

identity if everything commuted.

Now, for seme have an estimate

would be the

Explicitly,

depending only on R, we

420

CHAPTER

8

(8.106)

for any i. This is an easy consequence of (8.85) and Theorem 7.5. an easy application of Lemma 2.5 is needed to handle the

the

By (8.83), then, there are

depending only on I^, so that if (8.107) then for all : (The case

.

(8.108)

is not dealt with in (8.83). But then

could only be one of the fixed finite collection .

As we remarked

during the proof of Lemma for all

Also, as is clear from (4.6),

Thus, in our case,

must equal

where f is an element of a fixed finite collection of real analytic functions.

(8.108) now follows if

by the usual fact that, for seme

Now for (8.87), let us first assume L^ is even, and where

ANALYTIC

where

PARAMETRICES

421

Say i is

Then

We write the string

(8.109)

In the expression

we now replace each

string of the form of the left side of (8.109), for

odd,

by the corresponding string on the right side. We leave along any other factors in the expression.

In this manner

we have now rewritten the expression in the form where

for all i, where 3y (8.108) and Sublemma 8.7, for

some C ^ d e p e n d i n g only on R we have an estimate

as desired, if If L^ is odd, we put i

is even. , and rewrite

422 < C.R.

CHAPTER 8 (a+L)1

for an absolute constant C

certain C.,R.,C,,R. > 0 depending only on R.

and for We have used

the case where L. is even, as well as the fact that each A ~

P., being 3 the Fourier transform of an element of ~i'

AiC

, is a family of bounded operators.

This completes

the proof of lemma 8.5, and with it, the proof of Theorem 8.1.

9.

Applying the Calculus

In this chapter we derive a number of other results about the calculus, which are useful in applications.

The

chapter ends with a proof that the Kohn Laplacian, and a parametrix for it, lie in the systems analogue of the calculus after a contact transformation, under natural hypotheses. 1.

When using the calculus, it is frequently simplest to

work with formal sums.

To make this easier to do, we add

some elementary facts to (8.4) and Proposition 8.3, for later reference. Suppose K

€

Ck(U), K

Kf = 0 on U for all f

=

0 (K), u

o

c:

U, U open, and 0

Then K := 0 for all u € U . u 0 Indeed, the kernel of K : COO(U ) -+ Coo(U ) must be zero. Thus, coo for any u € U , the distribution K (uw- l ) must be zero for o u w € U. By analyticity, K := O. o u o

(ii)

For k

€

€ coo(U ).

0

so that for all N, all

we have (9.3)

434

CHAPTER

8

We define (9.4) and for f e C°°(U) we define c (9.5) Of course

depends on the choice of a,b and

: when we

(rarely) need to indicate the dependency we write

(9.6) Say

near u, and , then for v near u, (9.7)

so where

(9.8) We may use (9.8) to define The second term on the right

side of (9.8) is clearly smooth in v near u for any The point is that it can be estimated. Letting we have this: Suppose

Suppose

f has order L, and f is analytic near u. Then there are

and a neighborhood

of u so that

APPLYING THE CALCULUS

435

for all N,

(b)

Suppose We H n is open, g is a smooth function on W,

and for some C,R,L' > 0, _~M+L'

ID1···DMg(w)I < CKN-

whenever M;;; N - L', WE W (9.9)

and Dl, ... ,DM E S. Then g is analytic on W. Proof. (a) Clearly it is enough to shaw that for any c > 0, there exist C , R4 , V so that for all N 4 IDi ... DiDl ... DMGNV(W}! < C4R~+L where M;;; N - L, (9.10) I ;;; L, v E V, lu-lwj > c and Di, ... ,Di, Dl, ... ,DM E S.

Here the Dj, Dk are differen-

tiations with respect to wand v respectively.

But (9.10)

is clear from (9.3) and from iterating relations such as Xl[F(V,V)]

~

Fl (v,v) + F 2 (v,v) where Fl denotes the result

of applying Xl to F in the first variable, slinilarly F2 . (b)

We may assume w

= O.

Put UL

Note that (9.9) also holds with i f II~ II

= M.

=

(T,Xl,···,xn , Yl, ... ,Yn ).

a~(UL)

in place of Dl ... ~'

Indeed, a ~ (UL) is slinply an average of terms

of the form Dl ... DM.

Since o~(UL}

this completes the proof.

= d~at 0 (see (7.10B)),

CHAPTER 9

436

The utility of Proposition 9.3 was indicated on page 12 of the Introduction.

We will use it in the next

chapter. 7.

Of course, the calculus we have presented can be

transplanted onto any real analytic odd-dimensional manifold M, via an analytic diffecmorphism.

This is most natural if

M is a contact manifold, namely a 2n + 1-dimensional manifold on which there exists a one-form σ with oAdoA where.

(Here there are η factors of do.)

contact form.

An example is M = H ,

do / 0 every­

σ is called a

σ = τ = dt +

2^(x.dy .-y .dx .) . Note that τ is left invariant, since it satisfies τ(T) = 1, τ(X.) = τ(Y.) = 0 for all j. More gen­ erally, we may let M be any CR manifold with non-degenerate Levi form, with σ being a 1-form annihilating T at each point.

. ΘΤ. ,

(We use the notation and terminology of

[19].) A diffeomorphism between open subsets of two contact manifolds is called a contact transformation if it preserves the contact form at each point, up to a constant multiple (depending on the point).

Darboux's theorem implies that

if M is an analytic contact manifold with contact form σ, then (M, o) is locally diffeomorphic to (JH , τ) by an analytic contact transformation.

This observation will enable us,

in #9 below, to identify the Kohn Laplacian and a parametrix

437

APPLYING THE CALCULUS

for it as being within our calculus, on a non-degenerate analytic CR manifold, under Kohn's hypotheses. Since we do want to view our pseudodifferential operators as living on contact manifolds, it is only natural at this point to ask whether our classes of operators are invariant under analytic contact transformations.

(Then we would be

able to define our calculus, on an analytic contact manifold M, as that calculus which reduces to the one we have already seen, under any analytic contact transformation taking an open subset of M to K . Invariance under general diffeomorphisms is clearly out of the question because we are using a non-isotropic notion of homogeneity.)

Invariance

under an analytic contact transformation could certainly not be true without some qualification, since in (7.73) χ will in general be mapped to the characteristic function of an open set which is not a ball.

However, we have the

following result (which we will not use, so we only sketch the proof):

Proposition 9.4. Suppose ψ

Suppose U c H

is open, u

ε rj, r,s,R,R, > 0.

: U -»• U' is an analytic contact transformation.

Then there exist r',s',R',R! > 0, U u , as follows. Say K e C

^ U a neighborhood of

Let U' = ψ - 1 ( U ) . (U ;r,s,R,R,) . Then there exists

K' ε Ck(U^;r',s',R,,Rp so that

CHAPTER 9

438

(Kg) ,

K'g'

for all g

goW, K = O(K), K'

Here g'

E:

E' (0 ). o

O(K'), g'

goW, (Kg)'

(Kg) oW. If g

Proof (sketch):

coo(O'), we may write symbolically

E:

c

fg(v)K(W(u),v

(Kg)' (u)

-1

W(u))dv

fg ' (v)K(W(u), W(V) -lW(u))

Idet(DW(v)) Idv.

(9.11)

The crux of the matter is this lemma, which we shall eventually use with w = v Lemma 9.4.

E:

0'.

u:

Withw as in Propsition 9.4, v III (w)

for vw

-1

W(v)

-1

E:

0' fixed, put

W(vw)

Write lII(w) = (111 (w) ,1111 (w) , ... , 1112n (w)) inlli 0

coordinates, w

=

(t,x,y).

Then LIII

o

(0) = 0 i f

n

L is any

differential operator of degree less than or equal to 2 which involves only the x,'s and the y,'s. J

Proof.

J

Since left translations are clearly contact trans-

formations, III is also a contact transformation, which takes

o to

O.

The condition that III preserves

T

at each point w,

up to a constant multiple f(w), is equivalent to the following relations for all w near 0, and 1

~

k,t ~ n, where all

functions are to be evaluated at w = (t,x,y), and the sums

APPLYING

THE

CALCULUS

439

are from (9.12) (9.13) (9.14)

If deg

the lemma follows at once frcm (9.13), We also find

by differentiating (9.13) and differentiating (9.14). Finally, as we see by differentiating (9.13) with respect to y^ and (9.14) with respect to equations.

and adding the two resulting

This proves the lemma.

Keeping the notation of the lemma, but writing we now see from the lerana and the fact that f is a diffeomorphism near 0: For seme

In fact the error is bounded :

Frcm this

and the lemma, we see: (9.15)

In fact the error is bounded by allow v to vary in Lemma 9.4.

Now we In (9.15) we write

CHAPTER 9

440

Ψ = Ψ , c = c(ν), S = S ; these depend analytically on v. Putting w = v

u, writing c(v) = c(uu v ) , S = S _, , uu ν

and using a power series expansion as in the right-multi­ plication analogue of Proposition 7.12, we now see Ψ(ν) _ 1 Ψίω = (c(u)t,Sw·) +o(|v _ 1 u|) if ν

(9.16)

u = (t,w') and ν is sufficiently close to u. For

u e u, w = (t,w') ε IH n , write ρ (w) = (c(u)t,S w') . The point is that ρ (D w) = D (p (w)) if D

is a dilation. If

K is as in (7.73), it is then clear that X1(W)[JK1JJ11J(P11(W)) + Q C W U ) , P U ( W ) ) ]

(9.17)

is in C (U!) for any relatively compact open subset Ul of U' and X, the characteristic function of a sufficiently small ball.

(One must note loglwl - log]P (w)| is homogeneous

of degree zero in w.) This observation, (9.11) and (9.16) come pretty close to proving Proposition 9.4, but we must still explain what we are going to do with the o(|v u|) in (9.16) and the |det Οψ(ν)I

in (9.11).

The tern o(|v u|) is in fact an

analytic function of u and ν

u, say F(u,v u ) . Note also

|p (w)I ^ |w| and that the radius of convergence of the power series for KT. . at P (w) is also on the order of Iw |, by (7.79). For w small, then it is valid to expand

APPLYING THE CALCULUS

KT, , (p (w)+W) in power series about W = O W = F(u,w).

441

and to put

After one does this, it is then valid to

expand each F(u,w)

as a power series in w (αε (Z)

) .

The terms of this power series will all be o(|w|' ' ) .

In

this way one readily sees that one obtains an element K" of Ck(U") by replacing ρ (w) by ρ (w) + F(u,w) in (9.17), if U" is a sufficiently small neighborhood of Ψ

(u ) .

(One may need to reduce the support of X. to a smaller ball.)

If U' is sufficiently small, we obtain an element

K' of Ck(u') by multiplying K" (u,w) by ldet ο Ψ ^ - 1 ) ! , since one can also expand this function in power series about w = 0 as in the right-multiplication analogue of Proposition 7.12.

(Again one may need to reduce supp X,.)

This K 1 is as desired.

Further details are left to the

interested reader. Note that the principal cores are related by (K')°(w) = ldet ΕΨ (u)l Kj ( u ) (P u (w)). 8.

In the applications which follow, we shall frequently

suppose ourselves in the following situation.

We use the

notation and terminology of [19] or of [20], Sections 2 and 13. Let M be a 2n + 1-dimensional C°° CR manifold.

Suppose

ρ ε M, and that M is an analytic CR manifold near p, and

CHAPTER 9

442

that the Levi form is nondegenerate near p.

Thus M is an

analytic manifold near p, and Tl,O' TO,l are analytic subbundles of

[TM

near p.

Suppose we are given a smooth

Hermitian metric on

[TM

which is analytic in a neighbor-

hood of p, and which is compatible with the OR structure. (That is, Tl , oLTo , 1 and

= for Z,W

€

Tl,O.

In

particular, i f M is the boundary of a danain D in ([n+ l, with M=

aD

smooth, and analytic near p, we could use the re-

striction of the ambient metric to M.) metric a OR metr.ic. Ttal tions of Tl , each point.

\'I€

We call such a

may find W1 ' •.. , Wn analytic sec-

° near p, which are orthonormal in the metric at Conversely, if we were instead to begin by

choosing analytic sections W1' •.• 'Wn of Tl,O near p, which span Tl,O at each point near p, we could then choose a Hermitian metric on ([TIM, analytic near p, and compatible with the OR structure, such that W1' ... 'Wn are orthonormal at each point near p.

We let U be an open neighborhood of

P such that U is an analytic OR manifold,

is analytic

on U, and WI' ... 'Wn are orthonormal at each point of U, and such that there exists an analytic contact transformation I\J : U' -.U, U'c:

W,

I\J(U')

=

U.

We shall let AO,q, L~,q, (E,)O,q, (V,)O,q denote (O,q) forms which near any point have coefficients which are respectively smooth, in L2, in E' or in V', with respect

APPLYING

THE

CALCULUS

to a local basis of smooth is a basis for

443

forms.

Say now

and all are real analytic. We 1

may expand

in the form

I are multi-indices. We then write vector

where the to denote the row

all of whose entries are in

We abbreviate

the space of operators

with cores as in In the applications which follow, we shall frequently be seeking to establish that an operator properties:

has the following

for an open neighborhood and it-

Explicit ly, this last condition means that there exists a in the systems analogue of

(acting by convolu-

tion on the right) so that

We scmetimes refer as the

Frequently

also satisfy this:

Also, for for sane

for an open neighborhood

open, we let and we say

will

444

CHAPTER

9

a contact transformatior On the "germ" level, the definitions of the underlined terms in the last paragraph do not depend on the choice of Thus, suppose we are given (ii) with

satisfying

Say we are given another basis

for

. Then there exists an open neighborhood U ^ of

p and an operator K' with , and so that after the contact transformation is used in pulling back to

—where the basis

and so that

for

This is easy to see frcm the product Theorem 7.11(c), since the operator of multiplication by a real analytic function g on

with core g(u) (w).

(Since we are only interested in behavior near p , we can always reduce the support radius of the pullback

that the

hypotheses of Theorem 7.11(c) apply). Say

is an open neighborhood of

and

Then we say R is if there exist K ,

U satisfying (i), (ii), (iii) with

so that

for all f

locally in

near p , after

error, if there exist K , so that

We say R is

, up to an analytic regularizing satisfying (i), (ii), (iii), with

is analytic on

for all f

Note that, in these definitions, one can always assume that

APPLYING

THE

CALCULUS

445

U q and the support radius of the pullback of K to

are as

small as one likes. Also, when proving that an operator R has one of these properties, by producing a

, it is

not necessary to show that condition (iii) holds.

For one can

always shrink U q and the support radius of K to force it to hold. If R :

we say R is analytic

pseudolocal on on

if whenever f

and f is analytic

we then have Rf is analytic on U . Say now 'M) are both analytic pseudolocal on U.

also each R y is locally in

near p , after

Suppose up to an

analytic regularizing error is also analytic pseudolocal on U and

Then locally in error.

near p, after

up to an analytic regularizing

Indeed, choose the appropriate

for

reducing support radii and the size of the to of

we may assume is in C

, and by passing

Say that the pullback . By reducing

and

further, we may assume and that Then if f e

is analytic on

and Theorem 7.11(c) applies to complete the proof.

By

446

9.

CHAPTER

9

Next, we show that the Kohn Laplacian, and a para-

metrix for it, lie in our calculus after a contact transformation, under natural hypotheses. Let M,U be as in the second paragraph of #8; suppose that the Levi form is non-degenerate everywhere on M .

For

the time being, all considerations will be local, so we may assume M = U. Let the one-form a be an analytic section of near p, satisfying

in the induced metric on (CT*M.

As we have said in #5, o is a contact form. denotes Levi form.) is a self-adjoint matrix.

Then

Choose an orthogonal

trans format ion are also orthonormal at each point near p , and if at p .

Thus:

is diagonal

we may assume B is diagonal at p .

Let

the eigenvalues of the Levi form. We may assume that for some

for so that M is k-strongly pseudoconvex near p .

Suppose now S is the vector field dual to o , so that S is an orthonormal basis for (TIM. Denote the dual frame for

by

As in [20], we use the notation "E" for error terms. If

form (the J are multi-indices), then

APPLYING

THE

CALCULUS

447

E((6) will denote an expression of the form analytic.

with

will denote an expression of the

form

with

analytic; similarly

and we abbreviate

We then have (9.18)

for

functions f.

Following the computation of [20],

Sections 13 and 5, but using (9.18) instead of (13.3) of [20], we see

Here

is an analytic function

with (9.19) New let

Select an analytic

contact transformation Under

say

1

taking (M,o) (near p) to

and let us say

are mapped to vector fields These vector fields, being annihilated by t , are in the span of coefficients.

at each point, with analytic Thus, after

and fl ^ are in

respectively, on any neighborhood of p whose closure is contained in U. Of course

carries Sf to

so at p',

448

CHAPTER

9

Suppose, in fact, at Define

Then, at

for

But clearly

for sane constant

(9.20) Now let

a self-adjoint operator.

Then the system of cores which

is a diagonal matrix and has the core slot is clearly, at back of and

the principal core of the pull-

By Theorem 8.1 and its analogue for systems, above, the pullback of

in our calculus provided the Theorem 5.1 for all all I.

in the

will have a parametrix satisfy the conditions of

that is, if

is hypoelliptic for

By (9.20) the map

is a H e algebra autcmorphism of the Lie algebra of Exponentiating it, we obtain a Lie group autcmorphism of

APPLYING

which carries

THE

CALCULUS

449

onto

is hypoelliptic if and only if

is. Let

We check condition (e) of Theorem 5.1. We find

Since

are injective on

as they do not annihilate E q .

as long

This can happen only if

which by (9.19) can occur only if I = or

Thus if

Theorem

5.1(e) will be satisfied for all the In particular, by Theorem 8.1, on analytic hypoelliptic on q-forms for

must be or

This

was originally shown in [81], [77], and [78]. But Theorem 8.1, Proposition 9.1 and parametrix for

above also give us an analytic

of a very precise form:

Theorem 9.6.

Let M be a nondegenerate smooth CR manifold

of dimension

which is k-strongly pseudoconvex and

analytic on an open neighborhood

of a point p .

Suppose

we have a CR metric on M which is analytic on U.

Suppose

450

CHAPTER

9

Then p has open neighborhoods as follows. There exist operators and

, which are in respectively after an analytic contact

transformation \)j from

, and which have the following

properties: (a)

for

(b)

(c)

Say on W .

Say

and g is analytic

Then K g is analytic on W .

(d)

If f

, then

10.

Now let us suppose that

are analytic on are as in Theorem 9.6,

and additionally that M is compact. on (0,q) forms.

If

denote

then

and analytic hypoelliptic near p .

is closed, and ker

is hypoelliptic,

As is well-known [19],

[20], there is a "Hodge theory" for

Thus range

is finite-dimensional. ker

are real analytic on U. onto ker

Let

Of course

and the functions in ker Let

and let

be the projection in be the self-adjoint operator

APPLYING

annihilating ker

THE

CALCULUS

451

and inverting

on ran

so that

and (q1 suitable). since

Clearly

is hypoelliptic, so we may extend also we may extend

Since

is analytic hypoellilptic on

is analytic

pseudolocal on U. Let

denote the projection onto ker

(respectively) in

We then have the following corollary,

of which part (b) is well-known. Corollary 9.7.

In the situation of Theorem 9.6, assume M

is also compact. Again assume (a)

Then:

are locally ir (respectively) near p after

up to an analytic

regularizing error. (b)(i)

are analytic pseudo-

on U (in the sense explained in (ii) Suppose f Then

and

Proof.

is analytic on

f is analytic on U .

analytic on Select

we claim that for

then

above.)

If instead

is

is analytic on as in Theorem 8.11. we have

Then

452

CHAPTER

9

(9.21) where g is analytic on U Q .

This gives (a) for N^.

(9.21), note that for f

To see

we may write as desired, by the

analytic pseudolocality of N on

. For

we need only

note (9.22) on

Indeed, if f

and

then By (9.22),

we may extend

This gives (a) and

(b)(i) at once for

similarly for

of Sg, we also have (ii) for

similarly for

By self-adjointness on

giving (b)

10.

Analytic Pseudolocality of the Szego Projection and Local Solvability Let M be a smooth compact CR manifold of dimension 2n+l.

Suppose U Λ ' (M) is closed in the C°° topology, we shall show that the Szego projection S on M is "analytic pseudolocal" on U. (i)

That is:

If V C (M) continuously, then of course S : E' .> E', and, under the global hypothesis.

454

CHAPTER 10

we shall show that (i) and (ii) hold for f ε

E'.

(Cases in which it is known that S : C°°(M) -»• 0°°(Μ) con­ tinuously include:

M strictly pseudoconvex, η > 1 (see

equation (10.12) below); M the boundary of a bounded, smooth 2 strictly pseudoconvex domain in (C ([52]); M the boundary of 2 a bounded smooth pseudoconvex domain in !C , if each point is of finite type ([57]); M the boundary

of a bounded smooth

pseudoconvex domain in (Cn (n>2), if each point is of finite ideal type [57]. We do not need to assume M has any of these forms; but our theorem covers all these cases, since the range of 8 b : C°°(M) -> Λ ' (M) is closed in the C°° topology in all these cases, so the global hypothesis holds, as we shall explain presently.) Part of our global assumption is: For all f ε C°°(M) there exists u ε ( E ' ) 0 ' 1 so that Vbu=

(I-S)f.

(10.1)

This is a very weak version of a "closed range" hypo­ thesis for J. , since it heuristically states that the range of j , is the orthocomplement of the kernel of 3,.

We will

discuss conditions under which (10.1) holds, in a moment. Under condition (10.1), we shall show that the map f -> SfI (j is continuous from (T(M) to C°°((i) . Let V => U he any fixed open subset of M such that the map f -> Sf|

is

THE SZEGO" PROJECTION

continuous from Coo(M) to Coo(V). extend S : E' (V) + E'.

455

Then of course we may

Under the full global assumption,

stated below, we shall show that (i) and (ii) above hold for fEE' (V) + L2 (M).

Define O(S), the domain of S, to be

2 E' (V) + L (M) •

The full global hypothesis on

M

is the following:

For all f E O(S) there exists u E (E,)O,l so that

:7 b

= (I-S) f.

U

(10.2)

(10.2) follows if one knows that the range of 0b

Coo(M) + AO,l(M) is closed in the COO topology, or more gen-

erally, from the following criterion: Proposition 10.1. Say t

~

0.

Let M be a smooth, compact CR manifold. Suppose that there exist

sEE, C >

°

so that: t

00

for all u E C (M) there exists v E H (M) so that for some sequence S

in H

,

t {vm} E Coo(M) , vm+v in H , 0bvm + 0bu

(10.3)

and Ilvllt :s cllabull s .

Then for all f E O(S) with (I-S) f exists

W

E (E,)O,l so that Jbw

H- t , there (10.4)

= (I-S)f.

(Here Hs ,Ht denote Sobolev spaces.) Proof. Let - b Its denote the closure of

°b :

sidered as an operator from Ht to HS •

In this proof, we

°

rI'"

L

°

+ A ' 1 , con-

456

CHAPTER

simply write

10

Then, in (10.3), we may simply

write

If P is the projection onto ker

we may assume

since

clearly

But then

Thus we may assume is closed in H . Then

also has closed range.

that

Indeed, suppose

and g e dan for all h

some w e

and cong

sequently the range of

orthogonal to ker

in H1",

with

Select Then

so for

and we have

hence

proving that the range of

is closed.

Thus ran

for all

and the Proposition follows at once. Evidently, then, (10.4) would hold if one knew any of the following: The range o

f

i

s

closed in the (10.5)

topology; The range of the closure o

f

a

s

an operator (10.6)

on

is closed;

THE SZEGO

PROJECTION

457

OO

For some s £ t, C > O, we have: there exists ν ε H

s+1

For all u ε C (M)

— — (M) so that 3, ν = 3,u and b b

(10.7)

l|v||t s c||3-bu||s. Now we describe situations under which (10.5), (10.6) or (10.7) hold for all t i 0.

Hence, our theorem will hold

in any of these situations. (10.5) holds if M is strictly pseudoconvex, η > 1 ([19], page 88).

If M is the boundary of a bounded smooth pseudo-

convex domain in C 11

, we have (10.6), for all t £ 0 ([58]).

In the latter setting, (10.7) can be obtained more simply than (10.6) just by combining the main theorem of [55] with the method of Kohn-Rossi ([58], pages 540-1).

In fact, (10.7)

holds for all t S 0 if M is the boundary of a smooth pseudoconvex complex manifold D I=(=D', another complex manifold, provided there exists a nonnegative function λ on D' which is strongly plurisubharmonic in a neighborhood of M. again combine [55] with [58], pages 540-1.

(For this,

The latter argu­

ment gives, in this setting, 3,u = 3,v + Θ, where θ is a weighted harmonic form, and where Θ,, its restriction to CTM, is plainly in the range of 3, . It is then easy to express Θ, = 3, w where w satisfies good estimates, simply by using the fact that any two norms on a finite-dimensional spaces are equivalent.)

In fact, in any of these settings, we do

have (10.5), as an argument of Hormander [56], coupled with

458

CHAPTER 10

the arguments already cited, show. Here, then, is the theorem we shall prove. Theorem 10.2. Let M be a smooth compact CR manifold of dimension 2n+l.

Suppose U c M is open, and is a real analytic,

strictly pseudoconvex CR manifold.

Suppose is a smooth

Hermitian metric on ITM, analytic on U, which is compatible with the CR structure. Let S be the Szego projection (onto ker 3, in L ) . Assume (partial global assumption): For all f ε C (M) there exists u ε (E') ' „ J b u = (I-S)f.

so that (10.8)

Then: OO

ι

OO

The map f -> Sf |^ is continuous from C (M) to C (U).

(10.9)

Fix an open subset I/ => U of M so that the map f -»· Sf L is co

co

„

9

continuous from C (M) to C (1/), and let V (S) = E' (V) + L (M) . Assume (full global assumption): For all f ε P(S) there exists u ε ( E ' ) 0 ' 1 so that (10.10)

-i?,u = (I-S)f. b

Then: (a)

S : O(S) -> E' is analytic pseudolocal on (J. That is,

if f ε O(S) is analytic on an open subset V of U, then Sf is

THE SZEGO" PROJECTION

459

analytic on V. (b)

If g

(i)

0bf = g is locally solvable near p (for f

(ii)

E:

~bw

V (S), p

U, the following are equivalent:

E:

g is locally solvable near p (forW

E:

E:

E'); (E,)O,I);

(iii) Sg is real analytic near p.

(c)

Let

V

be an open subset of U.

If f

E:

V(S) and

3bf, or

even just 0bf, is analytic on V, then (I-S) f is analytic on V. In (d), (e) we assume U sufficiently small that there is an analytic contact transformat ion W : U I P

E:

U, U I

C

nP. Let

U. S is locally in

(d)

->-

a-2n-2 near p, after W, up to an analytic

regularizing error. (e)

For certain open neighborhoods U

occ U

exists an operator K so that K : EI

cc:::

U of p, there

) ->- E I (U) and E I (U) ->O V' (U), so that K is locally in 0-2n(w- I (U)) after W, and so that whenever f

E:

E' (U

(U

o) , (10.11)

on

O' where gl' g2 are analytic on COO(U) , and K = K* on C~(UD). U

U

O.

Also, K

C:(U )->-

o

(f) Say p E: U. Then there exist g E: E' (M) so that jbw = g is not locally solvable near p. (This part is well-known.)

460

CHAPTER 10

Corollary 10.3. Let D (iii) is clear since Sg = S(g-,1 ω) is analytic near ρ by analytic pseudolocality of S.

For (iii) =>(i), write g = Sg + (I-S)g.

term is locally in the range of G, Kowaleski.

(near p) by Cauchy-

Let φ ε c°°, φ = 1 near p.

is locally in the range of D

b

The first

Then (I-S) [(l-cp)g]

(near p) by (a) and Cauchy-

Kowaleski, while (I-S) (cpg) is locally in the range of D, (near p) by (e) and Cauchy-Kowalewski, for supp φ sufficiently small.

This proves (iii) =>(i).

Thus we need only prove (10.9), (a), (d), (e) and (f) of Theorem 10.2. We shall examine several cases: (1)

η > 1, M strictly pseudoconvex;

(ii) η = 1; M = 3D, D a smooth strictly convex domain in C

2

which is analytic in a neighborhood of U;

(iii) the general case. In our discussions of case (i) and (ii), we shall seek to establish only some of the desired conclusions. All the conclusions will be established in our discussion of Case (iii). Case (i). η > 1, M strictly pseudoconvex.

462

CHAPTER

10

In this case, we establish (10.9), (a), (d) and most of (e) . All of these conclusions follow easily from Kohn's formula (10.12)

where

is as in

verified on

of Chapter 9.

This is immediately

(Indeed, say Then

and

Applying

to both sides,

we see

Since we must have Thus and

follow at once.

proving (10.9); we extend still holds on

and (d) now

Further, it follows at once frcm (10.12) that (10.13)

with Since

(10.14) is locally in

near p , after

analytic regularizing error, we have not yet shown that K , in

up to an that we have

, may be chosen to be self-adjoint

on D a smooth strictly convex domain in

which is analytic in a neighborhood of

THE SZEGO" PROJECTION

463

In this case, we carry through the analysis just far enough to lay the groundwork for the general case. In this case, there is no "Hodge theory" for O~, and the approach through (10.12) is not available.

In fact, if

we merely assumed n = 1, M strictly pseudoconvex,we could not expect an analogue of (10.13) with B (everywhere) locally in the COO analogue of 0-2n after a contact transformation. For this would imply that the range of jb on L~,l(M) is closed.

known [57], [11], if this hOlds then M is embeddable in

As is

~ for some N, and this is not true in general [69]. We base our analysis instead on the work of Henkin [43]. We may write D = {p < O}, p smooth in a neighborhood of D and analytic in a neighborhood of p in ~2 p

is analytic on U.

L.\'

I >= ci W 12 awiaaP rW'W. Wj S 1 J

for all

~ € M,

p) (~)

for any

~ €

We may assume

Further, for some C > 0,

2

and (grad

We let a

=f. 0

w

€

(10.15)

T~ (M)

M.

= ita-alp, the natural contact form on M. We

may assume U is small enough that there is an analytic (0,1) form w on U, with = 1 at each point, and that there is an analytic contact transformation 1jJ:U' the basis 9).

{w}

+

U, U'

c:

E1.

We use

in forming pullbacks toE l (as in #8 of Chapter

We shall need the following Proposition.

Proposition 10.4.

There exist operators R,H so that

464

CHAPTER 10

continuously.

(iv) Ra b (v)

= I-H

on Coo(M) •

H:L 2 (M) ~ L2 (M) is a projection onto ker abo

(vi) R, acting on AO,I, is locally in 0-3 near p, after H, acting on Coo, is locally in 0- 4 near p, after Proof.

W.

W.

The main point is the verification of (vi).

(i)-(v)

were either proved by Henkin or follow from the methods of

[20).

At the end of the proof we shall sketch arguments for

(i)-(iii).

Note that (v) is an immediate consequence of (i)-

(iv) . We need the explicit formula for R. in an open neighborhood of

Di

P, again, is smooth

for s,z in this neighborhood,

Henkin sets 2

LPkW (sk-zk)'

k=1

where Pkm Then if f (Rf) (z) where

€

=

PW

~2-.(s) . dZ k

AO,I(M), Rf is the following function on M: (10.16)

THE

SZEGO

PROJECTION

465

(10.17) (This is equivalent to the formula right after equation (3.3) in [43].) We may assume

We first seek to show that there of 0 e 3H 1 with

exists a neighborhood exist

and there

as follows:

If (10.18)

then the sum converges absolutely for

and for

with supp (10.19) For (10.19), we may assume

For w

we examine

We examine the denominator in the expression for

obtained frcm (10.18).

show that, if

It will be enough to

then (10.20)

(10.21)

Here

is a positive definite quadratic form, and

other quadratic forms.

In general,

tion in a neighborhood of

are

denotes a smooth func-

of whose Taylor coefficients

of weighted degree less than m vanish; here, of course, will be analytic on

Once we know (10.20), (10.21), then

466

CHAPTER

10

since the expressions in square brackets are comparable to the denominator for

will have an expansion as

for w small.

Since the numerator arising

from (10.17) vanishes to first order as Jacobian arising from using is analytic near

and since the

to change variable in (10.16)

we shall have (10.19).

To see (10.20) and (10.21), for with

identify Then Re

and Re

(explaining why the

denominator in (10.17) cannot vanish, D being strictly convex). By Taylor's Theorem at 0 , (10.22) (10.23) Now

agrees with an element of small.

to second order in

Fran (10.15), (10.22) we now see small, for some

Thus,

for w small.

This at

once gives us the real part of (10.21). Also, by (10.22) and (10.23), Re as

vanishes to third order in C

so the claim for the real part in (10.20) follows

from that of (10.21) . As for the imaginary parts, it will suffice to show that they are of the form

plus terms vanishing to second

THE

order at

SZEGU

PROJECTION

467

It suffices to consider

since

vanishes to second order as

Define

(equivalently, Let

Then Im

Put

making a unitary change of coordinates in may assume

By

if necessary, we

then

Since

by the implicit function theorem we may use

as coordinates on M near

also use

By using

as coordinates on M near

we may

Now, writing

we have by Taylor's Theorem

plus terms vanishing to second order at 0. We need to show the change of coordinates

But - under we have at 0 that

At

Also

that the contact form

there.

so

Of course

Applying these relations to (10.24), and using the fact that is a contact transformation, we find the relations we need for the derivatives of

at 0 .

(10.20) and (10.21) follow,

hence (10.19) follows, r in (10.18) clearly depends only on

CHAPTER 10

468

the radii of convergence of I/J near 0

€

TIl

1

and of p near 0

and Ub need only be small enough that /wl We can now prove (vi) for R. -1

map I/J

If n


0, (10.31)

fIL(v,w) Idw < C, fIL(v,w) Idv < C. This implies R : L2 (V) We return to H.

+

L2 (V), which implies (ii) for R.

Using (10.30), and (10.29) for N

= 1,

and an analysis similar to that for R, we see that modulo operators with kerneffisatisfying conditions like (10.31), we have

Hf(~(v))

= q(v)f~*TPvfor

v near 0 and f supported near O.

It is now easy to use the Cotlar-Stein lemma, and to make simple changes in the proof of L2 boundedness of convolution with principal value distributions in [66], pages 18-20, to show (iii) for H.

This completes the proof of Proposition 10.4.

On a strictly convex domain, one could now get directly at S, in order to prove theorem 10.2, by using this observation of Kerzman and Stein [52]: since S and H project onto the same subspace, we have SH

= H,

HS

= S=7

SH*

= S,

so S(I+H-H*)

= H,

so S

= H(I+H-H*)-l.

(10.32)

I + H - H* is invertible on L2 because H - H* is skew-adjoint. Further, one may use Theorem 8.1 to show (I+H-H*)-l is locally

THE

in

near

SZEGO

after

PROJECTION

473

up to an analytic regularizing error.

One may then show Theorem Further,

and (d), by using since

this

is partway toward Theorem Rather than carry out this plan at this point, we will find it a little more convenient to work on the formal level, and to wait to carry out the construction until we deal with the general case of Theorem 10.2. We will need the analogue of Proposition 10.4 for We need another formula of Henkin [43]. We can find an ambient metric

in a neighborhood of M in

with the complex structure, so that whenever

compatible are co-

tangent vectors at a point We may assume We identify to

is analytic in a neighborhood of

in

with the orthoccmp1ement of dr with respect

at each point. By (10.15), for sane

strictly convex for all

the domain

is also

Then [43] (10.33)

where if f

we define (10.34)

where (10.35)

474

CHAPTER 10

We shall verify momentarily that Gex tf extends smoothly up to M.

If g denotes the boundary value, we let G1f

= Pg,

where P is the orthogonal projection perpendicular to dr at each point, in the ambient metric in ~*~2. The smoothness of Gextf up to M is verified in much the same manner as forH. Let V = 'Vp(hv(O», 2b = II'Vp(O)II. Identify h I;;

r.

~2 with E;, ElR4.

Observe (hv(O)

= - ~(Vo+iJVo). (E;,-hv(O» 2

+0 (h ) .

,1;;)

= - ~(Vh+iJVh)· (E;,-hv(O» 2

+ O(h)O(I;;) + O(h )

=

(O,I;;)+bh+O(h)O(I;;)

Using this with (10.21) we find analogues of (10.27),

(10.28) for (hv(O) ,f(w)).

The error estimates are to be

slightly modified; replace each occurence of 03f(f~1) in (10.29), (10.30) by (03+o30 (h)+O(h2 »i.

One still sees G tf extends ex

smoothly up to M, and that G : L~,l ~ L~,l 1 If n is the restriction to M of a form which is in A2 ,0 on a neighborhood of M in ~2, then fM3bn is extended smoothly to

= O. Indeed, if n

0, fM\n = fDdan = f Dd 2 n = O. Thus

if F(I;;) is holomorphic on a neighborhood of then f~(I;;)abgAdl;;lAdI;;2

D,

= fMab(Fgdl;;lAdC2) = O.

(10.35) we now see (Gextdbg)(z)

=

and g E COO(M) , From (10.34),

0 for ZED \.D, so by EO

continuity, (10.36) As in Proposition 10.4, we see G preserves AO,l(M), (E,)O,l(M)

1 and L~,l(M); that (10.33) and (10.36) hold on (E,)O,l(M); that

I

- G1 is a projection onto ran db

c:

0,1 (M), and that G L2 1

THE SZEGO PROJECTION

is locally in 0

475

near p, after ψ.

Using JjR* = I - G*, we could now show an analogue of Theorem 10.2(a) and (d) for j , , and the projection onto ker j , , in place of 3, and S, by a plan analogous to that we have sketched for S. Indeed, however, we immediately turn to the general situation. Case (iii).

The general case.

We now prove (10.9), (a), (d), (e) and (f) of Theorem 10.2, in the general case. We need the following well-known proposition. Proposition 10.5. Let U n e (C real analytic.

be open, and r : U. + K b e

Suppose M. = {ζ ε D. : r(z) = θ}, that dr φ 0

on M Q , that Ω η = U Q η {ζ : r(z) < θ}, and that M n , viewed locally as the boundary of Ω~, is strictly pseudoconvex at each point.

Say ρ ε M„.

Then there exist a biholomorphic map

Φ : U -»· U' for some open neighborhood U of ρ in I , U = U n , and a smooth, bounded, strictly convex domain D' so that Φ (MpOU) = 3D' 0 U", Φ(Ω OU) = D' 0 υ'.

Proof.

This is a simple version of Narasimhan's result [67].

To see it, we may assume ρ = 0.

Note that by the usual E. E.

Levi construction (see e.g. [20] pages 500-502), after a holomorphic change of coordinates, Ω

takes the form

476

CHAPTER

10

2

near

to at least third order at

and

where f vanishes We

may assume the summation here defines a strictly positive definite form on of replacing

(otherwise we may use the usual device

by

large). After the Cayley trans-

form

takes the

form where g vanishes to at least third order at p, and if

Choose

with

near 0.

real-valued,

Then for sufficiently small

we may

choose (Here It is now easy to see that (f) will follow once we have shown (10.9), (a), (d) and (e).

Indeed, as we have seen, these

statements imply the equivalence of (b)(ii) and (iii). for (f), we may assume that

Further,

as in Proposition 10.5.

Indeed, since M is real analytic near p, we may find [1] a real analytic CR embedding

of a neighborhood

of p into

and in fact, by Proposition 10.5 we may assume By choosing a suitable metric on M', we may clearly then assume Using the methods of [19], construct, for any near p , F holcmorphic in DQ. then

and we cannot solve

If

locally near p .

THE

SZEGO

PROJECTION

477

To get at S in the general case, we need to use an operator S' which is a "local analogue" of it, in the sense of the following lemma.

Let

be as after (iii) of

of

Chapter 9. Lemma 10.6.

In the situation of Theorem 10.2, say

and

is sufficiently small that there is an analytic contact transformation On seme open neighborhood

there exist operators

with so that: are in respectively, after

and

f is analytic on

for all f (ii) (iii) (iv) Proof.

is analytic on Say first

for any

Again we find a real analytic CR

embedding

a neighborhood of

Proposition 10.5.)

as in

We arrange that the metric on CCTM1 agrees

with that pulled over by Using M ' ,

on a neighborhood and our discussion in Case (i), we

produce K as in (e) (though it might not be self-adjoint on

478

CHAPTER

10

of an open set). We restrict

and the Szego projection on

M 1 to a suitably small neighborhood of

and use

them back to M , to produce the needed

When

to pull

In this case.

we use the same method of embedding and invok-

ing Proposition 10.5, to locally embed into a suitable It follows then that p has an open neighborhood

and

there exist (new) operators

and

so that (I)

and so that R , H are in after

(respectively)

The problem is that H does not satisfy the self-

adjointness condition (iii). To produce S' we will use the Kerzman^Etein

method, but at the germ level.

It would be

equally effective to pull over the Szego projection from SD^; however we shall find it somewhat more convenient to work entirely with formal series. We naturally denote the pullbacks to respectively. respectively.

These are in

Say that

We abbreviate the formal sums so that respectively.

Then

THE

SZEGO

PROJECTION

and

479

mod

(10.37)

since these relations need only be proved in an open neighborhood of each point of U', and we can apply Theorem 7.11(c) and (i) and (iv) of

of Chapter 9. Also, by associativity, and

(ii) and (v) of

of Chapter 9, mod FP, on U * .

(10.38)

(By (8.4), we actually have

such "improvements"

are irrelevant for our arguments.) We wish to use * to refer to adjoints taken with respect to the inner product on

We must therefore change our

previous notation.

to denote that mapping on

We use

operators and formal sums on

which we previously denoted

for example in Theorem Say, then,

and (vi) of where

neighborhood of p , and say K is in pullback

of Chapter 9. is an open

after

with

Then it is easy to see that there is an operator so that

for

and where we use the

inner product.

In

fact, it is easy to see that for f where

where

denotes multiplication by

a certain positive-valued analytic function on

with core

If now

define

Here

CHAPTER 10

480

L(Ki)~ = (¢-1(u)6(w» K2

# I(K~dj)~ # (¢;u)6(w».

If also

k2 I'-.ffi I'_.ffi I'_.ffi ~ m I' m C (V'), and LK2 # LKl = LK , note . (K*) = L(Ki) #

€

I(K )m, by associativity and (vi) of #1 of Chapter 9.

2

By the product formula Theorem 7.11(c), the principal core of h* is the same as the principal core of h formula Theorem 7.11(a), this is~. u

core of h.)

adj

By the adjoint O

(Recall H is the principal u

If a = h - h*, its principal core AO satisfies u

i..o

= _Ao . Thus AO(A) is skew-adjoint for all A # O. We may u u u therefore apply Theorem 8.1 and (viii) of #1 of Chapter 9 to

6 + a and 6 + a*, since the representation-theoretic criteria are met. Thus we find k l ,k 2 E FC- 4 (Ui) so that

Thus (6+a*)#ki = 6,

2

(6+a)#k =6 mod FP on

By the associative law applied to k # (6+a) #k l of Chapter 9,

ui.

2, and

(vii) of #1

(10.39)

mod FP on Ui.

Thus in fact k l #(6+a) = (6+a)#k l mod FP on Ui.

= 6,

We use the symbol (6+a)

k #(6+a*) 2 -1

= (6+a*)#k 2 = 6

, or (6+h-h*)

-1

, for

k , and (6+a*)-1 or (6+h*-h)-1 for k . l 2 Now

put S

I

h# ( Ma) -1

(10.40)

THE SZEGO" PROJECTION

4 which is in FC- (Oi). (o+h*-h)#h

Note that, on 0i,

= h*#(o+h-h*) (0+a*)-1#h*

so s'

so by (10.39).

481

(s')*

h*#h

=

(10.41)

mod FP

(0+a*)-1#h*#h#(0+a)-1 mod FP, (10.42)

s'#s'

Thus s' is a "fonnal

orthogonal projection".

Further, by (10.41), s'#h

=

(0+a*)-1#h*#h

s'#h

= h,

=

(0+a*)-1#(0+a*)#h

h# s'

Thus also (o-s')#(o-h) r'

s'

mod FP, so on 0i,

(10.43)

mod FP.

(o-s') modFP, onoi.

Put

(o-s')#r

Then by (10.37) and (10.40), r'#d

= 0-

s',

d#s'

o

mod FP,

By (x) of #1 of Chapter 9, we may select R'

S'

E

(10.44)

on 0i. E

0 -3 (0'),

0-4 (0'), with support radii as small as we please, so

00

if one of f 1 ,f2 is in E' (Oi) and the other is in Cc(Oi)' if the pairing ( , ) is the one induced fram the inner product on L2 (M). We may clearly assume that the support radii of the R',S' obtained through this construction are as small as we please,

482

CHAPTER

and different.

10

Thus we can pull

over to M , and Lemma

10.5(i), (ii), (iii) follow at once from Theorem 7.11(c) and (iii) of

of Chapter 9.

Lemma

(iii) and (ii), since if of

follows at once from

is sufficiently small, the kernel

is analytic on

This proves the Lemma.

With this preparation, we can now begin the proof in the general case. We retain, from the proof of the lemma, the notation R 1 , S ' for the pullbacks of In the notation of

say.

of Chapter

we may clearly find

as in (9.2), and open neighborhoods of say

so that if

then

(I)

(II) (m)

and

is analytic on an open set

then on (Choose

is analytic. near p 1 if

first so that

on W'; then choose V" small enough. Pull

over to

Use Theorem 7.11(b).)

call the pulled over operators and let

We will always be restricting

Then we can

actually define natural extensions of these operators, also denoted

simply by setting

THE SZEGO" PROJECTION

483

00

RNf = RN((jJf) , Si/ = SN((jJf) where (jJ

Cc(UO)' (jJ = 1 on W.

E:

By

property (II), this condition is independent of the choice of cutoff (jJ. Finally, we may tum to S. SNSf (on V, of course).

For f

d'" (M),

E:

Let I;; r: C~(V), I;;

=

1 on VI' an open

neighborhood of p, and otherwise arbitrary. global assumption (10.8), write Sf

=f

we may examine

-~bu,

Using the partial u

E:

E' (M); then,

on V,

where gl is analytic on V, by Lermna 10.6(iv).

At the same time,

on VI'

Sf + [(S'-I) + R'dbJ (I;;Sf) + [SN+RNdb][(l-I;;)SfJ

=

Sf + g2 + [SN + RNabJ [(l-~)SfJ

where g2 is analytic on VI' by Lemma 10.6(ii).

Equating these,

we see that on VI' Sf -

s'

(I;;f) - g

= S~

where g is analytic on VI' h W ;=

0 on V .. l

... .n

E:

(10.45)

+ RNw

E' (M), h

=

0 on VI' and

W E:

E' (M),

Let u be the left side of (10.45). Pulling back ,1,-1

tOltl , we see that on Vi = ~

(VI) ,

CHAPTER 10

484

where ~ E c~(UO)' ~

=1

on W.

Since (~h)1jJ' (~)1jJ

=0

near p' ,

p' has a neighborhood V so that for some L GO, we have for

2

2,

all N: IOl ••. OMu1jJ(v) I < CRMNM+L for M ~ N - L, v E V 0l,··.,OM E {T,Xl, ••• ,xn,Yl, ••• ,Yn }.

2.

9.3(b), u1jJ is analytic on V V2

= ,[,(V') 'l' 2

But then, by Proposition

Thus Sf -

s' (sf) is analytic on

for f E Coo(M).

In particular, since p is arbitrary in U, S : Coo(M) ~ Coo(U).

This map is continuous by the closed graph theorem for

Frechet spaces, since S : Coo(M) ~ L2 (U) is continuous and the 2

00

L (U) topology on C (U) is weaker than the usual topology. This proves (10.9). If we now repeat the above arguments for f E V (S), using the full global assumption (10.10), we see that Sf analytic on V2 for any S E C~(V), if s

s' (sf) is

= 1 on VI' In particular,

if f is analytic near p, so is sf, and hence (by (III) above) so is Sf.

This proves Theorem 10.2(a).

Theorem 10.2(d) also

follows; in fact: for all fEE' (VIIlV2), Sf """ S' f is .analytiC on VI n V2 • (10.46) That leaves only (e).

If n > 1, we may proceed as at the

beginning of the proof of Lerrma 10.6, and pull back the "K" for M' to M,

via~.

OgKf = (I-S')f + gl'

In this way we produce a new K with

KU~f =

(I-S')f + g2 on an open set

THE SZBSO PROTECTION

if f

and where

48 5

are analytic on

then obtain (10.11) on

if

We still must show that, when

can be shrunk and

K modified so that

We may assume that there

is an orthonormal basis that all

We

for (0,1) forms on

, such

are analytic, and such that the pullback of has been defined to be the vector As in the case n = 1, we select for

so that

Then the pullbacks

are related b

y

operator on h " .

With

w

h

e

r

e

and

adj is the adjoint

as just constructed, say

Let

Then it is easy to see from

(10.14), (9.21), and the analysis at the end of that

mod FP on

small support radius,

But then, if

of Chapter 9, has sufficiently

is analytic regularizing in a

neighborhood of p , by (iii) of

of Chapter 9.

Therefore,

is also analytic regularizing in a neighborhood of p.

Thus we may replace

Chapter 9. If

as in

This completes the proof of the theorem if with U' as in (10.37), we invoke (10.33) and

(10.36) to see that and for sane

of

mod FP,

(10.47)

The proof of (10.47) is entirely

analogous to that of (10.37).

486

CHAPTER

10

We may assume that there is an analytic

on

which is a unit vector at each point of U , and we may assume that the pullback of to be

has been defined

Then, again, the pullback of

simply

namely

is

Then (10.48)

Fran (10.47) we see (10.49) These relations are entirely analogous to (10.37), with in place of d,r,h.

The same analysis that led to (10.42),

(10.44) then shows we may find so that, mod FP, on

we have (10.50)

a

n

d

(

1

0

.

so also

5

1

) (10.52)

We finally see that, mod FP, on

(10.53) where (10.52), (10.51), (10.44).

We used, in order, (10.44), Note

THE SZBGO PROJECTION

48 7

k#(6-s') = k.

(10.54)

In (10.53), we take * to see Uk*

= δ - s' also; thus

k = k#(6-s') = k#-£#k* = (6-s')#k* = k*

(10.55)

by (10.54); thus