Lectures on Pseudo-Differential Operators: Regularity Theorems and Applications to Non-Elliptic Problems. (MN-24) 9781400870486

Table of contents :
Cover
Table of Contents
Preface
Introduction
I. Homogeneous Distributions
II. Basic Estimates for Pseudo Differential Operators
III. Further Regularity Theorems and Composition of Operators
IV. Applications
Appendix
References

Citation preview

LECTURES ON PSEUDO-DIFFERENTIAL OPERATORS: Regularity theorems and applications to non-elliptic problems

by ALEXANDER NAGEL and Ε. M. STEIN

Princeton University Press and University

of Tokyo Press

Princeton, New Jersey 1979

Copyright Q 19 79 by Princeton University Press All Rights Reserved

Published in Japan Exclusively by University of Tokyo Press in other parts of the world by PrincetonJUniversity Press

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

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

Table of Contents

Preface Introduction Chapter I §1 §2 §3

Further Regularity Theorems and Composition of Operators

Sobolev and Lipschitz spaces Non-isotopic Sobolev and Lipschitz spaces Composition of operators A Fourier integral operator; change of variables

Chapter IV §12 §13 §14 §15 §16

Basic Estimates for Pseudo Differential Operators

Examples of symbols The distance function p(x, ξ) LP estimates (p ^ 2) L estimates

Chapter III

§8 §9 §10 §11

Homogeneous Distributions

Homogeneity and dilations in IR n Homogeneous groups Homogeneous distributions on the Heisenberg group

Chapter II §4 §5 §6 §7

1

Applications

Normal coordinates for pseudoconvex domains Qi 3 and the Cauchy-Szego integral Operators of Hormander and Grushin The oblique derivative problem Second-order operators of Kannai-type

7 7 14 21 31 31 35 46 55

76 76 82 89 98 104 104 109 115 122 138

Appendix

144

References

156

Preface

The theory of pseudo-differential o p e r a t o r s (which originated a s singular integral operators) was largely influenced by i t s appli­ cation to function theory in one complex variable and regularity p r o p e r t i e s of solutions of elliptic partial differential equations. It is our goal h e r e to give an exposition of some new c l a s s e s of pseudo-differential o p e r a t o r s relevant to s e v e r a l complex variables and c e r t a i n non-elliptic problems. As such this monograph contains the details of the r e s u l t s we announced e a r l i e r in [34], together with some background m a t e r i a l . Wnat is presented below was the subject of a course given by the second author at Princeton University during the Spring t e r m of 1978. We a r e very happy to acknowledge the a s s i s t a n c e given us by David J e r i s o n . He prepared a draft of the lecture notes of the c o u r s e and made several valuable suggestions which a r e incorpo­ rated in the text. We should also like to thank Miss Florence Armstrong for h e r excellent job of typing of the manuscript.

1 INTRODUCTION

T h e o b j e c t of t h i s m o n o g r a p h i s to p r e s e n t t h e t h e o r y of c e r t a i n n e w c l a s s e s of p s e u d o - d i f f e r e n t i a l o p e r a t o r s .

T h e s e c l a s s e s of o p e r a t o r s

i n t e n d e d t o s a t i s f y two g e n e r a l r e q u i r e m e n t s . b e r e s t r i c t i v e e n o u g h t o b e b o u n d e d in spaces.

are

On t h e o n e h a n d , t h e y s h o u l d

L i p s c h i t z s p a c e s , and S o b o l e v

On t h e o t h e r h a n d , t h e y s h o u l d b e l a r g e e n o u g h to a l l o w f o r a

d e s c r i p t i o n of t h e p a r a m e t r i c i e s of s o m e i n t e r e s t i n g n o n - e l l i p t i c d i f f e r e n t i a l and p s e u d o - d i f f e r e n t i a l o p e r a t o r s and o t h e r o p e r a t o r s s u c h a s t h e C a u c h y Szego and H e n k i n - R a m i r e z i n t e g r a l s f o r s t r i c t l y p s e u d o - c o n v e x d o m a i n s . B e f o r e d i s c u s s i n g t h e s e n e w c l a s s e s h o w e v e r , we w i l l b r i e f l y r e c a l l s o m e b a s i c d e f i n i t i o n s , and o u t l i n e t h e s i t u a t i o n in t h e c l a s s i c a l

"elliptic"

case. A p s e u d o - d i f f e r e n t i a l o p e r a t o r , defined initially on the Schwartz class

has the f o r m :

(1)

where

i s t h e F o u r i e r t r a n s f o r m , and t h e s y m b o l

i s s m o o t h , and h a s a t m o s t p o l y n o m i a l g r o w t h in i m p o s e s additional d i f f e r e n t i a l inequalities on this s y m b o l . t h e " c l a s s i c a l e l l i p t i c s y m b o l s " of o r d e r

m

One also For example,

a r e defined by:

(2)

One i n t e r e s t in t h e s e c l a s s e s a r i s e s f r o m t h e f a c t t h a t a p a r a m e t r i x f o r a n

- Z -

an elliptic differential operator of order differential operator

m can be written as a pseudo-

with symbol in

.

The appearance of the Fourier transform in the definition (1) of a pseudo-differential operator often makes this the appropriate form for proving of

2

L

estimates.

x, the operator

the boundedness of

For example, if the symbol

a(x,D)

is then just a Fourier multiplier operator, and

2 η a ( x , D ) f r o m L (]R ) t o i t s e l f i s e q u i v a l e n t t o t h e

uniform boundedness of the symbol a(£). to prove that if from

a(x, ξ) is independent

a(x, ξ) e

0

^ then a(x, D)

More generally, it is fairly easy extends to a bounded operator

l3(IRn) to itself. However, if one wants to prove that pseudo-differential operators

are bounded on

Lp ( ρ Φ 2 ) o r o n L i p s c h i t z s p a c e s , o n e n e e d s t o r e p r e s e n t

these operators in another way as singular integral operators.

These are

operators of the form (3)

f—> Kf(x) =

C

K(x, x-z) f (z) dz .

IR Here the kernel K(x,y)

will in general be singular when

integral in (3) must be taken in a principal value sense.

y = 0, so the By explicitly

writing out the Fourier transform in (1), and formally interchanging the order of integration, one sees that the kernel K(x,y)

of the operator is

related to the symbol a(x, ξ) by the formula: (4)

K(x,y) = (2ir)"n C e l ( y ' J ,n IR If

a(x, ξ) ε

Q

?)

a(x, ξ) d4 .

i s a classical elliptic symbol, one can use (4) to

-3-

o b t a i n e s t i m a t e s o n t h e a s s o c i a t e d k e r n e l , of t h e f o r m

(5) at l e a s t when

( F o r f u t u r e r e f e r e n c e , n o t e t h a t (5) i s

e q u i v a l e n t to (5') where

is the b a l l c e n t e r e d at

Once one has e s t i m a t e s

x with radius

, one can apply the c l a s s i c a l

C a l d e r o n - Z y g m u n d t h e o r y to o b t a i n

estimates

singular integral o p e r a t o r with k e r n e l

for the

F o r e x a m p l e , it f o l l o w s

e a s i l y f r o m (5) o r (5') t h a t o n e h a s t h e w e l l - k n o w n c o n d i t i o n : (6)

for a kernel

K(x,y)

c o r r e s p o n d i n g to a s y m b o l

It i s t h i s

a p p r o a c h t h a t w e t r y to i m i t a t e in s o m e n o n - e l l i p t i c In s t u d y i n g a v a r i e t y of n o n - e l l i p t i c p r o b l e m s ,

situations. several more general

c l a s s e s of p s e u d o - d i f f e r e n t i a l o p e r a t o r s h a v e b e e n d e v e l o p e d by H o r m a n d e r ,

C a l d e r o n and V a i l l a n c o u r t ,

B e a l s a n d F e f f e r m a n , and B e a l s .

- for example

Boutet de Monvel, Sjostrand,

(See [ 1] a n d [25] f o r r e f e r e n c e s . ) In

t h e s e c l a s s e s , o n e a g a i n i m p o s e s c e r t a i n c o n d i t i o n s o n t h e s i z e of t h e d e r i v a t i v e s of a s y m b o l , b u t t h e c o n d i t i o n s a r e d i f f e r e n t f r o m t h o s e i n (2). F o r e x a m p l e , o n e c a n a l l o w a c e r t a i n l o s s in t i v e , a n d a s i m i l a r g a i n in

with e v e r y

with every derivative.

x

deriva-

In a l l of t h e s e

c a s e s , i t i s p r o v e d , a m o n g o t h e r t h i n g s , t h a t s y m b o l s of o r d e r z e r o g i v e rise to operators which are bounded on

2

L .

(The proofs, however, are

considerably more delicate than in the classical elliptic case. ) However, boundedness in

I? (p^2) or in Lipschitz spaces is in general false, and

one does not obtain appropriate estimates for the associated singular integral operators. O n t h e o t h e r h a n d , F o l l a n d a n d S t e i n [16], a n d R o t h s c h i l d a n d S t e i n [39] have shown that parametricies for certain hypoelliptic differential

operators can be approximated, in an appropriate sense, by singular integral convolution operators on nilpotent Lie groups.

Since a version

of the Calderon-Zygmund theory is available in that context, they are able ρ to prove sharp L and Lipschitz estimates for these parametricies. However, these operators were not realized as pseudo-differential operators. W e c a n n o w e n u n c i a t e a b a s i c g u i d i n g p r i n c i p l e of o u r w o r k :

To treat

only those classes of symbols for which one can prove that the correspond­ ing operators have singular integral realizations with kernels having properties analogous to (5), (5'), and (6).

In this sense our approach to

pseudo-differential operators is essentially different from the generaliza­ tions which have been studied in the last dozen years.

What the more

g e n e r a l f o r m s o f ( 2 ) , ( 5 ' ) , a n d (6) m i g h t b e i s n o t a s i m p l e m a t t e r , b u t i t is in part motivated by the theory of singular integral operators on nilpotent groups, and the background for this is presented in Chapter 1. Our theory then proceeds along the following lines: _}_·

A ρ function is introduced in the

( χ , ξ) s p a c e w h i c h r e f l e c t s t h e

geometry of each particular situation and in t e r m s of which we will control the size of symbols and their derivatives.

This ρ function leads by

duality to a basic family of " b a l l s " in the χ space.

It i s the pseudo-

distance defined by these balls, and their volume in t e r m s of which we estimate the kernels of our operators (when realized as (3)). 2.

In this setup one can apply the Calderon-Zygmund theory (via a variant

ρ of (6)) to prove L estimates for our operators, assuming they a r e bounded 2

in L .

But here we must emphasize an important point.

work with a preliminary symbol c l a s s

S m (defined v e r y roughly by the P

requirement that the analogue of (5'') holds). 2

broad to allow L to

,). 1,1

estimates

At this stage we

Butthis c l a s s i s too

(in fact in the c l a s s i c a l c a s e it corresponds

So we must refine the c l a s s

S m ; the resulting c l a s s , P

cannot be defined in t e r m s of simple differential inequalities.

Sm , P

The actual

motivation for the definition we give i s in t e r m s of the explicit examples presented in Chapter 1.

The c l a s s

also has the further property that

it allows Lipschitz space and Sobolev space estimates - and this is carried out in Chapter 3.

There a r e two types of estimates of this kind, isotropic

and non-isotropic ones. 3. ~

We then show that operators whose symbols belong to S m a r i s e in P

various applications such a s : (i)

The Cauchy-Szego integral and Henkin-Ramirez integral for

strictly pseudo-convex domains. (ii)

The parametricies for

on boundaries of strictly pseudo-

convex domains, in the sub-elliptic c a s e .

(iii)

The p a r a m e t r i c i e s of o p e r a t o r s of Hormander

in the " s t e p 2" c a s e .

X^+ L· X. , j =1 J

(The higher step c a s e needs a generalization of our

theory; in this connection, s e e the announcement [35]. ) (iv)

The "oblique derivative" problem.

of our symbol c l a s s e s i s

Here a further extension

needed, because in general the elliptic symbols

do not belong to S m , and the p a r a m e t r i c i e s a r e a mixture of S m symbols P P with elliptic symbols. (v)

The p a r a m e t r i c i e s for the second-order singular o p e r a t o r s of

the type f i r s t studied by Kannai, e . g . ,

-7-

C h a p t e r I.

Homogeneous distributions

C h a p t e r I m a y b e t h o u g h t of a s a r e v i e w of s o m e k n o w n f a c t s w h i c h a r e b a s i c in m o t i v a t i n g o u r t h e o r y .

P r o o f s a r e for the m o s t p a r t only

sketched. §1.

H o m o g e n e i t y and d i l a t i o n s in Denote

.

Fix positive exponents

define

and

Observe that

(1)

(2)

(3) (More g e n e r a l l y , one might let where

A

be g i v e n b y m u l t i p l i c a t i o n b y t h e m a t r i x

is a r e a l m a t r i x w h o s e e i g e n v a l u e s all h a v e p o s i t i v e

real part. ) T h e c h a n g e of v a r i a b l e f o r m u l a f o r d i l a t i o n s

where

(In g e n e r a l ,

a = trace

is g i v e n by

A. )

D e n o t e E u c l i d e a n n o r m by P r o p o s i t i o n 1. (a) (b)

There exists a n o r m function

satisfying

if and o n l y if i s e v e r y w h e r e c o n t i n u o u s and

(c) (d)

(This is j u s t a n o r m a l i z a t i o n . )

-8-

Proof.

Define

To o b t a i n s m o o t h n e s s , apply the

implicit function t h e o r e m .

T h e r e s t of t h e p r o p o s i t i o n i s o b v i o u s .

D e f i n e p o l a r c o o r d i n a t e s by

Remark.

where

since this holds when

and both s i d e s

t r a n s f o r m the s a m e way u n d e r the dilations P r o p o s i t i o n 2.

where

on the unit s p h e r e and

to i s a p o s i t i v e

dcr d e n o t e s t h e u s u a l m e a s u r e f u n c t i o n on the s p h e r e .

p r o o f , a s i m p l e c a l c u l a t i o n of t h e J a c o b i a n , i s l e f t t o t h e r e a d e r . F a b e s and R i v i e r e Corollary.

The (See

p.20 . )

There exists a constant

such that for all

measurable,

Remark.

The c o r o l l a r y i m p l i e s that

i s l o c a l l y i n t e g r a b l e iff

i s i n t e g r a b l e a t i n f i n i t y iff Let

We s a y t h a t Let

f i s h o m o g e n e o u s of d e g r e e

K be a d i s t r i b u t i o n .

a l w a y s m e a n t e m p e r e d d i s t r i b u t i o n . ) If

if

(By d i s t r i b u t i o n w e s h a l l

K were a function homogeneous

of d e g r e e

It

i s t h e r e f o r e n a t u r a l to c a l l a d i s t r i b u t i o n

K h o m o g e n e o u s of d e g r e e

for every test function

X if

-9A distribution

K is

in a n o p e n s e t

if t h e r e i s a f u n c t i o n

such that class

We w i l l c a l l a d i s t r i b u t i o n

if it i s h o m o g e n e o u s of d e g r e e

K of

and A

Theorem 1.

K i s a d i s t r i b u t i o n of c l a s s X. if and o n l y if

K is a d i s t r i b u -

t i o n of c l a s s Proof.

Because

that

i t ' s e a s y to s e e

i s h o m o g e n e o u s of d e g r e e Let

there.

denote the

Choose

f u n c t i o n on

such that

has compact support,

in a n e i g h b o r h o o d of

so i t s F o u r i e r t r a n s f o r m i s

Denote

K

0.

(even analytic).

e v e r y w h e r e and

For sufficiently large

M, u s i n g h o m o g e n e i t y , we s e e t h a t

q u i c k l y e n o u g h a t i n f i n i t y so t h a t i s c o n t i n u o u s , and

d e r i v a t i v e of Example 1.

decreases

Therefore,

is c o n t i n u o u s e x c e p t at the o r i g i n .

i s h o m o g e n e o u s (of d e g r e e

for large

Similarly,

x , so t h a t a n y

is continuous outside the o r i g i n . Suppose

Re X; t h e n

K defines a distribution.

a r i s e in t h i s w a y .

In f a c t , if

qed. function away f r o m 0

t h a t i s h o m o g e n e o u s of d e g r e e fore,

that a g r e e s with

(by P r o p o s i t i o n 2 ) .

There-

C o n v e r s e l y , a l l d i s t r i b u t i o n s of c l a s s

K is s u c h a d i s t r i b u t i o n , let

be as above

-10Then

i s a d i s t r i b u t i o n of c l a s s X s u p p o r t e d at t h e o r i g i n . i s a s u m of d e r i v a t i v e s of t h e d e l t a f u n c t i o n a t t h e o r i g i n .

e a s y to c h e c k t h a t

E x a m p l e 2. 0, of d e g r e e sphere.

has homogeneity

Suppose -a

Thus It i s Therefore

In o r d e r t h a t a f u n c t i o n

away f r o m

d e f i n e a d i s t r i b u t i o n , it m u s t h a v e m e a n v a l u e z e r o o n t h e

C o n v e r s e l y , e a c h d i s t r i b u t i o n of c l a s s

- a i s t h e s u m of s u c h a

f u n c t i o n and a c o n s t a n t m u l t i p l e of t h e d e l t a f u n c t i o n a t P r o p o s i t i o n 3.

More generally,

g e n e o u s of d e g r e e

0.

suppose

is h o m o -

Then:

(a)

There exists a distribution

(b)

K c a n b e c h o s e n t o b e of c l a s s \

f o r all multiindices

K that a g r e e s with

such that

tion is vacuous u n l e s s

(Notice that this condi-

l i e s at c e r t a i n e x c e p t i o n a l points on the n e g a t i v e

real axis (c)

The distribution

t i o n s of

Proof sketch. degree zero.

K of c l a s s

for those

Write

is unique up to l i n e a r c o m b i n a -

for which

i s h o m o g e n e o u s of

Fix

converges absolutely for

Re

It c a n b e c o n t i n u e d a n a l y t i c a l l y to b e

-11m e r o m o r p h i c in

It h a s a t m o s t s i m p l e p o l e s a t t h e p o i n t s

a n d t h e s e p o l e s v a n i s h u n d e r t h e c o m p a t i b i l i t y c o n d i t i o n s of b ) .

is e n t i r e .

T h e m a i n p a r t of

i s (by a n a l y t i c c o n t i n u a t i o n )

T h u s p o l e s a r i s e only when if w e i m p o s e t h e c o n d i t i o n ( s )

In f a c t ,

and in t h a t c a s e t h e p o l e v a n i s h e s w h i c h a r e e q u i v a l e n t to t h o s e stated

in b). We c a r r y o u t t h e a r g u m e n t i n m o r e d e t a i l f o r t h e c a s e P r o p o s i t i o n 3'. away f r o m

Proof.

Suppose

0.

e x t e n d s to a d i s t r i b u t i o n of c l a s s - a

Suppose

definition:

i s h o m o g e n e o u s of d e g r e e

has m e a n value zero.

We g i v e a n a l t e r n a t i v e

We w i l l d e f i n e

exists works because

if a n d o n l y if

Note that t h e r e

such that

. It f o l l o w s t h a t

(In f a c t ,

b = l/maxaj

-12c o n v e r g e s absolutely as

Thus

is w e l l d e f i n e d .

Its h o m o -

geneity is also obvious. Now s u p p o s e

d o e s not h a v e m e a n v a l u e z e r o .

T h e r e is a c o n s t a n t

such that

i s h o m o g e n e o u s of d e g r e e Since

Kj

-a

and h a s m e a n v a l u e z e r o on the s p h e r e .

d e f i n e s a d i s t r i b u t i o n of c l a s s - a w e a r e r e d u c e d to s h o w i n g

that (4)

d o e s not.

Denote

i s a d i s t r i b u t i o n and a g r e e s w i t h exists a distribution

K

of c l a s s

Then

t h a t o c c u r in t h e s u m .

for all

t > 0.

Assume there a w a y f r o m 0.

so t h a t

The f o r m u l a implies f o r all

By h o m o g e n e i t y , w e a l s o h a v e

0.

- a that a g r e e s with Choose

for all other

away f r o m

t > 0.

i for all

t > 0.

Therefore

B u t a c h a n g e of v a r i a b l e in (4) s h o w s t h a t

a contradiction. Remark.

Let

i m p l i c i t y t h a t if

b e h o m o g e n e o u s of d e g r e e

-a.

We h a v e s h o w n

h a s m e a n v a l u e z e r o w i t h r e s p e c t to o n e h o m o g e n e o u s

-13norm

, t h e n it h a s m e a n v a l u e z e r o w i t h r e s p e c t to any h o m o g e n e o u s

norm

( s a t i s f y i n g P r o p o s i t i o n 1 a ) , b ) , c) b u t n o t n e c e s s a r i l y t h e

n o r m a l i z a t i o n d) ).

In f a c t , w e c a n s e e t h i s d i r e c t l y by o b s e r v i n g t h a t

T h e l e f t - h a n d side is b o u n d e d , but the r i g h t - h a n d s i d e would g r o w like -logb

if

Examples. 1.

d i d n o t h a v e m e a n v a l u e z e r o w i t h r e s p e c t to

(Recall the convention

(See S t e i n [41], C h a p t e r I I I . ) L e t

when

t a k e n in t h e s e n s e of P r o p o s i t i o n

3 b). M o r e g e n e r a l l y , if of d e g r e e

is a h a r m o n i c polynomial on

homogeneous

k, then

* F o r t h e o t h e r v a l u e s of these identities can be suitably i n t e r p r e t e d if o n e t a k e s into a c c o u n t t h e p o l e s and z e r o e s of

-14-

2.

with

Thus

K is the f u n d a m e n t a l solution to the h e a t e q u a t i o n

A f u n d a m e n t a l fact connecting h o m o g e n e o u s d i s t r i b u t i o n s and o p e r a t o r s is the following: T h e o r e m 2. of d e g r e e

Suppose

- a ) and i s

- a (or m o r e g e n e r a l l y

away f r o m the origin.

Then

T h e P r o o f of T h e o r e m 2 i s i m m e d i a t e f r o m t h e f a c t t h a t

is

Tf = K *f

with Re

K i s h o m o g e n e o u s of d e g r e e

is a bounded o p e r a t o r on

h o m o g e n e o u s of d e g r e e 0 , a n d h e n c e b o u n d e d . Remark. We w i l l s e e l a t e r t h a t T e x t e n d s to a b o u n d e d o p e r a t o r o n

§2.

Homogeneous groups We s h a l l n o w s k e t c h t h e b a c k g r o u n d f o r a n i m p o r t a n t g e n e r a l i z a t i o n

of T h e o r e m 2. C o n s i d e r Lie group.

w i t h a g r o u p m u l t i p l i c a t i o n t h a t m a k e s it a

A s s u m e (for s i m p l i c i t y ) that

* S e e a l s o F a b e s a n d R i v i e r e [12],

are canonical

coordinates.

(This means that they are the coordinates given by the

exponential map.

In particular,

χ ^ = -x and a straight line through the

origin is a one-parameter subgroup. ) Suppose we have a one-parameter a

set of dilations 5 t (x) = (t t group.

I

a

x., . . .,t i

n

χ ) that are automorphisms of the n

We will call a group with dilations a homogeneous group.

This situation arises often.

If

G is a semi-simple Lie group then

G = KAN, where K i s a maximal compact subgroup, A

abelian, and N

nilpotent.

acts on N by

(This is the Iwasawa decomposition of G. ) A 2

dilations, and L (N) is of interest in representation theory.

For example,

let G = SU(n, 1), the biholomorphic self-mappings of the unit ball O

a n d t h e s y m b o l of t h i s i n v e r s e . Several words of caution need be stated.

F i r s t (2) h a s a n o n - t r i v i a l 2 ( i . e . , L?) n u l l s p a c e , n a m e l y t h e c o n s t a n t m u l t i p l e s o f e , and so only a right inverse can be found.

Moreover, we need the exact symbol

c o r r e s p o n d i n g t o t h i s i n v e r s e of ( 2 ) , b e c a u s e e v e n if w e m a k e v e r y g o o d approximations (for fixed X), these approximate symbols will not behave right under differentiation with respect to

X , and e s t i m a t e s o f t h i s k i n d

are crucial in what follows. W e s e e k t h e s y m b o l of t h e o p e r a t o r (3) and the range of

D K = I, A A

K

Λ

K

Λ

determined by

2, - X t /2 . is orthogonal to e

-124-

O n e w a y of o b t a i n i n g t h i s o p e r a t o r a n d i t s s y m b o l i s t o o b s e r v e t h a t ( s e e (14) of t h e p r e v i o u s

section).

T h e n if w e u s e (20) w e s e e t h a t t h e s y m b o l of

s y m b o l of

is e x a c t l y the

and so e q u a l s

(4) where

i s g i v e n b y (18) of However,

it i s p o s s i b l e t o w r i t e t h e s y m b o l of

in a m o r e

p a c t f o r m and to g i v e a m o r e e l e m e n t a r y d e r i v a t i o n f o r it.

This

d e r i v a t i o n a l s o a p p l i e s to t h e h i g h e r o r d e r o b l i q u e d e r i v a t i v e

com-

simpler

problem.

( T h e r e we need to i n v e r t the o p e r a t o r

The r e s u l t s we shall d e r i v e below for the c a s e h i g h e r v a l u e s of P r o p o s i t i o n 3.

k.

See [35], whe r e the c a s e

The operator

has

k = 1

also hold for t h e s e

k = 2 is d i s c u s s e d in d e t a i l . )

symbol

(5)

Also, (6)

Proof.

W e n o t e f i r s t t h a t if

Therefore,

then

and so

for some constant

* T h e f o r m u l a w e s t a t e d in [34] h a s a n e r r o r in it. correct form.

The above is the

-125-

Next,

is d e t e r m i n e d by the a s s u m p t i o n that

to

C a r r y i n g o u t t h e c o m p u t a t i o n of

u(t)

is o r t h o g o n a l

we have

Thus

(7)

F r o m t h i s t h e s y m m e t r y p r o p e r t y (6) i s o b v i o u s .

Next, replace the

i n n e r i n t e g r a l i n (7) b y u s i n g c o n t o u r i n t e g r a t i o n in t h e z-variable

around

t h e r e c t a n g l e (if x < t)

H e n c e (7) l e a d s t o t w o i n t e g r a l s w i t h t h e i n n e r i n t e g r a l s , t a k e n a l o n g t h e vertical ray

x + is,

and along the v e r t i c a l r a y

t+is,

So

(8)

T h e s e c o n d i n n e r i n t e g r a l i s i n d e p e n d e n t of out the

x

x , a n d s o if w e

carry

i n t e g r a t i o n w e g e t t h e s e c o n d t e r m of ( 5 ) .

T o d e a l w i t h t h e f i r s t t e r m of (8) r e q u i r e s o n l y a n e v a l u a t i o n of t h e

F o u r i e r t r a n s f o r m of

at

T h i s g i v e s t h e f i r s t t e r m of ( 5 ) ,

-126-

and the p r o p o s i t i o n is p r o v e d . We now define a We l e t

c l a s s a p p r o p r i a t e to t h e s y m b o l

be given by the q u a d r a t i c f o r m

g i v e n by (8).

(in t h e

v a r i a b l e s ) and s e t W e t h i n k of

as the d u a l v a r i a b l e to

of

l i m i t e d to

and with

(t,x), with everything

independent

P r o p o s i t i o n 4.

T o m a k e the n e c e s s a r y e s t i m a t e s we n e e d the following two

simple

lemmas.

Lemma 1.

L e m m a 2.

Jtf

If

m

is an integer

and

Re

is bounded and r a p i d l y d e c r e a s i n g on

then

then

w fsothal elnotrthw e, eF sTraohnbredy tfashionre sm ts e-ecfsot nli dm d ailnteetm e gin mr aLtIiw neom ent hm debiayvfiip1draesirtstcsw ot,reni svif uiidsaw el retashtui einsocenfea csi tni ntcthw eaot c a s e s t: h ei n stehccoaotnn-d

-127-

case.

In t h e s e c o n d c a s e w e h a v e

constant

and so w e

since

in t h a t c a s e .

get a s an e s t i m a t e

In p r o v i n g the e s t i m a t e s f o r region where

it s u f f i c e s t o r e s t r i c t a t t e n t i o n t o t h e

i n v i e w of t h e s y m m e t r y i n

i n t e g r a l r e p r e s e n t a t i o n (7) s h o w s t h a t

a

g i v e n by (6);

the

is s m o o t h a c r o s s

We p r o v e f i r s t t h a t

(8)

F o r t h e s e c o n d t e r m i n (5) w e u s e L e m m a 1, w i t h

m=l,

This gives as e s t i m a t e

and

which is

if w e c o n s i d e r t h e c a s e s w h e r e

and

F o r t h e f i r s t t e r m i n (5) w e o b s e r v e t h a t

and

t h e n u s e L e m m a 2, with T h e i n e q u a l i t i e s of t h e d e r i v a t i v e s of ( a s r e q u i r e d b y (15) i n s u m m a r i z e d as follows. derivative gains

(9)

(10)

w i t h r e s p e c t to

a n d t h e b e g i n n i n g of Each

of C h a p t e r 2) c a n b e

derivative gains Thus we want

and

and e a c h

-128-

T h e r e a d e r s h o u l d h a v e n o d i f f i c u l t y v e r i f y i n g (9) a n d (10) ( a n d t h e corresponding inequalities for higher

and

d e r i v a t i v e s ) in t h e

w a y w e p r o v e d (8).

T h e d i f f e r e n t i a t i o n c o n d i t i o n w i t h r e s p e c t to

(17) i n

that

where

requires

a

estimates,

satisfies e s t i m a t e s like

and

and

that

and

don't appear,

t

(see

satisfying

similar

and

and

but i m p r o v e d by a f a c t o r In f a c t t h e t e r m s

same

and the f a c t

c a n b e e a s i l y v e r i f i e d u s i n g L e m m a s 1 and 2.

This,

and the c o r r e s p o n d i n g r e s u l t s f o r h i g h e r d e r i v a t i v e s c o m p l e t e the proof that We a r e n o w in a p o s i t i o n to w r i t e d o w n t h e r i g h t p a r a m e t r i x f o r (Recall that

where

is a r e a l s y m m e t r i c p o s i t i v e d e f i n i t e m a t r i x v a r y i n g

s m o o t h l y in

x. ) T h e p a r a m e t r i x is

where

(11)

with The

c l a s s associated with

P

is defined as follows:

is the q u a d r a t i c f o r m

Now unfortunately b e c a u s e the

P

P

g i v e n b y (5) a n d ( 6 ) .

i s n o t of t h e c l a s s

i s t h e p r e s u m p t i v e i n v e r s e of

and

this is not

surprising, and this is

-1 29-

n o t of t h e c l a s s the class

since

i t s e l f i s n o t of

W h a t i s n e e d e d i s a n e x t e n s i o n of t h e c l a s s e s

w h i c h we

now define.

Definition.

b e l o n g s to the " e x t e n d e d c l a s s

every integer

N

if f o r

we can write

(12)

where

(a)

(b) (c)

b e l o n g s to

w h e n t e s t e d i n t e r m s of d e r i v a t i v e s

(in x and

of o r d e r

N.

Remark.

S i n c e a l l t h e e s t i m a t e s w e h a v e m a d e f o r o p e r a t o r s of o u r

s y m b o l c l a s s e s d e p e n d o n l y o n a f i n i t e n u m b e r of d e r i v a t i v e s , t h e r e g u l a r i t y p r o p e r t i e s w i l l h o l d if i n e a c h c a s e w e m a k e T h u s e . g . , the o p e r a t o r w ho se s y m b o l is if

N

P r o p o s i t i o n 5.

N

large

will m a p

required enough. to

i s s u f f i c i e n t l y l a r g e ( i n t e r m s of p , k a n d m ) .

If

then for each

N

we can w r i t e

(12')

where

and

t e s t e d i n t e r m s of d e r i v a t i v e s of o r d e r

b e l o n g s to N.

when

-1 30It s u f f i c e s t o p r o v e t h e p r o p o s i t i o n i n t h e c a s e with

N o w if

then clearly

the symbolic calculus holds for

and

The Kohn-Nirenberg

formula

( s e e H o r m a n d e r [24], p . 147) s t a t e s t h a t

remainder

w h e r e t h e r e m a i n d e r b e l o n g s to

O b s e r v e that each so we c a n i t e r a t e t h i s

process, M

u n t i l w e f i n a l l y o b t a i n (12 ), w i t h

l a r g e enough we then see that N

are

If w e c h o o s e a s f a r a s d e r i v a t i v e s of o r d e r

concerned.

O u r m a i n r e s u l t f o r the oblique d e r i v a t i v e is a s f o l l o w s : Theorem.

(a)

The symbol

(with

b y (5) a n d (6)) b e l o n g s t o

(b)

P

given

and

is a r i g h t - p a r a m e t r i x f o r

( w h e n r e s t r i c t e d t o a c o m p a c t s u b s e t of t h e x - s p a c e ) . T o p r o v e t h e t h e o r e m w e s h a l l n e e d a s e r i e s of

L e m m a 3.

Let and

be a

function on when

so t h a t

lemmas.

when

Then

T h i s r e s t r i c t i o n is n e c e s s a r y b e c a u s e we have always a s s u m e d all o u r s y m b o l s to have c o m p a c t x s u p p o r t .

-131-

Recall that

where

is s u c h t h a t

when

The condition on

and

when

T h e p r o o f of t h e l e m m a f o l l o w s e a s i l y f r o m t h e f a c t t h a t (see the Appendix).

L e m m a 4.

Suppose

T h e n on the set w h e r e

s a t i s f i e s a l l t h e d i f f e r e n t i a l c o n d i t i o n s of

Proof.

O b s e r v e that

Thus

for large

Moreover,

g i v e s a g a i n of

e a c h d e r i v a t i v e w i t h r e s p e c t to

since

p r o v e d t h a t if

then

So w e h a v e

on the s e t w h e r e

x d e r i v a t i v e s of a c a n b e h a n d l e d a s f o l l o w s .

where

Write

Then

and by w h a t we have just p r o v e d on the set w h e r e works for higher

L e m m a 5. of

Now the

and The same

argument

x-derivatives .

If

qed.

then

satisfies the

condition

on the set w h e r e

Recall that s y m m e t r i c positive definite.

where

is

real

-132Proof.

We s t a r t w i t h a s y m b o l

and

and s u b s t i t u t e

L e t u s f i r s t v e r i f y t h a t t h e r e s u l t i n g s y m b o l i s of c l a s s in the set w h e r e

Observe that

s at i s -

f i e s the c o r r e c t bound.

gives two t e r m s ,

Next c o n s i d e r the

derivative.

namely

Taking

and

But in b o t h c a s e s t h e d e c r e a s e is (at l e a s t )

t h a t condition is s a t i s f i e d .

If w e f o r m

so

then we get

a n d a g a i n t h e d e c r e a s e i s of t h e

o r d e r , this time the s a m e way,

Higher

derivatives are treated

s i n c e w e a r e d e a l i n g w i t h p r o d u c t s of s y m b o l s of t h e

kind with f a c t o r s like

so b e h a v e s l i k e a n

T h e l a t t e r i s of c o u r s e i n

s y m b o l in t h e set w h e r e

that the s a m e way.

right

in same and

and so we

D e r i v a t i v e s w i t h r e s p e c t to

x

are

see

treated

For example

So

where

where

and

We c a n w r i t e

Moreover

and

-1 33s a t i s f i e s the c o n d i t i o n (17) in w o r k s f o r (higher)

x

The s a m e argument

d e r i v a t i v e s and the l e m m a i s p r o v e d .

We n o w c o m e to the p r o o f of p a r t (a) of the t h e o r e m .

A c c o r d i n g to L e m m a s 3 and 5, However,

and so a l s o

is s u p p o r t e d in the set w h e r e since

where

We w r i t e

and so by l e m m a s ,

We next d e c o m p o s e

f u r t h e r , as

c o m e f r o m the t e r m

and

f r o m the t e r m

(on m a k i n g the s u b s t i t u t i o n We a s s e r t that

b e c a u s e d e r i v a t i v e s with r e s p e c t to

c o n t r i b u t e e i t h e r t e r m s c o n t r o l l e d by the c o r r e c t d e c r e a s e since t i v e s act o n

o r t e r m s w h i c h a r i s e when the

o r its d e r i v a t i v e s .

a gain of o r d e r

deriva-

F o r e a c h s u c h d e r i v a t i v e we get

and so f o r a d e r i v a t i v e of o r d e r

a r e led to e s t i m a t i n g

m

we

h o w e v e r , this is

clearly

s i n c e we a r e r e s t r i c t e d to the

set w h e r e hold f o r the

c a l c u l u s and lead to the

where x

Similar

arguments

derivatives.

F i n a l l y we c o m e to the t e r m

w h i c h i n v o l v e s c o n s i d e r a t i o n of the

-134integral

where

and

Re

We c l a i m the f o l l o w i n g a s y m p t o t i c e x p a n s i o n

(13)

with

real,

in the s e n s e that

as

(13) c a n be p r o v e d by w r i t i n g

where

f u n c t i o n w h i c h is = 1 n e a r z e r o , and v a n i s h e s o u t s i d e

a c o m p a c t set.

The second integral contributes z e r o

asymptotically

( s e e the p r o o f of L e m m a 1), w h i l e if we u s e the f a c t that

we get a f t e r i n t e g r a t i o n by p a r t s (13) with T h i s a l l o w s us to w r i t e

remainde

T h e s e r i e s can b e w r i t t e n as

Collecting t e r m s

we g e t l i n e a r c o m b i n a t i o n s of t e r m s of the f o r m

When we substitute

we o b s e r v e that we can w r i t e it in

-1 35the f o r m

where

a r e s y m b o l s in

Moreover

while

and

b e l o n g to

r e s t r i c t e d to the set w h e r e

In a d d i t i o n ,

w e have that as we have a l r e a d y pointed o u t .

A l t o g e t h e r then e x c e p t f o r the r e m a i n d e r t e r m , p r o d u c t s s y m b o l s of the c l a s s

is a finite s u m of

and

Now the r e m a i n d e r t e r m i n s o f a r as it and its d e r i v a t i v e s a r e c o n c e r n e d

b e h a v e s no w o r s e than

w h i c h b e h a v e s l i k e a s y m b o l of

class

m

when d e r i v a t i v e s of o r d e r

are c o n s i d e r e d ,

(with o f c o u r s e

T h i s c o m p l e t e s the p r o o f o f the f a c t that

(The r e a d e r should k e e p in m i n d that in v i e w of the s y m m e t r y p r o p e r t y (6) f o r

it s u f f i c e s to c a r r y out o u r c o m p u t a t i o n s f o r

P

when

as long as the d e c o m p o s i t i o n s we h a v e c a r r i e d out r e s p e c t s this s y m m e t r y . tion.

N o t i c e that this is the c a s e f o r

Now

and

a r e r e a l and the e x p o n e n t s

(and

by t h e i r d e f i n i -

has this s y m m e t r y , b e c a u s e c o e f f i c i e n t s are odd.

w e have

Since and this is e x t e n d e d

by the s y m m e t r y (6) to We shall c o m p u t e We have o b s e r v e d that

in o r d e r to p r o v e p a r t (b) of the t h e o r e m . and s i n c e

we have that

P

also

-1 36b e l o n g s to

T h u s we c a n apply the s y m b o l i c c a l c u l u s and o b t a i n

( s e e (1))

(14)

where

is a s u m of t e r m s i n v o l v i n g p r o d u c t s of the f o r m

(15) with p r e a s s i g n e d

m.

However,

is the s y m b o l of a r i g h t i n v e r s e of

thus

we have (16)

Substituting

g i v e s us

Now f r o m the f o r m u l a (5) it is e v i d e n t that where

and

is of the f o r m

h a v e s i m i l a r e x p r e s s i o n s to

Thus

but

and

by the s a m e a r g u m e n t u s e d to t r e a t u s i n g the fact that where

and

Moreover, and

|

we s e e that

By the

s a m e a r g u m e n t the t e r m s (15) a l s o b e l o n g to

.

s u f f i c i e n t l y l a r g e , then e v e r y s y m b o l in

b e l o n g s to

t e s t e d a g a i n s t d e r i v a t i v e s of o r d e r

F i n a l l y , if we take

T h i s s h o w s that

when

m

-1 37and the t h e o r e m is p r o v e d .

Concluding (1)-

remarks

We can o b t a i n a s i m i l a r l e f t - p a r a m e t r i x f o r

T h i s c a n b e d o n e by taking ad j o i n t s of the b a s i c r e l a t i o n and noting that all the s y m b o l c l a s s e s u s e d a r e e s s e n t i a l l y i n v a r i a n t u n d e r adjoints.

A l t e r n a t i v e l y we c a n f o l l o w a p a r a l l e l d e r i v a t i o n to that f o r

by f i r s t c o m p u t i n g the s y m b o l fies

of the o p e r a t o r

which satis-

and w h i c h is u n i q u e l y s p e c i f i e d by the f u r t h e r

f a c t that

One c a n s h o w (with an a r g u m e n t s i m i l a r to that

of P r o p o s i t i o n 3), that

(5')

(6')

(2).

The assumption

in this c a s e .

f o r the s y m b o l s

(as g i v e n in

In f a c t if we take

T h e o r e m 21 in need o n l y take

14 this o p e r a t o r h a s a l e f t p a r a m e t r i x

holds

then a c c o r d i n g to

E

in

and we

while

It is a l s o to be noted that the n o n - i s o t r o p i c S o b o l e v s p a c e s

-138Sob

d i s c u s s e d h e r e a r e then e q u i v a l e n t to the s p a c e s

f o r the o p e r a t o r

t r e a t e d in R o t h s c h i l d - S t e i n [ 3 9 ] . (3).

A s a r e s u l t of the a b o v e , and in p a r t i c u l a r the f i r s t r e m a r k ,

w e c a n a s s e r t the f o l l o w i n g l o c a l r e g u l a r i t y r e s u l t f o r the o p e r a t o r (the o p e r a t o r

has a c o r r e s p o n d i n g e x i s t e n c e s t a t e m e n t ) .

Suppose

(17) and

g

b e l o n g s ( l o c a l l y ) to e i t h e r

or

( l o c a l l y ) to

Then

respectively.

f r o m the r e g u l a r i t y r e s u l t s of the c l a s s w h o s e s y m b o l s b e l o n g to not know if

i m p l i e s that

f

belongs

This follows

and the f a c t that o p e r a t o r s

preserve these c l a s s e s .

a t o r s of the s t a n d a r d c l a s s

1 6.

Sob

H o w e v e r , we do

b e c a u s e it is not t r u e that o p e r preserve

S e c o n d - o r d e r o p e r a t o r s of K a n n a i - t y p e We s h a l l c o n s t r u c t the p a r a m e t r i c i e s f o r o p e r a t o r s

of the f o r m

(1)

where

and

a r e s m o o t h r e a l f u n c t i o n s , with the

nxn

matrix

s y m m e t r i c and p o s i t i v e d e f i n i t e . Kannai [ 2 6 ] s h o w e d that b a s i c e x a m p l e s of the o p e r a t o r s type (1) a r e u n s o l v a b l e , y e t h y p o e l l i p t i c . e x t e n d e d by s e v e r a l p e o p l e ,

see e . g . ,

of the

T h i s r e s u l t has s i n c e b e e n

B e a l s and C . F e f f e r m a n [ 4 ] w h e r e

s o m e e a r l i e r r e f e r e n c e s may be found.

-1 39We s h a l l s h o w h o w to c o n s t r u c t a r i g h t p a r a m e t r i x f o r similarly a left p a r a m e t r i x for

(and

T h e m e t h o d w i l l b e s i m i l a r to that

u s e d f o r the o b l i q u e d e r i v a t i v e p r o b l e m in the p r e v i o u s s e c t i o n , but the d e t a i l s w i l l turn out to be m u c h s i m p l e r .

Let us d e a l f i r s t with

We

b e g i n by d e s c r i b i n g the s y m b o l c l a s s e s a p p r o p r i a t e f o r this p r o b l e m . We take

w h i c h w e c a n w r i t e as

where

a r e a spanning set of l i n e a r f o r m s f o r

the s u b s p a c e g i v e n by function

equals

and w h i c h d e p e n d s m o o t h l y on and we set

x. The

t i o n is d e f i n e d by

The func-

hence

(2)

where T h e n a t u r a l c a n d i d a t e f o r the s y m b o l

of

of a r i g h t p a r a m e t r i x

is

(3)

with

g i v e n by (5) and (6) of

Theorem.

15.

When r e s t r i c t e d to c o m p a c t

x

subsets

(a)

(b)

Proof.

R e c a l l that

and if we m a k e the

-140s

u

b

s

t

i

t

u

t

i

o

n

w

e

see

that (4) Thus by (8) of

15, we s e e that

(5) L e t us n o w c o n s i d e r the P

b e l o n g s to

d e r i v a t i v e s of

is that an a p p l i c a t i o n of

w h i l e an a p p l i c a t i o n of

P.

T h e r e q u i r e m e n t that g i v e s a gain o f

should g i v e a gain of

Thus we e x p e c t

(6)

and this f o l l o w s f r o m (9) in

15, s i n c e

We a l s o e x p e c t

(7)

However,

is

The t e r m

b e c a u s e of (9) in and

15 and

is bounded by

while

While .

Now

This gives

us (7). The higher c o n s i d e r the

x

d e r i v a t i v e s a r e t r e a t e d in the s a m e w a y . derivatives.

We c l a i m that

L e t us n o w

-141(8) where

and

and

(More precisely

In f a c t ,

and so (8) f o l l o w s f r o m what we have p r o v e d in

and

15 f o r

t o g e t h e r with the o b s e r v a t i o n s that

while

Of c o u r s e (8) is of the a p p r o p r i a t e f o r m as r e q u i r e d b y d e f i n i t i o n (20) in

7.

By the s a m e a r g u m e n t we s e e that

(9)

where again enter.

Higher

s i n c e h e r e the t e r m x

derivatives are treated similarly.

Thus, finally,

T o p r o v e p a r t (b) of the t h e o r e m we u s e the identity (16) of the s y m b o l i c c a l c u l u s .

Thus

is a s u m of t e r m s of the f o r m

d o e s not

15 and

-142(10)

and

m u l t i p l i e d by s m o o t h f u n c t i o n s of and

x.

w e s e e that while

By ( 9 ) , and s i n c e a c t u a l l y and

( R e c a l l that

T h e a n a l o g u e of (9) f o r

( e x c l u d i n g d e r i v a t i o n with r e s p e c t to

where

second-derivatives

is

and

However, in (10) a r e in Further (1)

and

so a l l the t e r m s w h i c h a p p e a r

p r o v i n g the t h e o r e m .

remarks Suppose

u(t,x)

i s a s o l u t i o n of the heat e q u a t i o n

with

Then

(11)

is a s o l u t i o n of

(f) = 0, in the c a s e

then

Moreover,

w h i c h s h o w s that this

In this s p e c i a l c a s e of o f the o p e r a t o r w h i c h is the r i g h t i n v e r s e of and the r a n g e of

L

if

is not h y p o e l l i p t i c . is the e x a c t s y m b o l

u n i q u e l y d e t e r m i n e d by is o r t h o g o n a l to the null

-143s o l u t i o n s of (2)

L

of the f o r m (11).

By s i m i l a r m e t h o d s (using

instead o f

we c a n o b t a i n a

left parametrix for

w h o s e s y m b o l a l s o b e l o n g s to

special c a s e when

then

is the

e x a c t s y m b o l of the o p e r a t o r w h i c h is the l e f t i n v e r s e of d e t e r m i n e d by

A g a i n , in the

uniquely

a n n i h i l a t e s the f u n c t i o n s

of the f o r m ( 1 1 ) . (3)

T h e p a r a m e t r i x f o r the g e n e r a l

theorem. or

Suppose

then

respectively.

and

f b e l o n g s l o c a l l y to

p r o v e s the f o l l o w i n g r e g u l a r i t y

g b e l o n g s ( l o c a l l y ) to

-144Appendix:

S o m e c o m p u t a t i o n s c o n c e r n i n g the c l a s s

In o r d e r not to have d i s c o u r a g e d the i n t e r e s t e d r e a d e r we p o s t p o n e d to this a p p e n d i x s o m e of the m o r e e l e m e n t a r y but t i r e s o m e involving our s y m b o l s .

computations

R e c a l l the d e f i n i t i o n s of the s y m b o l c l a s s

the p r e l i m i n a r y c l a s s e s

g i v e n in

the i n c l u s i o n r e l a t i o n s

and

and

A m o n g t h e s e hold

(of w h i c h the f i r s t is o b v i o u s ,

and the s e c o n d is p r o v e d b e l o w ) . 1.

Preliminaries Let

x,

A(x)

Since

A(x)

be a real symmetric

nx n m a t r i x , and s u p p o s e f o r e a c h

is either positive o r negative s e m i - d e f i n i t e .

A(x)

is s e m i - d e f i n i t e we have

Hence:

Next, we have a l r e a d y c h e c k e d that if

Put

-145T h e n ( P r o p o s i t i o n 7)

Since

Q

is a q u a d r a t i c f o r m d e p e n d i n g s m o o t h l y on

Lemma A.

x

we a l s o h a v e :

where each

A s s u m e that

We c o n s i d e r p o s s i b l e v a l u e s f o r

I

and

m:

follows since f o l l o w s f r o m (iv) s i n c e We have by (v)

since

and

T h i s c o m p l e t e s the c a s e

-146-

We w i l l skip the c a s e

since

until the e n d .

-147We s k i p t h i s a l s o until the e n d .

since When to p r o v e .

the d e r i v a t i v e is i d e n t i c a l l y z e r o ,

and

H e n c e it o n l y r e m a i n s to c h e c k the c a s e s

H e r e we u s e

We d e a l with the s e c o n d t e r m f i r s t .

When

so t h e r e is nothing

is a s u m of s q u a r e s .

A s in P r o p o s i t i o n 7,

-148-

When

we get

H e n c e we have to take c a r e of

where we get

and this is bounded by

-149-

For

w h i c h is b o u n d e d by

L e m m a B.

If

t h e r e is nothing to p r o v e .

If

t h e r e is

-1 50a l s o nothing to p r o v e .

Now:

The c l a s s

a r e all c l e a r .

T o c h e c k (d), note that

is a s u m o f t e r m s of the f o r m :

where

We e s t i m a t e e a c h s u c h t e r m in a b s o l u t e v a l u e

by

a r e the u s u a l f a c t o r s )

-151-

provided

Corollary. N o t i c e that by L e m m a s A and B ,

Corollary.

Theorem.

is a c o m p l e x v e c t o r

are again o b v i o u s . To prove

space

follows f r o m

We m u s t e s t i m a t e

-152But

is a s u m of t e r m s of the type

where

H e n c e we h a v e to e s t i m a t e t e r m s of the f o l l o w i n g t y p e :

where

Now

H e r e a t y p i c a l t e r m is b o u n d e d by

where

Therefore,

Finally, H e n c e , o n e t e r m we have to e s t i m a t e i s :

-153-

since

and t h e r e f o r e

T h e o t h e r t e r m w e have to e s t i m a t e i s :

-154since This proves (c). T o p r o v e ( d ) , let

Then

(by the p r o p o s i t i o n ) .

T o s h o w that

s i d e r the b e h a v i o r of

x

detail

we s h a l l have to c o n -

d e r i v a t i v e s of

aZ (see

7).

L e t u s c o n s i d e r in

A t y p i c a l t e r m r e s u l t i n g f r o m the d i f f e r e n t i a t i o n is

a m u l t i p l e of

where Since

e a c h d e r i v a t i v e o c c u r r i n g a b o v e is a s u m of t e r m s ;

( s e e the identity f o l l o w i n g (2) in

7).

T h i s l e a d s us to t e r m s of the f o r m

where Now

is a p o l y n o m i a l of d e g r e e

while

T h e r e f o r e by the

proposition

and h e n c e where

w h i c h is the d e s i r e d i n c l u s i o n f o r the p r o o f of (d). F i n a l l y , to p r o v e (e) note that f o r p o l y n o m i a l (in £), so s a t i s f i e s the r e q u i r e m e n t s .

is a q u a d r a t i c On the o t h e r hand

-1 55-

so it r e m a i n s to c h e c k that

so t h i s h o l d s .

-1 56References 1

R. Beals, "A general calculus of pseudo-differential operators," Duke Math. J . (1975) 42, 1-42.

2

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3

R. Beals and C. Fefferman, "Spatially inhomogeneous pseudodifferential operators," Comm. Pure Appl. Math (1974) 27, 1-24.

4

, "On the hypoellipticity of second-order operators," Comm. Partial Diff. Equations (1976) 1, 73-85.

5

L. Boutet de Monvel, "Hypoelliptic operators with double character­ istics and related pseudo-differential operators," Comm. Pure Appl. Math (1974) 27, 585-639.

6

L. Boutet de Monvel and J. Sjostrand, "Sur la singularity des noyaux de Bergman et de Szego,'' Asterisque (1976) 34-35, 123-164.

7

L. Boutet de Monvel and F. Treves, "On a class of pseudo-differential operators with double characteristics," Inventiones Math (1974) 24, 1-34.

8

A. P . Calderon, "Lebesgue spaces of differentiable functions and distributions," Amer. Math. Soc. Proc. Symp. Pure Math 5(1961), 33-49.

9

A. P . Calderon and R. Vaillancourt, "A class of bounded pseudodifferential operators," P r o c . Nat. Acad. Sci. (1972) 79, 1185-1187.

10

R. R. Coifman and G. Weiss, "Analyse harmonique non-communica­ tive sur certains espaces homogenes," Lecture Notes in Mathematics (1971) no 242, Springer Verlag.

11

V. Yu, Egorov and V. A. Kondrater, "The oblique derivative problem," Math. USSR Sbornik (1969) 7, 1 39-169.

12

E . B. Fabes and Ν. M. Riviere, "Singular integrals with mixed homogeneity," Studia Math. (1966) 27, 19-38.

13

C. Fefferman, "The Bergman kernel and biholomorphic mappings of pseudo-convex domains," Invent. Math. (1974) 26, 1-66.

14

G. B. Folland, "Subelliptic estimates and function spaces on nilpotent Lie groups," Arkiv f. Mat. (1975) 13, 161-207.

-1 57-

15.

G. Β. Folland and J. J. Kohn, "The Neumann problem for the Gauchy-Riemann complex," Annals of Math. Studies (1972) no. 75, Princeton University Press.

16.

G . B . F o l l a n d a n d Ε . M . S t e i n , " E s t i m a t e s f o r t h e "§ complex and analysis on the Heisenberg group," Comm. Pure and Appl. Math (1974) 27, 429-522.

17.

L. Garding, Bulletin Soc. Math. France (1961) 89, 381-428.

18.

R. Goodman, "Nilpotent Lie groups," (1976) no 562, Springer Verlag.

19.

P . C. Greiner, J. J . Kohn, and Ε. M. Stein, "Necessary and sufficient conditions for solvability of the Lewy equation," Proc. Nat. Acad. Sci. (1975) 72, 3287-3289

20.

P. C. Greiner and Ε. M. Stein, " Estimates for the ^-Neumann problem," Mathematical Notes (1977) no 19, Princeton University Press.

20a.

V. V. Grushin, "On a class of hypoelliptic pseudo-differential operators degenerate on a sub-manifold," Math. USSR Sbornik (1971) 1 3, 1 55-185.

21 .

Lecture Notes in Mathematics

S. Helgason, "Differential geometry and symmetric spaces," (1962) Academic P r e s s , New York.

22.

I. I. Hirschman Jr. , "Multiplier transformations I," Jour. (1956) 26, 222-242.

23.

L. Hormander, "Pseudo-differential operators and non-elliptic boundary problems," Ann. Math. (1966) 83, 129-209.

23a.

, "Hypoelliptic second-order differential equations," ActaMath. (1967)119, 147-171.

24.

, "Pseudo-differential operators and hypoelliptic equa­ tions," Amer. Math. Soc. P r o c . Symp. Pure Math. (1967) no. 10, 138-183.

25.

,

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to appear. 26.

Y. Kannai, "An unsolvable hypoelliptic differential operator," Israel J . Math. (1971) 9, 306-315.

-1 5827

N. Kerzman and Ε. Μ. Stein, "The Szego kernel in terms of CauchyFantappie kernels," Duke Math. Jour. (1978) 45, 197-224.

28

A. W. Knapp and Ε. M. Stein, "Intertwining operators for semisimple groups," Ann. of Math. (1971) 93, 489-578.

29

A. Koranyi and S. Vagi, "Singular integrals in homogeneous spaces and some problems of classical analysis," Ann. Scuola Norm. Sup. P i s a (1971) 25, 575-648.

30

S. Krantz, appear.

31

P. Kree, "Distributions quasi-homogenes ," C.R.A. Sci. Paris (1965) 261, 2560.

32

J. L. Lions and J. Peetre, "Sur une classe d'espaces d'interpolation," Publ, Math. Inst. Hautes Etudes Sci. (1964) 1 9, 5-68.

33

W. Madych and N. Riviere, "Multipliers of Holder classes," of Funct. Analysis (1976) 21, 369-379-

34

A. Nagel and Ε. M. Stein, "A new class of pseudo-differential operators," Proc. Nat. Acad. Sci (1978) 75, 582-585.

35

, "Some new classes of pseudo-differential operators," Proc. Symp. Amer. Math. Soc. held in Williamstown, Summer 1978, to appear.

36

R. O'Neil, "Two elementary theorems on the interpolation of linear operators," Proc. Amer. Math. Soc. (1966) 17, 76-82.

37.

D. H. Phong and Ε. M. Stein, "Estimates for the Bergman and Szego projections on strongly pseudo-convex domains," Duke Math. Jour. (1977) 44, 695-704.

38.

Ν. M. Riviere, "Singular integrals and multiplier operators," Arkiv f. Mat. (1971)9, 243-278.

39.

L. P. Rothschild and Ε. M. Stein, "Hypoelliptic differential oper­ ators and nilpotent groups," Acta Math. (1976) 137, 247-320.

40.

J . Sjostrand, "Operators of principal type with interior boundary conditions," ActaMath. (1973) 1 30, 1 -51.

41 .

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"Generalized function spaces of Campanato type,"

to

Jour,

-1 5942.

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43.

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44.

S . W a i n g e r , " S p e c i a l t r i g o n o m e t r i c s e r i e s in K d i m e n s i o n s , " M e m . A m e r . Math. S o c . (1965) no 5 9 .

45.

N. W i e n e r , " T h e F o u r i e r i n t e g r a l and c e r t a i n of i t s a p p l i c a t i o n s , (1933), C a m b r i d g e Univ. P r e s s .

L i b r a r y of C o n g r e s s Cataloging in Publication Data

Nagel, Alexander, 1945Lectures on pseudo-differential operators. Includes bibliographical references. 1 . Pseudodifferential operators. I . Stein, Elias M . , 1951joint author. II. Title. Q A 3 2 9 . 7 . 5 1 5 ' . 7 2 ISBN 0-691-082147-2

79-19388