Estimates of the Neumann Problem. (MN-19), Volume 19 9781400869220

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ESTIMATES FOR THE

-IIEUMANN PROBLEM

P. C. GREINER and E. M. STEIN

Princeton University Press 1977

Copyright (c) 1977

by Princeton University Press

Published by Princeton University Press, Princeton, New Jersey In the United Kingdom:

Princeton University Press, Guildford, Surrey All Rights Reserved

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

Published in Japan exclusively by University of Tokyo Press in other parts of the world by Princeton University Press

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

Preface

The g-Neumann problem is probably the most important and natural example of a non-elliptic boundary value problem, arising as it does from the Cauchy-Riemann equations. It has b e e n k n o w n f o r s o m e t i m e , g o i n g b a c k t o t h e w o r k of K o h n , h o w t o d e t e r m i n e s o l v a b i l i t y a n d p r o v e r e g u l a r i t y of s o l u t i o n s i n t h e s t u d y of t h i s p r o b l e m . T h e m a i n t o o l t h a t w a s u s e d were various estimates. What had been lacking was a m o r e e x p l i c i t c o n s t r u c t i o n of t h e o p e r a t o r ( o r a t l e a s t a n approximation) giving the solution, and the study of estimates in other norms, all of which would be needed in order to have a b e t t e r u n d e r s t a n d i n g of t h e s t r u c t u r e o f t h e ^ - N e u m a n n p r o b l e m It is our primary purpose here to describe the work we have done in this direction in the last few years. 1 We construct parametricies and give sharp estimates in appropriate function spaces . Our secondary aim in writing this monograph is didactic. W e p r e s e n t n o t o n l y t h e r e q u i r e d d e t a i l s of t h e p r o o f s of o u r main results, but also certain prerequisites and some additional m a t e r i a l w h i c h w e h o p e w i l l g i v e t h e r e a d e r a c l e a r e r v i e w of the whole subject. It is with pleasure that we take this opportunity to express our appreciation to Miss Florence Armstrong for her excellent job of typing the manuscript.

November 1976

A p r e l i m i n a r y a n n o u n c e m e n t of o u r m a i n r e s u l t s w a s g i v e n i n t h e 1 9 7 4 M o n t r e a l C o n f e r e n c e [12],

T a b l e of Contents

Preface Introduction Part I

1

A n a l y s i s on the H e i s e n b e r g g r o u p Guide to P a r t I

10

1. 2.

13 27

S y m b o l s on the H e i s e n b e r g g r o u p A comparison

3. P a r t II

on f u n c t i o n s and the s o l v a b i l i t y of the L e w y e q u a t i o n P a r a m e t r i x f o r the

Neumann p r o b l e m

Guide to P a r t II 4. 5. 6. 7. 8. 9. 10. P a r t III

36

44

A d m i s s i b l e c o o r d i n a t e s on s t r o n g l y p s e u d o - c o n v e x CR. manifolds 48 Levi metrics 64 on (0, 1 ) f o r m s 70 L o c a l s o l u t i o n of the D i r i c h l e t p r o b l e m f o r 79 R e d u c t i o n to the b o u n d a r y 1 0 1 A parametrix f o r n e a r b M , 1 1 0 A parametrix for n e a r bM, 118 The estimates

Guide to P a r t III 11 . 12. 13.

R e v i e w of the theory The B e s o v s p a c e s The s p a c e s and

14. 15. 16.

The s p a c e s B ? , L ? , a n d o n Main results Solution of _

17.

Concluding r e m a r k s

130 132 134 141 M

and

bM

149 153 172 177

S u m m a r y of N o t a t i o n

190

References

192

Introduction

Why the g-Neumann problem?

Let M

be an open relatively compact subset of a complex manifold

M' of dimension

n+ 1, and assume that the boundary of M,

smooth and strongly pseudo-convex.

bM,

is

The g-Neumann problem for M

arises when one tries to solve the Cauchy-Riemann equations on M (1)

i u = f,

where f

is a given (0,1) form with

(2)

3f = 0 ,

and in particular when one wishes to have control on the behavior of

U

near the boundary in t e r m s of similar control over f. P a r t of the difficulty of this problem is connected with the fact that (1) is overdetermined, and also without some further conditions the solution of (1) is not uniquely specified.

There is however a general

formalism due to Spencer (applicable also in many other situations) which gets around these initial difficulties.

Instead of (1) and (2) one

considers the second-order equation

(3)

Du= (Ia*+a* I)u=f.

3* i s t h e a d j o i n t o f

§ (which is defined once one chooses a fixed Her-

mitian metric on M). write (1) as

To explain (3) more precisely we temporarily

= f, and (2) as

g^f = 0, where the subscripts indicate

that the first § acts on functions, and the second Strictly speaking • should be written as

5 on

3*^+ 5* 5 ^ .

(0,1)

forms.

And now it is

clear that the equation (3) comes naturally equipped with a pair of boundary conditions

(4)

u ε domain (δ*)

(5)

S1 u ε domain (a* )

_

The equation (3), which is essentially Laplace's equation, together w i t h t h e t w o b o u n d a r y c o n d i t i o n s ( 4 ) a n d (5) g i v e u s t h e J - N e u m a n n problem.

It can be shown that if

u

is a solution of this problem, with

gf = 0, then U = §*u solves our original problem (1), and is in addition uniquely specified by the property that

U is orthogonal to holomorphic

functions on M

Kohn's solution, and some further problems The analytic difficulty of the problem is due to the fact that while the differential operator Π is elliptic, the boundary conditions are not. 2 Nevertheless, using L

estimates, Kohnwas able to prove existence

and make a systematic analysis of the regularity properties of solutions of this problem.

One of his main results is the estimate

(6)

< A k ( I )DuI l

Ilull2 L

for u gain of

k+1

2 L

k

+ Hull

2

),

n=0, 1, 2 , . . .

L

s a t i s f y i n g t h e b o u n d a r y c o n d i t i o n s (4) a n d ( 5 ) . 1, and not

Thus there was a

2 as in the standard elliptic boundary value problems.

The problems that were left open were as follows: (i)

To understand more fully these regularity properties, even in

the context of the (ii)

2 L norm,

To find the corresponding estimates for other function spaces,

Lp s p a c e s , L i p s c h i t z s p a c e s , e t c . ,

e.g., (iii)

To give a more explicit construction of the operator (the Neumann

operator), which expresses the solution

u in terms of f.

To deal with (i) - (iii) is the main task of this monograph.

Three principles We shall be guided by three principles, the first of which is by now well-understood. First, the solution of a boundary-value problem for a differential operator which is elliptic can, by the use of the theory of pseudo-differential operators, be reduced to the inversion of a pseudo-differential operator acting on the boundary. pseudo-differential operator.

So our first task is to isolate this

This is the operator

D+

(see Chapter 8),

and since it is not elliptic we need to know it rather precisely: Q+

is of

order 1, but its zero order terms are not negligible. In inverting that the inverse of

G+ we are guided by the second principle, namely should be modeled on the inverse of

special case corresponding to the Heisenberg group.

Q+

in the

This accounts for

the key role of the Heisenberg group in our analysis, which incidentally * Background material that might be useful for the reading of this monograph is contained in the survey [35].

is closely related to the similar role it plays in the case of the boundary analogue of (1) o r (3). The third principle follows from the second.

All estimates which

are sharp will reflect the structure of the Heisenberg group (or what amounts to the same thing, the complex structure of

M).

Thus there

are "good" directions which are singled out, and in terms of these direc­ tions we have a gain of two, as in the usual elliptic case.

Let us now

describe these things in greater detail.

Reduction to a boundary problem We first solve the Dirichlet boundary-value problem for • , and our procedure here uses well-known techniques. operator

We construct Green's

G which solves the inhomogeneous problem with zero Dirichlet

boundary conditions, and the Poisson integral P, which solves the homogeneous problem with given Dirichlet boundary conditions. (7)

where

Thus

u = G(Qu) + P(ub)

U^

is the restriction of u to the boundary.

Actually we only construct an approximate version of (7), valid in appro­ priate coordinate patches near the boundary (see Theorem 7.66), but this suffices for our purposes.

Because of the non-elliptic nature of

the boundary-value problem for which we want to use (7), it is crucial that we keep track of the symbols one order less than the top order,

*See Folland and Stein [9]·

- S -

and this makes our calculations somewhat elaborate. Now let

B— b e t h e b o u n d a r y o p e r a t o r g i v i n g t h e s e c o n d g - N e u m a n n 9

boundary condition (i.e., (5)).

Then the basic boundary operator we have

to deal with (and whose symbol we determine rather precisely) is D+ = B _ P

(8)

Inversion of D+ Using the symbolic calculus we find another operator, • , in many ways similar to Π+, so that (9)

Dj 3 = - • D +

approximately.

Here • is the Kohn Laplacian for the b

§ b

complex (acting on

(0,1) forms on bM). Now when η > 1 , •

has an approximate inverse

K , given as

an integral operator modeled on a convolution operator on the Heisenberg group. when η > 1

( T h i s i s o n e of t h e m a i n r e s u l t s of t h e p a p e r [ 9 ] ). an approximate inverse to

Thus

is -KD ; with this it is a

s t r a i g h t f o r w a r d m a t t e r to w r i t e d o w n a n a p p r o x i m a t i o n to t h e N e u m a n n operator, (giving the solution to the KQ ,

B_ , 3

The case

P

and

G.

g - N e u m a n n p r o b l e m ) i n t e r m s of

(See Proposition 9. 26. )

n=l

W h e n n=l

the operator • is not invertible, and therefore a b

further analysis is required.

The idea is as follows.

Near the charac­

t e r i s t i c v a r i e t y o f Π + , t h e o p e r a t o r Π, b e h a v e s in t h e c a s e b

n=l

like

it does in the case

η >1 .

Hence near the characteristic variety of

we can write an inverse of

O+

similar to that for

away from its characteristic variety,

Q+

η > 1.

However,

is elliptic, and so here we

can find an inverse by the use of the standard calculus of pseudo-differ­ ential operators. Π

Now the required analysis for several ways. (10)

b

(when n=l ) can be done in

The most elegant approach is via the identity

K Db = DbK = I - Cfc

on the Heisenberg group which was obtained in a joint work [11] with Kohn. Here K

i s a n ( e x p l i c i t ) c o n v o l u t i o n o p e r a t o r (of t y p e 2 ) i n t h e

Heisenberg group, and Cb

is the Cauchy-Szego projection.

Incidentally

the identity (10) leads to the necessary and sufficient conditions for local solvability of

π

b

when n= l , for

t h e L e w y o p e r a t o r , w h e n n=l . The inverse of

Q+

Π

b

on functions for any

n, and for

(For further details see Chapter 3. )

(when n=l) can then be given two alternative

(but roughly equivalent) forms in terms of

K

and

C, ; see Lemma 10. 25, b

and Lemma 10. 32. The estimates Once the analysis of the inverse of for the solution of the

Q+

is concluded, the estimates

J-Neumann problem (i.e.. control of the Neumann

*Our original approach, sketched in [12], was more complicated. It used the material in Chapters 1 and 2.

- T -

o p e r a t o r ) c a n b e g i v e n in t e r m s of c o r r e s p o n d i n g e s t i m a t e s f o r f o u r c l a s s e s of o p e r a t o r s : The r e s t r i c t i o n o p e r a t o r , s t a n d a r d 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 , convolution o p e r a t o r s of the H e i s e n b e r g t y p e s t u d i e d in [ 9 ] , and P o i s s o n o p e r a t o r s . w e n e e d a r e known.

F o r the f i r s t two c l a s s e s of o p e r a t o r s the e s t i m a t e s

F o r the o t h e r two c l a s s e s w e need to invoke the

r e s u l t s of [ 9 ] ( s e e a l s o [ 2 9 ] ), and s o m e n e w e s t i m a t e s need to b e m a d e . T h e r e q u i r e d w o r k i s done in C h a p t e r s 1 2 to 1 5 . O u r m a i n c o n c l u s i o n s a r e then a s f o l l o w s .

S u p p o s e u i s the

solution of the p r o b l e m (3) w i t h b o u n d a r y condition (4) and (5). fslij(M),

for some p,

1 1

and when the dual variable lies in a half-space).

n-of>0,

Next one takes the

Fourier transform in the other variables, obtaining the symbol of the o p e r a t o r i n a h a l f - s p a c e , w h e n R e ( n - a ) > 0 ; s e e P r o p o s i t i o n (1 . 3 ) .

For other values of

a (excluding the singular values where

0 , - 1 , - 2 , . . . )) t h e f u l l s y m b o l i s o b t a i n e d b y a n a l y t i c c o n t i n u a t i o n , which requires replacing an integral over a segment by a loop integral. The final result is in Theorem 1.21 and its corollary.

The fact that this

result agrees with the original fundamental solution is expressed in Theorem (2.4) and Corollary (2.22). The singular case

a = ±n

The problem of 1-forms in 2 complex variables leads to the operator

s£ , a

where a = n-2q = -1,

with Lewy's equation. α

,

when a = ±n,

n=l .

This situation is closely connected

There no longer exists a fundamental solution of

but one can find a relative fundamental solution

involving the Cauchy-Szego projection. Lemma (3 18).

The main identity is given in

An alternative method of deriving this identity is indicated

in Proposition (3.26).

The identity (3.18) leads to necessary and suffi­

cient conditions for the local solvability of equation.

(u) = f,

and for the Lewy

-13C h a p t e r I.

S y m b o l s on the H e i s e n b e r g g r o u p s

Let

(1.1) be the u s u a l l e f t invariant v e c t o r f i e l d s on the H e i s e n b e r g g r o u p F o l l o w i n g the notation and t e r m i n o l o g y of F o l l a n d - S t e i n [ 9 ] w e d e f i n e

(1.2)

T h e p u r p o s e of this c h a p t e r is to c o m p u t e the s y m b o l of the f u n d a m e n t a l s o l u t i o n of

.

Let

denote a (presumptive) fundamental solution

and set

(1.3)

Taking

u n d e r the F o u r i e r t r a n s f o r m w e o b t a i n the o p e r a t o r

(1.4)

Assume

(The c a s e

will follow by replacing

by

-14- a . ) Since w e want

to a c t on the H e i s e n b e r g g r o u p b y c o n v o l u t i o n

w e shall t r y a k e r n e l of the f o l l o w i n g f o r m

(1.5) w h e r e w e u s e d the notation

The c h o i c e ( 1 . 5 )

is d i c t a t e d by the f o l l o w i n g c o n s i d e r a t i o n s .

b y ( 0 . 3 ) it s u f f i c e s to c o n s i d e r the s p e c i a l c a s e when Next

(when

w = 0).

is i n v a r i a n t u n d e r u n i t a r y l i n e a r t r a n s f o r m a t i o n of the z - v a r i -

a b l e s , and s o one m a y l o o k f o r a o b s e r v e that

w h i c h d e p e n d s o n l y on

annihalates any f u n c t i o n of

a r e led to the f o r m (1. 5) (when w= 0).

For general

g r o u p law ( 0 . 2 ) to r e d u c e m a t t e r s to the c a s e

if

y =0

First

A bit of a l g e b r a y i e l d s

w=0.

[z].

|z] .

Also

Thus w e

w w e then u s e the Next w e shall s o l v e

-15We s e t

H e n c e w e need to s o l v e

(1.6)

if

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

set

w h i c h r e d u c e s the e q u a t i o n ( 1 . 6 ) to the f o l l o w i n g b e t t e r known f o r m

(1.7) if

.

The identity

i m p l i e s that

is a s o l u t i o n of if u > 0 and (1.8)

Re a > 0.

T h e r e f o r e w e a s s u m e that

i. * S e e [ 6 ] , C h a p t e r 6.

-16W e s t i l l need to d e t e r m i n e the unknown f u n c t i o n

T o this end w e

note that

as long as

if

and

and

n > 1.

|z-w|

In p a r t i c u l a r

is s m a l l .

Thus to have the

"correct"

f u n d a m e n t a l s i n g u l a r i t y w e set (1.9) We w o u l d like to point out that w e a r e s t i l l in the p r o c e s s of t r y i n g to find the s y m b o l

by h e u r i s t i c c o n s i d e r a t i o n s .

Once found, we

shall p r o v e its c o r r e c t n e s s f o r all We continue b y applying the o p e r a t o r induced b y the k e r n e l

to

A f t e r i n t e r c h a n g i n g the o r d e r of i n t e g r a t i o n , w e obtain the k e r n e l

-17-

(1.10)

Now

where

and w e set

(1.11) T set Fh i ni sa l lyyThus i,e l to d s o(b1t.a1i0n) the b e c soand ymmebsnote o l , wthat e multiply ( 1 . 1 1 ) by

- 1 8 -

T h e i n t e r c h a n g e of the o r d e r of i n t e g r a t i o n is j u s t i f i e d b y F u b i n i ' s t h e o r e m b e c a u s e the s e c o n d i t e r a t e d i n t e g r a l is e a s i l y s e e n to be c o n vergent.

Now

which yields

(1. 12)

L e d b y o u r h e u r i s t i c s , w e state o u r f i r s t r e s u l t as f o l l o w s : 1.13.

Proposition.

U

then

-19-

(1.14)

i n d u c e s the o p e r a t o r

w h i c h has the p r o p e r t y that Ji

f

is in the S c h w a r t z s p a c e and supp Proof.

(1 . 14).

We r e p l a c e

by

W e w i l l show that if

is c o n t a i n e d in

s

in (1 . 12) and o b t a i n the f o r m

then

is a f u n d a m e n t a l s o l u t i o n

of if

f b e l o n g s to the S c h w a r t z s p a c e of f u n c t i o n s , s u c h that This requires

w h e r e w e set

supp

-20T o s i m p l i f y m a t t e r s w e note that

A simple computation yields

f o r every function

h.

Next w e r e d u c e the p r o b l e m to s o l v i n g an i n h o m o -

g e n e o u s o r d i n a r y d i f f e r e n t i a l e q u a t i o n as f o l l o w s .

Therefore

W e set

and r e q u i r e that the f o l l o w i n g d i f f e r e n t i a l e q u a t i o n

-21is s a t i s f i e d (1.15)

F o l l o w i n g p r e v i o u s c a l c u l a t i o n s we s e t

T h e n the l e f t - h a n d side of ( 1 . 1 5 ) b e c o m e s

which yields

This proves Proposition

Next w e continue Let

1.13.

a n a l y t i c a l l y on the c o m p l e x a - p l a n e .

D d e n o t e the c o n t o u r

In o t h e r w o r d s

D

and r e t u r n s to

-1.

s t a r t s at

- 1 , e n c i r c l e s the o r i g i n o n c e c o u n t e r c l o c k w i s e

Consider

-22-

(1.16)

A g a i n w e apply

as w e did in ( 1 . 1 5 ) , w e o b t a i n

(1.17)

On

D

we set

where

logs

is the p r i n c i p a l b r a n c h of the l o g a r i t h m ,

r e a l if

s

on the p o s i t i v e

1.18.

(1.19)

is

Proposition.

W Re

p a r t of the r e a l a x i s

i.e.,

logs

is

Therefore

then f o r

-23-

can be continued a n a l y t i c a l l y on the c o m p l e x a - p l a n e to all s u c h that

Proof.

In (1. 19) w e d e f o r m the path of i n t e g r a t i o n , D , into i n t e -

grating along the r e a l a x i s f r o m o n the c i r c l e of r a d i u s

5 around the o r i g i n and, f i n a l l y r e t u r n i n g f r o m

On the f i r s t p a r t

where

Therefore

then i n t e g r a t i n g

is r e a l , on the l a s t p a r t

-24 If

we l e t

on the c i r c l e v a n i s h e s .

and the c o n t r i b u t i o n of the i n t e g r a l

Thus we a r e l e f t w i t h

(1.20)

We set

and ( 1 . 2 0 ) c o m b i n e d w i t h (1 . 1 9) y i e l d s (1 . 14).

This

p r o v e s the p r o p o s i t i o n . A n a l o g o u s c a l c u l a t i o n s y i e l d the s y m b o l

i

as w e have a l r e a d y r e m a r k e d . W e c o l l e c t the r e s u l t s of this c h a p t e r in the f o l l o w i n g f o r m . 1.21

Theorem.

The s y m b o l

is g i v e n by

(1.22) (1.23) where

for

(1.24)

and

is g i v e n b y

of a f u n d a m e n t a l s o l u t i o n of

-25-

Here

Moreover,

b e continued a n a l y t i c a l l y in the c o m p l e x

can

plane to

a c c o r d i n g to the f o r m u l a (1.25)

w h i c h is a l s o valid f o r all

with

O b s e r v e that w h e n

sign then

is d e f i n e d b y e i t h e r

( 1 . 2 4 ) o r ( 1 . 2 5 ) as long as

1 . 26

Corollary.

c a n b e e x t e n d e d b y continuity to

as f o l l o w s (1.27)

Thus d e f i n e d Proof. e x a m p l e , if

is

o u t s i d e of the o r i g i n .

T h i s f o l l o w s f r o m a s i m p l e i n t e g r a t i o n by p a r t s . Re

,

then

For

-26(1 . 28)

This proves Corollary 1.26.

-27C h a p t e r 2.

A comparison

Set

(2.1)

where (2.2)

In (2. 1) w e a s s u m e

A c c o r d i n g to P r o p o s i t i o n 7 . 1

of [ 9 ] the o p e r a t o r

defined by

(2.3)

is i n v e r s e to

2.4

Theorem.

where

(2.5)

whenever

Let

as l o n g as

a be a d m i s s i b l e .

a

is a d m i s s i b l e ,

i.e.,

Then

is g i v e n b y ( 1 . 2 2 ) .

In o t h e r w o r d s ,

-28Proof.

Starting w i t h

(2.6)

w e shall c o m p u t e (2.7)

We shall do the c o m p u t a t i o n o n l y if u s e the notation

and r e c a l l that

F i r s t we compute

(2.8)

It is s i m i l a r if

We

-29H e n c e w e need to evaluate

(2.9)

Now

where we set

z = 2u.

If

from

where

L

d e n o t e s the f o l l o w i n g path

w e can change the path of i n t e g r a t i o n

-30 T h i s d e f o r m a t i o n is j u s t i f i e d by noting that f o r

R l a r g e , the

i n t e g r a n d is bounded by

since

Finally

, where

log

b r a n c h of the l o g a r i t h m , i . e . .

d e n o t e s the p r i n c i p a l This easily yields

T h u s , w e have d e r i v e d

2.10

Lemma.

Let

Then

T h e next r e s u l t c o n c e r n s changing the p a r a m e t e r s 2.11

Lemma.

Suppose

(2. 12)

Let Then

a, c

be r e a l n u m b e r s .

n and

a•

-31-

Proof•

Erdelyi:

v . 1. p. 256 p r o v e s the f o l l o w i n g f o r m u l a

(2.13)

where

m i n ( 0 , - c ) . We r e p l a c e

a by

, and

c

by

in

( 2 . 1 3 ) , we obtain

(2.14)

as long as

Actually, since

w e m a y a s s u m e that

Setting

and

s a t i s f i e s the m o r e r e s t r i c t i v e c o n d i t i o n

in the r i g h t - h a n d side of (2. 14) w e obtain that the l e f t -

hand s i d e of (2. 14) is e q u a l to

w h i c h g i v e s the r i g h t - h a n d side of ( 2 . 1 2 ) , s i n c e This p r o v e s L e m m a 2 . 1 1 .

-32Next w e r e t u r n to the c o m p u t a t i o n of and

or,

.

Set

T h e n the h y p o t h e s e s of L e m m a 2 . 1 1 c a n b e put in the f o r m

equivalently,

(2.15)

A c c o r d i n g to L e m m a 2 . 1 1 w e have

Thus L e m m a 2 . 1 0 y i e l d s 2.16

Lemma.

Let

Then

(2.17)

Now w e a r e r e a d y to c o m p l e t e the p r o o f of T h e o r e m 2 . 4 . (2. 7) and ( 2 . 1 7 ) y i e l d

(2.18)

First

-33-

Next the a n a l y s i s that y i e l d s (1 . 14) f r o m ( 1 . 1 1 ) a p p l i e s and w e o b t a i n (2. 19)

as long as

and

T o r e m o v e the r e s t r i c t i o n on

a , w e note that

and b y i n t r o d u c i n g p a r a b o l i c c o o r d i n a t e s with i n t e g r a b l e at the o r i g i n and e n t i r e in

a .

is

Therefore

(2.20)

is h o l o m o r p h i c in a w h e n e v e r

ly is a d m i s s i b l e .

Next w e note that

g i v e n b y ( 2 . 1 9 ) if

can be e x t e n d e d to all a d m i s s i b l e the e x t e n s i o n g i v e n in T h e o r e m 1 . 2 2 , a l s o f o r ( 1 . 2 3 ) it f o l l o w s i m m e d i a t e l y that if

and holomorphically, F r o m the f o r m u l a

s a t i s f i e s the h y p o t h e s i s of

-34T h e o r e m 2 . 4 then

given by

(2.21)

is h o l o m o r p h i c in a g r e e when

a when

S i n c e ( 2 . 2 0 ) and ( 2 . 2 1 )

is in an o p e n i n t e r v a l of the r e a l a x i s , t h e y m u s t a g r e e

f o r all a d m i s s i b l e 2.22

is a d m i s s i b l e .

Corollary.

This p r o v e s T h e o r e m 2 . 4 . Set

Then

where

is d e f i n e d in (1 . 2 4 ) . A s a b y - p r o d u c t of t h e s e c o n s i d e r a t i o n s w e found a s o l u t i o n of an

inhomogeneous confluent h y p e r g e o m e t r i c differential equation.

We r e c a l l

that in W h i t t a k e r ' s s t a n d a r d f o r m the c o n f l u e n t h y p e r g e o m e t r i c d i f f e r e n t i a l e q u a t i o n is g i v e n b y

(2.23)

L e t u s c o n s i d e r the d i f f e r e n t i a l e q u a t i o n ( 1 . 1 5 )

W e substitute

-35-

T h e n an e l e m e n t a r y c a l c u l a t i o n y i e l d s the f o l l o w i n g r e s u l t . 2.24

Proposition.

Then

(2.25)

for

Re

and its a n a l y t i c c o n t i n u a t i o n f o r all o t h e r

n+4, . . . y i e l d s a s o l u t i o n of the f o l l o w i n g i n h o m o g e n e o u s W h i t t a k e r differential equation

(2.26)

T h e a n a l y t i c c o n t i n u a t i o n is g i v e n b y

(2.27)

if

where

D

is the Hankel c o n t o u r a p p e a r i n g in(1.16)-

-36C h a p t e r 3.

on f u n c t i o n s and the s o l v a b i l i t y of the L e w y equation

A q-form

f on

is a s u m

(3.1)

where

The

a r e c o m p l e x - v a l u e d f u n c t i o n s on

o p e r a t o r (mapping q - f o r m s to

indexed b y

q + 1 - f o r m s ) is then d e f i n e d b y

(3.2)

where

s e e ( 1 . 1 ) , and the f o r m a l adjoint

is g i v e n by (3. 3)

The interior product

is d e f i n e d a f t e r f o r m u l a (6. 10).

One then d e f i n e s

the L a p l a c i a n c o r r e s p o n d i n g to this c o m p l e x ; it is (3.4) We d e n o t e the r e s t r i c t i o n of that on the H e i s e n b e r g g r o u p

to q - f o r m s b y

.

It turns out

takes a particularly elegant f o r m ,

is d i a g o n a l , m o r e p r e c i s e l y

(3. 5)

with

where

i

is d e f i n e d by ( 1 . 2 ) .

The r e g u l a r i t y and the

-37existence theory for

has b e e n studied in [ 9 ] , using the f u n d a m e n t a l

s o l u t i o n d i s c u s s e d in C h a p t e r 2.

This yields a fundamental solution f o r

Such a f u n d a m e n t a l s o l u t i o n d o e s not e x i s t f o r or

q = n - - t h e s e c o r r e s p o n d to

these c a s e s .

q=0

We shall now c o n s i d e r

A c t u a l l y w e shall do the w o r k o n l y f o r

q = 0, b e c a u s e

q=n

f o l l o w s by c o m p l e x conjugation. A s i m p l e c a l c u l a t i o n s h o w s that

(3.6)

t h e r e f o r e annihilates the b o u n d a r y v a l u e s o f h o l o m o r p h i c f u n c t i o n s on

and s o is f a r f r o m h y p o e l l i p t i c .

M o r e o v e r (we shall

s e e b e l o w ) the e q u a t i o n (3.7) is g e n e r a l l y not e v e n l o c a l l y s o l v a b l e .

T o get to the s o l u t i o n of this

p r o b l e m let us r e c a l l s o m e of the b a s i c t e r m i n o l o g y . upper half-plane

The g e n e r a l i z e d

is g i v e n b y

(3. 7') w i t h its b o u n d a r y (3.8) The mapping

given by identifies

has the s t r u c t u r e of a g r o u p , the H e i s e n b e r g g r o u p

with in

-38p a r t i c u l a r the l e f t - i n v a r i a n t v e c t o r f i e l d s on the H e i s e n b e r g g r o u p

a r e r e s t r i c t i o n s to

vectorfields

of the h o l o m o r p h i c and the l a t t e r a r e tangential

at Any

( o r a d i s t r i b u t i o n with c o m p a c t s u p p o r t ) l e a d s to

the C a u c h y - S z e g o i n t e g r a l

C(f),

d e f i n e d in

by

(3.9) with (3. 10)

where

f

of the l a t t e r with The r e s t r i c t i o n of

and C(f)

is d e f i n e d on

v i a the i d e n t i f i c a t i o n

c o r r e s p o n d s to the L e b e s g u e v o l u m e of to

(taken on the H e i s e n b e r g g r o u p ) is

g i v e n by (3.11)

where

and the c o n v o l u t i o n in (3. 11) is w i t h

r e s p e c t to the H e i s e n b e r g g r o u p . in

When

the l i m i t in (3. 11) e x i s t s

n o r m ( f o r f u r t h e r d e t a i l s s e e K o r a n y i and V a g i [ 2 2 ] ). Next w e shall d e r i v e a " r e l a t i v e f u n d a m e n t a l s o l u t i o n "

We write

Therefore

of

|

-39(3.12)

where

(3.13) and

(3.14)

T h e s e f o r m u l a s c a n be found in C h a p t e r 2. (3. 12) with r e s p e c t to

F o r m a l d i f f e r e n t i a t i o n of

a y i e l d s the f o l l o w i n g e q u a t i o n

(3.15)

Set

(3.16)

w h e r e w e set D e f i n e the o p e r a t o r

K

by

(3.17) 3.18

Lemma

(3.19) w h e n acting on d i s t r i b u t i o n s of c o m p a c t s u p p o r t . * H e r e the l o g a r i t h m of the quotient m e a n s the d i f f e r e n c e of the c o r r e s p o n d i n g logarithms.

-40 Proof.

The a r g u m e n t is s i m i l a r to the p r o o f of T h e o r e m 6 . 2 of [ 9 ]

W e shall o n l y show that transposition.

W e set

the s e n s e of d i s t r i b u t i o n s , as

On the o t h e r hand

Therefore

A L ce ct toirndgi n g to p p . we 4 4 1obtain - 4 4 3 of [ 9 ]

T h e o t h e r identity f o l l o w s b y • it. .

W e shall then take the l i m i t in

Now

-41-

where we set

Hence

which proves L e m m a 3.18. In addition to the l e m m a one should o b s e r v e that Thus mutes with

is an o r t h o g o n a l p r o j e c t i o n w h i c h c o m -

T h e m e a n i n g of L e m m a 3 . 1 8 is that

K

inverts

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

3. 20

Theorem.

if and o n l y if

3.21

Corollary.

is s o l v a b l e in a n e i g h b o r h o o d of a is r e a l - a n a l y t i c in a n e i g h b o r h o o d of

If the a b o v e c o n d i t i o n f o r

f

P.

is s a t i s f i e d and if

f

belongs

to one of the s p a c e s ( s e e r e f . [ 9 ] f o r d e f i n i t i o n s ) or

then w e c a n find a u w h i c h b e l o n g s to or

h o o d of

respectively, where

Q is an a p p r o p r i a t e n e i g h b o r -

P. T h e o r e m 3 . 2 0 is p r o v e d in [ 11] as a c o n s e q u e n c e of the identity

(3.19).

The c o r o l l a r y f o l l o w s f r o m [ 9 ] if w e note that

d i s t r i b u t i o n on

of d e g r e e

O b s e r v e that w h e n equation

K

is a h o m o g e n e o u s

-2n.

n = l , then

is a s o l u t i o n of L e w y ' s

-42(3.22) if

Thus w e have

3.23

Theorem.

Given

f , then e q u a t i o n ( 3 . 2 2 ) has a s o l u t i o n in a

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

if and o n l y if the C a u c h y - S z e g o i n t e g r a l

is r e a l - a n a l y t i c in a n e i g h b o r h o o d of

3.24

Corollary.

P.

If the a b o v e c o n d i t i o n f o r

f

is s a t i s f i e d and if

to one of the s p a c e s

f

belongs

then w e can

find a v w h i c h b e l o n g s to respectively, for appropriate neighborhoods

of

P.

A c t u a l l y , the n e c e s s i t y p a r t of the c o n d i t i o n of T h e o r e m 3 . 2 3 g i v e s the f o l l o w i n g s t a t e m e n t : of the point

P ji

C(f)

L ( v ) = f has no s o l u t i o n in any n e i g h b o r h o o d

cannot be continued a n a l y t i c a l l y a c r o s s

p a r t i c u l a r , if

f

f u n c t i o n in

w h i c h has no a n a l y t i c e x t e n s i o n a c r o s s

P ; in

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

The n e x t r e s u l t " e x p l a i n s " "relative fundamental solution"

for

P.

the e x i s t e n c e and d e r i v a t i o n of the .

We define

by

(3.25)

See the w o r k of Sato, Kawai , and K a s h i w a r a [ 3 0 ] , f o r a r e l a t e d r e s u l t . F o r this s e e G r e i n e r , Kohn, and Stein [ 1 1 ] .

-43and set

3.26

T h e n w e have

Proposition.

(i)

f o r all

in s o m e n e i g h b o r h o o d

of (ii)

w h e r e the l i m i t is taken p o i n t w i s e , and in the s e n s e

of d i s t r i b u t i o n s . Proof.

(i) f o l l o w s f r o m a s i m p l e c o m p u t a t i o n .

A s f o r ( i i ) , we

write

T h i s c o m p l e t e s the p r o o f . F o r future r e f e r e n c e w e state 3.27

where

Proposition.

The s y m b o l ,

is g i v e n by

is f i r s t u s e d in (1 . 11). T h i s f o l l o w s e a s i l y b y taking the F o u r i e r t r a n s f o r m of the f u n c t i o n , and using (3. 11).

P a r t II.

P a r a m e t r i x f o r the g-Neumann p r o b l e m

Guide to P a r t II T h e p u r p o s e of t h i s p a r t i s to r e d u c e the g-Neumann p r o b l e m to the i n v e r s i o n of 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 •*" d e f i n e d o n t h e b o u n d a r y ; thereby one obtains a parametrix, an approximate "Neumann" operator, which gives a solution for our original problem, modulo controllable error terms . S i n c e t h e g i s t of t h e m e t h o d i s t o r e d u c e t h e q u e s t i o n t o t h e b o u n d a r y , bM, and there to approximate by the Heisenberg group, certain prelim­ i n a r y p r o b l e m s m u s t b e d e a l t w i t h , w h i c h w e now d e s c r i b e . Admissible coordinates Under the assumption that bM is strongly pseudo-convex we can i n t r o d u c e , f o r e a c h fixed

£ebM, a basic mapping

η

ζ , T)) of a

neighborhood in bM ( c e n t e r e d a t ξ) to a neighborhood of the o r i g i n in the Heisenberg group.

We c a n v i e w t h i s m a p p i n g a s g i v i n g a n a d m i s s i b l e

coordinate system.

It p l a y s m a n y r o l e s .

s y s t e m a b a s i s of the v e c t o r f i e l d s in

F o r e x a m p l e , in t h i s c o o r d i n a t e '^'(bM) is well approximated

( a t ξ) b y the c o r r e s p o n d i n g s t a n d a r d b a s i s in the H e i s e n b e r g g r o u p ; s e e Proposition (4.3).

M o r e o v e r if

f — > C k(y ^ x ) f ( y ) d y i s t h e e x a c t f o r m H η of an o p e r a t o r in the H e i s e n b e r g g r o u p d e r i v e d f r o m the c o m p l e x s t r u c t u r e ,

the o p e r a t o r f — > { k ( u ( £ , η)) f (ξ)((p) = 1,

then

h a s a known p a r a m e t r i x K , and s o the a p p r o x i m a t e i n v e r s e to c f

t u r n s out to b e The c a s e

-KQ .

n=l

When n = l ,

Q

b

i s not i n v e r t i b l e .

T h i s f a c t i s i n t i m a t e l y connected

w i t h the n o n - s o l v a b i l i t y of the L e w y equation. of C h a p t e r 3 , one c a n c o n s t r u c t an o p e r a t o r approximately, where

H o w e v e r u s i n g the r e s u l t s K,

i s e s s e n t i a l l y the p r o j e c t i o n o p e r a t o r on the

b o u n d a r y v a l u e s of a n t i - h o l o m o r p h i c f u n c t i o n s . acteristic variety D+ + E ,

where E

order

0 whose symbol is

We c a n a l s o w r i t e

i s a n o r d i n a r y 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 of

E

=1

and

C

n e a r t h e c h a r a c t e r i s t i c v a r i e t y of Q + .

b

are approximately orthogonal since

t h e i r s y m b o l s ( r e s p e c t i v e l y of t y p e s

and

support.

D+

T h u s a n a p p r o x i m a t e i n v e r s e of

K+ =E KD - E Q ,

Now a w a y f r o m i t s c h a r ­

is elliptic, and hence invertible.

I=E

The "projections"

s o that K Q , = I - C 1 b b

with Q

acteristic variety, i.e.,

a n i n v e r s e of

Qn+=E

^iave disjoint i s g i v e n by - K + , w h e r e Q1"

away from its char­

approximately.

The approximate Neumann operator A s a r e s u l t of t h e a b o v e a n a l y s i s w e g e t t h a t t h e a p p r o x i m a t e N e u ­ m a n n o p e r a t o r s (the i n v e r s e t o o u r o r i g i n a l p r o b l e m ) i s g i v e n b y ( 9 . 2 3 ) , (or (10.29)). T h e r e g u l a r i t y p r o p e r t i e s of t h e v a r i o u s o p e r a t o r s t h a t m a k e u p t h e N e u m a n n o p e r a t o r a r e t h e s u b j e c t of P a r t I I I .

-48C h a p t e r 4.

A d m i s s i b l e c o o r d i n a t e s on s t r o n g l y

pseudo-convex CR manifolds Let

be a CR m a n i f o l d , i . e . , a real oriented

dimension

m a n i f o l d of

t o g e t h e r with a subbundle

c o m p l e x i f i e d tangent bundle

of the

satisfying

(a) (b) (c)

is i n t e g r a b l e in the s e n s e of F r o b e n i u s , i . e . ,

a r e s e c t i o n s of

if

then so is t h e i r L i e b r a c k e t

Now w e r e s t r i c t o u r attention to a l o c a l c o o r d i n a t e p a t c h d e n o t e a b a s i s f o r the tangent bundle that

yield a basis for

on

Let such

where

(4. 1) If

is s t r o n g l y p s e u d o - c o n v e x w e m a y a s s u m e , as w e shall s e e

l a t e r , that

have b e e n c h o s e n to s a t i s f y the f o l l o w i n g c o m -

mutation relations. (4.2) where

denotes " e r r o r t e r m s , "

V i a the e x p o n e n t i a l m a p p i n g F o l l a n d and Stein [ 9 ] c o n s t r u c t e d n o r m a l c o o r d i n a t e s in

U and s h o w e d , that in t h e s e c o o r d i n a t e s

a r e equal to t h e i r a n a l o g u e s on the H e i s e n b e r g g r o u p m o d u l o " e r r o r

terms."

-49See [ 9 ],

14.

In the s a m e p a p e r they gave a m o r e d i r e c t

c o n s t r u c t i o n f o r the n o r m a l c o o r d i n a t e s in the c a s e when

"geometric" is the s t r o n g l y

p s e u d o - c o n v e x b o u n d a r y of a c o m p l e x analytic m a n i f o l d ; s e e [ 9 ] ,

§18.

The p u r p o s e of this c h a p t e r is to show that the " g e o m e t r i c "

con-

s t r u c t i o n of n o r m a l c o o r d i n a t e s can b e c a r r i e d out on a r b i t r a r y s t r o n g l y pseudo-convex CR manifolds. 4.3

Proposition.

Let

H e r e is the m a i n r e s u l t .

U b e a c o o r d i n a t e n e i g h b o r h o o d on ^

coordinates

with

T h e r e exists a smooth mapping

(4.4) s u c h that, if

and

then w e can w r i t e

(4. 5)

and

u has the f o l l o w i n g p r o p e r t y :

fix

Then

g i v e s a c o o r d i n a t e r e p r e s e n t a t i o n of T h e n in this c o o r d i n a t e s y s t e m ( w h i c h is c e n t e r e d at g) w e can w r i t e

(4.6)

(4.7)

amd

-50-

(4. 8)

w h e r e w e u s e d the notation

with (4.9)

Proof.

Let

(4.10)

Fix

as a b a s e p o i n t .

W e i n t r o d u c e new c o o r d i n a t e s

L as f o l l o w s (4.11)

where

C o n s e q u e n t l y w eW have e substitute

Now we have

-51-

and o b t a i n

F o r the sake o f s i m p l i c i t y w e s e t

(4.12)

Then we can write

(4.13)

(4.14)

i = l , . . . , 2n.

W e i n t r o d u c e new v a r i a b l e s

(4.15)

(4.16) Then

and

We a s s u m e that

as f o l l o w s

-52-

Substituting this into

w e obtain

The c o n c l u s i o n s of the p r o p o s i t i o n r e q u i r e that w e c h o o s e the so that

that is

-53-

N o w , in

the c o e f f i c i e n t of

v a n i s h e s , b e c a u s e the s p a c e of a n t i - h o l o m o r p h i c

v e c t o r f i e l d s is c l o s e d u n d e r b r a c k e t s .

T h e r e f o r e (4. 13) and (4. 14) e a s i l y

yield

Similarly,

in

the T h e rc oe ef of friec i e n t of

is

In o t h e r w o r d s

- 54a o 0(;) = a 00(;)' JP PJ

for

j=l,ooo,n,

and

a (0+ )0(;) - a(O ) 0(;) = 460 , P J n J+n P JP 1

~

-

j, P

~

-

no

From the above we obtain

j,k=l,ooo,n o Since

a 0 0(;) JP

=a

00(;) PJ

as long as

j,pool,ooo,n,

we can write

(4017)

1 do (;) = --2 (a o 0(;) + a 00(;))' JP JP PJ

j,p=l, 0 0 0,2n,

with j:f. p+n, P f: j+no

P

I- j+n and j I- p+n,

On the other hand

aO(O )O(~)=a(O )00(~;J+4, J J+n J+n J and we need do(o )(~) = 2 - aO(o+ )O(~), J J+n J J n d(o )0(;) = -2-a(0 )00(;)0 J+n J J+n J This implies do(o+ )(~) -d(o+ )o(~) = 4 - (aO(O )0(;) - a(O )00(;)) = 4 - 4 = 0, J J n J n J J J+n J+n J I

==>

dj(j+n)(;) = d(j+n)/;) = -Z(aj(j+n)O(;) + a(j+n)jO(;))o

Thus, finally we have (40 18) j,k=I,000,2n o

-55Now we a r e r e a d y to d e r i v e (4. 5). o r d e r t e r m s in

For

a r e g i v e n by

the f i r s t Since

w e have (4. 19) F r o m ( 4 . 1 3 ) w e m a y take (4.20) T o find

we set

(4.21)

w h i c h is equal to (4. 15) m o d u l o

) where

T h i s c h a n g e d o e s not a f f e c t the p r e v i o u s c o m p u t a t i o n s and it w i l l g r e a t l y simplify our final f o r m u l a s .

Therefore (4.22)

Now

Thus w e c o n s i d e r

-56-

Thus (4.23)

Thus the

(4.24)

a r e d e t e r m i n e d and we have p r o v e d P r o p o s i t i o n 4. 3

Corollary.

Then

w h e r e we s e t Next w e d i s c u s s the b e h a v i o r of

u when interchanging

and

T h e r e s u l t i n g s y m m e t r y , o r m o r e p r e c i s e l y , the a p p r o x i m a t e v e r s i o n g i v e n b e l o w , p l a y s a u s e f u l r o l e in v a r i o u s e s t i m a t e s .

See [ 9 ],

§15.

-57( 4 . 2 5)

Lemma.

W e have

(4.26) (4.27)

Proof.

w h i c h p r o v e s (4. 27).

Next

e

A s f o r (4. 2 6 ) , u s i n g e i t h e r (4. 15) o r ( 4 . 2 1 ) w e obtain

-58-

where

Now

hence

4T.h2u8s if C owreo lwant l a r y . w(4. h i c26) Ifh isto just hold ( 4w.e1 8 r) .eand q uTi h rU ei s p isr osvuefsf i cLi e nmt m l ya s m ( 4a.l2l 5, ) .then

-59-

is e q u i v a l e n t to

4.29

Definition.

s a t i s f y i n g ( 4 . 2 ) in

Let U.

r e p r e s e n t o u r f i x e d b a s i s of W e shall s a y that

is an a d m i s s i b l e c o o r d i n a t e s y s t e m in whenever

U c e n t e r e d at

e x p r e s s e d in the c o o r d i n a t e s

has the

f o r m (4. 6 ) , ( 4 . 7 ) and ( 4 . 8 ) . 4 . 30

Proposition.

in the s e n s e that, if

All admissible coordinate s y s t e m s are equivalent U is s u f f i c i e n t l y s m a l l and

admissible coordinate system,

e(^,|j)

is a n o t h e r

then

(i)

(ii)

There exist

s u c h that

w h e r e w e set

similarly for

Proof. w e have

Therefore

(i) = > ( i i )

so w e have to p r o v e (i).

A c c o r d i n g to the h y p o t h e s e s

- 6 0 -

where e

coordinates.

Since

e

stands f o r any d e r i v a t i v e w i t h r e s p e c t to the

To continue

is a d m i s s i b l e

and s i n c e w e have A s i m i l a r c o m p u t a t i o n s h o w s that

T h i s p r o v e s (i) of P r o p o s i t i o n 4 . 30. A n i m p o r t a n t set of c o o r d i n a t e s a r e the n o r m a l c o o r d i n a t e s , a r e d e f i n e d in an invariant f a s h i o n , and c o n s t i t u t e an a d m i s s i b l e system.

T o be m o r e p r e c i s e , f i x

and l e t

which

coordinate

d e n o t e the e x p o n e n t i a l

-61m a p at

b a s e d on the f r a m e

That i s , f o r

s u f f i c i e n t l y c l o s e to the o r i g i n in endpoint

define

to be the

of the i n t e g r a l c u r v e with

T h e n , a c c o r d i n g to [ 9 ],

dinate m a p p i n g admissible.

of the v e c t o r f i e l d 14,

the n o r m a l

, in s o m e s u f f i c i e n t l y s m a l l n e i g h b o r h o o d of

cooris

F r o m o u r P r o p o s i t i o n 4 . 30 and T h e o r e m 14. 10(d) of [ 9 ]

w e i m m e d i a t e l y have (4.31)

Proposition.

borhood

U in_

Let Let

there exists constants

vary over a sufficiently small neigh-

u be an a d m i s s i b l e c o o r d i n a t e s y s t e m .

Then

s u c h that

(4. 32) (4. 33) F i n a l l y , w e c o m p u t e the v o l u m e e l e m e n t . m e t r i c on basis for hood,

i.e.,

(4.34) (4.35) (4.36) Set

s u c h that

Suppose we are given a is an o r t h o n o r m a l some coordinate neighbor-

-62T h e n (4. 34) and ( 4 . 3 5 ) y i e l d

and

C o n s e q u e n t l y w e have

Hence

is an o r t h o n o r m a l b a s i s f o r

where we set

Then

where we set or detg(y) 4.37 Proposition.

In o t h e r w o r d s Thus w e have d e r i v e d Suppose w e a r e g i v e n a m e t r i c on

s u c h that

-53and

s a t i s f y (4. 34), (4. 35) and (4. 36).

(4.38)

in s o m e c o o r d i n a t e n e i g h b o r h o o d

(4.39)

Then

Set

Chapter 5.

Levi metrics

Let M be a sub-domain with smooth boundary bM of a complex manifold M'.

Then to each P point of

a Hermitian form on T ' ^(bM)|p.

bM one can assign a Levi-form,

(This Levi-form is not unique, but

is determined up to a positive multiple. )

The assumption that M

is

strongly pseudo-convex means that this form is strictly positive definite at each point P s bM.

The purpose of this chapter is to give an explicit

construction of a Hermitian metric on M, which restricted to coincides with the Levi form (i. e ., (5.7) is satisfied). metric a Levi metric.

We always

T^ ' "(bM)

We call such a

work with such a metric because the

boundary theory in [ 9 ] requires it. Let

r

be a C

CO

real-valued function on an

n+1

dimensional complex

analytic manifold M' and let M= { r ' > 0 } c M' be a relatively compact domain with boundary bM. dimensional real

C

CO

of

'

manifold.

p seudo-convex, that is v point P

If

d r ' 4- 0 o n b M t h e n b M

is a

2n+l

We shall assume that M is strongly

f r' _ V . , , is positive definite at every y z.z. Ji, j= l , . . . ,n+1 ^ ι J bM on vectors (a, , . . . , a , ) which satisfy the side condition 1 n+1 1

n+1 Σ a.r' (P) = 0. j=l 3 j

(5.1)

This condition is invariant under holomorphic coordinate transformations. Here

(zj> · · · >z

) denotes an arbitrary analytic coordinate system in

some neighborhood of

PsbM.

According to Proposition 4 . 4 of [21] there

* See however the remarks in the concluding section of the Introduction.

-65exists a positive number

A , s u c h that

induces a positive definite Hermitian m e t r i c ,

namely

(5.2)

in s o m e n e i g h b o r h o o d

V of

bM.

(This is only a p r e l i m i n a r y m e t r i c ,

not the f i n a l one c o n s t r u c t e d . ) L e t

d e n o t e the i n n e r p r o d u c t induced

b y this m e t r i c on the c o m p l e x i f i e d tangent bundle V.

Choose a neighborhood

(p on

M'

s u c h that

U on

of

bM, and

of

and a p o s i t i v e v a n i s h e s o u t s i d e of

function V.

Let

d e n o t e an a r b i t r a r y p o s i t i v e d e f i n i t e H e r m i t i a n m e t r i c on

M'

and set

(5.3)

We shall show that the m e t r i c to the L e v i f o r m on Let to

bM, i.e.,

reduces

bM. d e n o t e the set of h o l o m o r p h i c v e c t o r f i e l d s tangent

in a n a l y t i c l o c a l c o o r d i n a t e s

-66The Levi f o r m

is an H e r m i t i a n i n n e r p r o d u c t on

d e f i n e d as f o l l o w s .

Set

(5.4) Clearly

on

bM.

Let

We define

(5.5) w h e r e the r i g h t - h a n d s i d e d e n o t e s the u s u a l c o n t r a c t i o n o p e r a t i o n b e t w e e n tangent and c o t a n g e n t v e c t o r s and (• , • )

We denote by

the inner p r o d u c t g i v e n by o u r m e t r i c (5. 3).

5. 6 P r o p o s i t i o n .

Let

.

Then

(5. 7) f o r all

P e bM.

We also have

consequently (5.8) f o r all Proof.

W e c h o o s e an a p p r o p r i a t e c o o r d i n a t e s y s t e m in w h i c h to

c a r r y out o u r c a l c u l a t i o n s . is tangent to

We m a y as w e l l a s s u m e that the h y p e r p l a n e bM

at P

and

P

is the o r i g i n of the c o o r d i n a t e

-67system.

Thus

w h e r e f r o m T a y l o r ' s f o r m u l a w e have

Next w e m a k e the f o l l o w i n g a n a l y t i c change of v a r i a b l e s

In t h e s e c o o r d i n a t e s

a s s u m e s the s i m p l e r f o r m

Thus w e m a y as w e l l a s s u m e that

is of the f o r m

(5.9) where

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

p r o v e (5. 7) it s u f f i c e s to c o m p u t e the two s i d e s at P = (0, . . . , 0) c a s e that

and

b e l o n g to a b a s i s of

To in the

Locally such a

b a s i s is g i v e n b y

That this is a l o c a l b a s i s f o l l o w s by noting that is a b a s i s of

at

P.

Now at

P , that is at

and z = (0,...,0).

- 6 8 -

the r e f o r e

On the o t h e r hand a s i m p l e c o m p u t a t i o n y i e l d s

H e r e w e evaluated the c o e f f i c i e n t s at the o r i g i n .

Finally,

Therefore Next w e p r o v e (5. 8). type

(1,0)

and

T o this end w e note that if

in l o c a l a n a l y t i c c o o r d i n a t e s ,

(5.10)

where

(5.11)

f

is the i n v e r s e of

'

,

that is

is a f o r m of then

-69where,

similarly,

is the i n v e r s e of

Therefore

and

and b y the i n v a r i a n c e u n d e r h o l o m o r p h i c c o o r d i n a t e c h a n g e s w e have p r o v e d Proposition (5.6). Examples We give two e x a m p l e s of L e v i m e t r i c s , (i)

Let

b e the unit b a l l in

T h e n the usual E u c l i d e a n m e t r i c

is a L e v i m e t r i c . (ii)

Let

M

b e the d o m a i n

!

T h e n t h e r e is a L e v i m e t r i c on ties, among others: of

M

(a)

w h i c h has the f o l l o w i n g p r o p e r -

The m e t r i c is i n v a r i a n t u n d e r the t r a n s l a t i o n s

c o m i n g f r o m the H e i s e n b e r g g r o u p

(b)

The d i s t a n c e of a

point f r o m the b o u n d a r y is g i v e n by a r e d e s c r i b e d in the e x a m p l e at the end of C h a p t e r 17.

Further details

-70C h a p t e r 6.

on ( 0 , 1 ) - f o r m s

F r o m now on w e a s s u m e a f i x e d H e r m i t i a n m e t r i c w h i c h s a t i s f i e s ( 5 . 7 ) on

b M , (a " L e v i " m e t r i c ) .

h o o d of (6.1)

bM,

M

The following analysis

is done in a f i x e d b o u n d a r y c o o r d i n a t e n e i g h b o r h o o d geodesic distance f r o m

on

U.

Let

p denote

b M , at l e a s t in s o m e s u f f i c i e n t l y s m a l l n e i g h b o r -

in M

and

o u t s i d e of

d e n o t e an o r t h o n o r m a l b a s i s f o r d e n o t e the dual b a s i s f o r

in .

Let U and let

We have

(6.2)

or,

equivalently

(6.3) for

functions

f.

( 6 . 3 ) i m p l i e s that in l o c a l c o o r d i n a t e s

c o m p l e x c o n j u g a t e of

N o w , if

and t h e r e f o r e

a r e tangential h o l o m o r p h i c v e c t o r f i e l d s .

F u r t h e r m o r e so is (6.4)

, n,

is the

because

then

-71Finally

(6.5)

since

W e shall a s s u m e a b o u n d a r y c o o r d i n a t e s y s t e m in

U in w h i c h the m e t r i c has the f o r m

T h e v o l u m e e l e m e n t is (6.6)

where (6.7)

Then

Next w e c o m p u t e the ad j o i n t s .

* W e p o i n t out that

is not

Let

u be a

of (5. 3).

f u n c t i o n in

U

and

-72w h e r e we used (6. 5) and set (6.8)

Thus w e have

(6.9)

and the f i r s t Neumann b o u n d a r y c o n d i t i o n is g i v e n by

(6.10)

w h e r e the i n t e r i o r p r o d u c t is d e f i n e d as f o l l o w s . Then

otherwise

if ( 6 . 1 1 ) Next w h e r e w e set

(6.12) We note that

Let

-73because T o c o m p u t e the adjoint w e set

Then

Choosing

w e obtain the f i r s t N e u m a n n b o u n d a r y

c o n d i t i o n on

(0,2) forms,

namely,

(6.13)

T o s i m p l i f y the notation w e s e t

and

S i m i l a r l y w e set

Then

-74-

(6.14)

F i n a l l y w e shall t r a n s l a t e the f i r s t Neumann b o u n d a r y c o n d i t i o n ( 0 , 2 ) f o r m s into the s e c o n d Neumann b o u n d a r y c o n d i t i o n forms.

y

on

on (0, 1)

T h i s r e q u i r e s that ( 6 . 1 1 ) s a t i s f i e s ( 6 . 1 3 ) , that is

(6.15)

j=l , 2 , . . . , n .

If w e a s s u m e that

a l r e a d y s a t i s f i e s the f i r s t

Neumann b o u n d a r y c o n d i t i o n

then

(6.16)

a r e v e c t o r f i e l d s tangential to

bM.

Next w e c o m p u t e the c o m p l e x L a p l a c i a n forms.

T h i s is a lengthy but not d i f f i c u l t c a l c u l a t i o n .

the m a i n s t e p s .

F o r s i m p l i c i t y of notation w e shall f r o m now o n d r o p

the s u p e r s c r i p t f r o m Let

We shall only give

be a

and w r i t e it as (0,l)-form.

Then

-75-

where (6.17) where

is d e f i n e d b y ( 6 . 1 2 ) .

of the v e c t o r f i e l d s boundary) with

( w h i c h a r e tangential at the c o e f f i c i e n t s and

i . e . , multiplication by

functions.

a r e d e f i n e d in (6. 8). as f o l l o w s .

stands f o r l i n e a r c o m b i n a t i o n s

stands f o r t e r m s of o r d e r z e r o , Similarly

Together these f o r m u l a s yield

-76-

W e a r e g o i n g to put

•

into d i a g o n a l f o r m , m o d u l o

t e r m s , but f i r s t w e m a k e a f e w r e m a r k s

"remainder"

We write

(6.18) where

T

therefore

is a r e a l v e c t o r f i e l d .

T

has unit l e n g t h .

d u a l to e a c h o t h e r ,

because

Then

N e x t w e n o t e that

T

and

are

-77If

r

is d e f i n e d b y ( 5 . 4 ) then

(6. 19) on

b M , b e c a u s e b o t h s i d e s annihilate

have unit l e n g t h on (6.20)

Lemma.

Proof.

both sides

b M - - s e e (5. 8 ) - - a n d they point in o p p o s i t e d i r e c t i o n . For

w e have

Clearly

and w e have (6.21)

where we used Proposition 5 . 6 . T o put o u r f o r m u l a s in f i n a l f o r m let us r e c a l l o u r notation f o r "remainder terms. " is a m a t r i x of

functions,

d e n o t e s a l i n e a r c o m b i n a t i o n of coefficients, also let

stands f o r stand f o r an

with

f u n c t i o n s w h i c h v a n i s h on

bM.

We

m a t r i x of t a n g e n t i a l d i f f e r e n t i a l

-78operators,

i . e . , v e c t o r f i e l d s that do not i n c l u d e

is z e r o e x c e p t , p o s s i b l y , in its The t e r m f a c t that vector field.

6.22

(6.23)

s u c h that

column or

c o m e s f r o m the t e r m s on the b o u n d a r y e q u a l s

row. and the

an ( i m a g i n a r y ) tangential

T h e n L e m m a 6 . 2 0 and the p r e v i o u s c a l c u l a t i o n s y i e l d

Proposition.

Let

Then

-79C h a p t e r 7.

L o c a l s o l u t i o n of the D i r i c h l e t p r o b l e m f o r

Let

b e the i n v a r i a n t l y d e f i n e d " c o m p l e x n o r m a l "

holomorphic covector near from

•

bM.

Here

p stands f o r g e o d e s i c d i s t a n c e

bM.

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

bM,

where

(7.2)

C tions to

l

e

a

r

l

y

,

bM.

where

stands f o r the r e s t r i c -

s o l v e the f o l l o w i n g " D i r i c h l e t "

problem

(7. 3) (7.4) (7.5)

T h e p u r p o s e of this c h a p t e r is to c o n s t r u c t , l o c a l l y . operator

G : (f;h, g ) — > u .

Green's

The c o n s t r u c t i o n is quite t e c h n i c a l ,

therefore

w e b e g i n w i t h a q u i c k s k e t c h of the m a i n idea behind it (and is not intended to b e p r e c i s e ) . Let • (v) = f .

E

be a f u n d a m e n t a l s o l u t i o n of

We set w = u - v ,

u

g i v e n in (7. 3).

and (7. 5) w e o n l y need to s o l v e f o r (7.6) (7.7)

i.e.,

v = E(f)

solves

T h e n to s o l v e (7. 3), (7. 4)

-80(V.

8)

In g e n e r a l s u c h a w

is not u n i q u e ,

c o n s i d e r a t i o n s w e s h a l l a s s u m e that it i s .

but f o r the sake of h e u r i s t i c a l T h e n the o p e r a t o r

P,

(7.9) is c a l l e d the P o i s s o n o p e r a t o r f o r f i r s t a fundamental solution 5

d e n o t e the 5 - f u n c t i o n of

E. bM,

•.

To construct

P

one c o n s t r u c t s

Next let

Let

i.e.,

(7.10)

Then bM.

can be c o n s i d e r e d as a (0, l ) - f o r m on

w i t h s u p p o r t in

Consequently

(7.11) D e f i n e the o p e r a t o r

by

(7.12) Then (7.13) is an e l l i p t i c 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 on its i n v e r s e .

bM.

T h e n , a c c o r d i n g to ( 7 . 1 1 ) the o p e r a t o r

Let P

denote defined by

(7.14) i s the P o i s s o n o p e r a t o r .

In the r e s t of this c h a p t e r w e s h a l l u s e t h e s e

i d e a s to c o n s t r u c t e x p l i c i t l y l o c a l v e r s i o n s of the o p e r a t o r s * since

has z e r o - o r d e r

terms.

E , G and

P

The existence and properties of these operators are by now well known.

The purpose of the following lengthy calculations is to obtain pre­

cise local expressions for these operators which will be essential for the construction of the approximate Neumann's operator in Chapters 9 and 10. We use the calculus of pseudo-differential operators and boundary layer potentials.

As such our treatment is based on the approach developed in

detail by A. Calderon [ 5 ], L. Hormander [18], R. Seeley [31] and L. Boutet de Monvel [ 2 ]. From the known a priori estimate for the Q-Neumann problem it is clear that we have to keep track of the principal part of the operators involved and also of those t e r m s whose degree of homogeneity is one less than that of the principal part. fixed local coordinates.

This makes sense as long as we work in

Our final results concerning regularity and

existence of solutions will, of course, be stated invariantly. We fix, once and for all, a boundary coordinate neighborhood U c M' with coordinates a subset of

M

2n , 2

(χ, ρ), χ = (χ , . . . , χ ). X Ht 1

so that

U

is identified with

U Π bM = {(χ, ρ) € U; ρ= 0}.

L e t u s now b r i e f l y r e c a l l the m a i n f a c t s about p s e u d o - d i f f e r e n t i a l operators.

In v i e w of the notation w e h a v e adopted i t w i l l b e s i m p l e r if

w e d e s c r i b e m a t t e r s in the s e t t i n g of

bM, i . e . , l o c a l l y in ] R ^ n + ^ .

c o r r e s p o n d i n g s t a t e m e n t s f o r the n e i g h b o r h o o d s of

The

M r e q u i r e then o n l y

the notational addition of two v a r i a b l e s . By a "classical" pseudo-differential operator of order an operator

T

whose symbol

p ( x , £) b e l o n g s t o t h e c l a s s

k, we mean

k S =

k

;

-82i.e.,

p

is j o i n t l y

and s a t i s f i e s f o r all

The o p e r a t o r

T

in x a

and

and

d e f i n e d by

p

is g i v e n as

We s h a l l s o m e t i m e s w r i t e If

and

has c o m p a c t s u p p o r t in the x - v a r i a b l e

T

as

so as to i n d i c a t e its o r d e r .

a r e two s u c h 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

to s y m b o l s o p e r a t o r of o r d e r

and

(corresponding

then t h e i r p r o d u c t is a p s e u d o - d i f f e r e n t i a l whose symbol

po q

has an a s y m p t o t i c d e v e l o p m e n t

in the s e n s e that

F o r a rapid and h i g h l y r e a d a b l e i n t r o d u c t i o n to 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 s e e N i r e n b e r g [ 26]. o u r o p e r a t o r s o p e r a t e on

It -w-ll be c l e a r f r o m the c o n t e x t w h e t h e r o r on

bM.

w i l l d e n o t e the s y m b o l of

A pseudo-differential operator w h o s e s y m b o l is independent of

w i l l be d e n o t e d b y

d e f i n e d on W o r k i n g in

l o c a l c o o r d i n a t e s w e w i l l not d i s t i n g u i s h b e t w e e n f o r m s and v e c t o r f u n c t i o n s W e b e g i n b y c o m p u t i n g the s y m b o l ,

of

•.

-83-

and

Similarly

T h e r e f o r e , using the notation of P r o p o s i t i o n 6. 22 we have 7.15.

Lemma.

The s y m b o l

of

is g i v e n by

(7.16)

Here

stands f o r m u l t i p l i c a t i o n by f u n c t i o n s .

-84Next w e c o m p u t e the s y m b o l of the f u n d a m e n t a l s o l u t i o n in

U.

E

for

Let

(7.17) d e n o t e the usual a s y m p t o t i c e x p a n s i o n of the s y m b o l

e

of

E.

Let (7. 18) b e the d e c o m p o s i t i o n of the s y m b o l (7. 16) in d e c r e a s i n g o r d e r of h o m o g e neity (in

and

so that

has d e g r e e

.

Write

(7. 19) where Then we c a n obtain the f i r s t t e r m of the a s y m p t o t i c e x p a n s i o n as the i n v e r s e of the h i g h e s t o r d e r t e r m in (7. 18), i e . ,

(7.20)

F o l l o w i n g the c o m p o s i t i o n f o r m u l a f o r s y m b o l s to the next o r d e r , we have

Therefore

-85T h e t e r m s in the b r a c k e t can b e e a s i l y c o m p u t e d as f o l l o w s ,

and

where

and

have c l a s s i c a l s y m b o l s of o r d e r one and t w o ,

t i v e l y , w h i c h do not d e p e n d on

respec-

C o l l e c t i n g t e r m s w e obtain

(7.21)

R e c a l l that (7.22)

F o l l o w i n g the h e u r i s t i c d i s c u s s i o n c u l m i n a t i n g in (7. 14) w e shall apply

E

to f o r m s s u p p o r t e d on

bM,

T h e r e f o r e we

-86i n t e g r a t e out the n o r m a l c o m p o n e n t

of the c o t a n g e n t b u n d l e ,

i.e.,

(7.23)

is the s y m b o l o f

E

o p e r a t i n g on

that w e s e p a r a t e the t e r m s in a r e independent f r o m

.

.

These considerations

that d e p e n d on

suggest

f r o m the t e r m s w h i c h

T o this end w e set

(7.24)

and (7.25) Then (7.26)

+ m u l t i p l i c a t i o n by f u n c t i o n s . W e c o l l e c t t h e s e c a l c u l a t i o n s in the f o l l o w i n g f o r m . 7.27.

Lemma.

U o f the o r i g i n in i.e.,

Let

E

b e a f u n d a m e n t a l s o l u t i o n of

•

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

, with coordinates w i t h s u p p o r t in

U.

Then

E

-87is 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 with s y m b o l

i

,

where

and if (7.28) stands f o r the usual a s y m p t o t i c e x p a n s i o n ,

then

(7.29) and (7.30)

Here

and

Remark.

a r e g i v e n b y (7. 24) and ( 7 . 2 5 ) ,

respectively.

L e m m a 7 . 2 7 d e t e r m i n e s the f i r s t two t e r m s of the a s y m p t o t i c

e x p a n s i o n of the s y m b o l of the o p e r a t o r tion that s u c h an o p e r a t o r e x i s t s . p r o v e the e x i s t e n c e of s u c h

E.

E.

T h i s is d o n e u n d e r the a s s u m p -

With a little e x t r a w o r k one can a l s o W e s k e t c h the i d e a .

F i r s t by following

out the c o m p l e t e a s y m p t o t i c f o r m u l a f o r the c o m p o s i t i o n of two s y m b o l s w e can c o n s t r u c t 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 where

S is an o p e r a t o r w h o s e k e r n e l is ( j o i n t l y )

s o that By r e s t r i c t i n g

c o n s i d e r a t i o n to a s u f f i c i e n t l y s m a l l n e i g h b o r h o o d , the n o r m of l e s s than o n e , satisfies

S is then

e x i s t s and thus the o p e r a t o r I.

Since

the l o c a l r e g u l a r i t y of the o p e r a t o r

-88•

i n s u r e s that

an o p e r a t o r with a

kernel.

We now c o m e to the o p e r a t o r s of P o i s s o n t y p e . 7. 31 j

Definition.

is a s y m b o l of P o i s s o n type of o r d e r

j,

an i n t e g e r , if it s a t i s f i e s the f o l l o w i n g c o n d i t i o n s .

(i) t (ii)

has c o m p a c t s u p p o r t in the

(x, p) v a r i a b l e s (in

(iii)

W e note that

p

c a n be a s c a l a r function

o r a m a t r i x function.

a l s o n o t e , that as a r e s u l t of this d e f i n i t i o n , fixed 7. 32

p,

T h e n the m a p p i n g in

is, for each

a s y m b o l of the standard c l a s s

Definition.

,

Let

We

, and is s o u n i f o r m l y .

b e a s y m b o l of P o i s s o n type of o r d e r

P = f — ^ F , of a f u n c t i o n

f

in

j.

to a f u n c t i o n F

g i v e n by

(7. 33)

a s s u m i n g the i n t e g r a l m a k e s s e n s e , is c a l l e d an o p e r a t o r of P o i s s o n type of o r d e r

j.

In p a r t III w e shall study the b e h a v i o r of o p e r a t o r s of P o i s s o n type

-89of o r d e r z e r o on

• and L i p s c h i t z s p a c e s .

T h i s s u f f i c e s to y i e l d

• and

L i p s c h i t z e s t i m a t e s f o r o p e r a t o r s of P o i s s o n type of a r b i t r a r y o r d e r , s i n c e d i f f e r e n t i a t i n g t h e m on the l e f t o r m u l t i p l y i n g t h e m on the right by e l e m e n t s of

again l e a d s to s y m b o l s of P o i s s o n t y p e .

The next point is the r e s u l t that an o p e r a t o r of P o i s s o n t y p e .

acting on f o r m s on

is

We shall p r o v e it is and w e shall c o m p u t e

the r e l e v a n t p a r t of its s y m b o l . Let

E

T h i s is a c c o m p l i s h e d in a s e r i e s of l e m m a s .

when

and v a n i s h in

and l e t

Suppose w i t h s u p p o r t in

V.

Let

E^

d e n o t e the

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 induced b y the s y m b o l We set (7. 34) where (7.35)

7.36.

Lemma.

Let

b e d e f i n e d by ( 7 . 3 4 ) . - 1 .

More precisely

(7. 37)

Then

is an o p e r a t o r of P o i s s o n type of o r d e r

-90-

with s u p p o r t in

where

(7.38)

(ii)

and v a n i s h e s if

(iii)

is a s y m b o l of P o i s s o n type of o r d e r

and -co.

The g i s t of the p r o o f of this l e m m a is the s i m p l e identity (i).

The

d e t a i l e d p r o o f of a m o r e g e n e r a l f o r m of this l e m m a can b e found in H o r m a n d e r [18] , T h e o r e m 2. 14. T o s i m p l i f y m a t t e r s w e u s e the T a y l o r e x p a n s i o n of r e s p e c t to 7. 39.

where

p about

T h i s e a s ily y i e l d s ,

with

(with

Lemma.

and

are symbols

-91of P o i s s o n type of o r d e r

-2

and

-3, respectively,

and

are

evaluated at In a s i m i l a r m a n n e r let

b e induced by the s y m b o l

Define (7.40)

with s u p p o r t in

.

induced b y the s y m b o l

Then

is of P o i s s o n

type, where

(7.41)

and

is a s y m b o l of P o i s s o n type of o r d e r

To calculate

m o r e p r e c i s e l y w e need the f o l l o w i n g e v a l u a t i o n of i n t e g r a l s . that

and

(Note

-92Thus we h a v e the f o l l o w i n g r e s u l t .

7.42.

Lemma.

is induced by the f o l l o w i n g s y m b o l of P o i s s o n type

where

The c o e f f i c i e n t s of

j

in

a r e evaluated at

is a s y m b o l of P o i s s o n type of o r d e r We r e m a r k that an o p e r a t o r of P o i s s o n type of o r d e r

and

-3.

applied to f o r m s on

induces

- 3 , h e n c e its p r e c i s e f o r m is

irrelevant. F i n a l l y , b y s u m m i n g the r e l e v a n t t e r m s in L e m m a s 7. 36, 7. 39 and setting

in the c o e f f i c i e n t s of

and

the n e c e s s a r y e x p r e s s i o n f o r the s y m b o l

we obtain of the o p e r a t o r

as f o l l o w s

-937.43.

Proposition.

The operator

Let

g i v e n by

(7.44) is induced by a s y m b o l

of P o i s s o n type g i v e n by

(7.45)

where (i) (ii)

(iii) of o r d e r (iv)

and and

a r e s y m b o l s of P o i s s o n type

respectively, and v a n i s h e s in

-94and Furthermore (7.46) Let

d e n o t e the r e s t r i c t i o n of

to

i.e., (7.47)

w i t h s u p p o r t in

Then

is an e l l i p t i c 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 w i t h s y m b o l

(7.48)

Let s u f f i c i e n t l y s m a l l , then i.e.,

suppose

we have, (7.49) (i)

Lemma

where

If has an i n v e r s e , with s u p p o r t in

,

V

is

in Then

-95(ii)

(iii)

is 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 w i t h s y m b o l where

(7. 50)

Proof.

The r e s u l t f o l l o w s e a s i l y f r o m the a b o v e by the c l a s s i c a l

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 c a l c u l u s and then by an a d d i t i o n a l a r g u m e n t , as in the R e m a r k f o l l o w i n g L e m m a 7 . 2 7 . W e d e f i n e the o p e r a t o r

P,

(7.51) by (7. 52) Then a c o m b i n a t i o n of ( 7 . 4 5 ) and ( 7 . 5 0 ) g i v e s (7. 53) w h e r e P and

a r e induced b y the s y m b o l s p and

(7.

4

(7.55)

5

)

w

h

e

of p o i s s o n t y p e , and r

e

-96(7.56)

Also

(i) order

and

a r e s y m b o l s of P o i s s o n type of

-2,

(ii) (iii) (7 46) and L e m m a 7 . 4 9 can now be put in the f o l l o w i n g f o r m . 7. 57.

Proposition.

Let

Then (i) (ii)

With a slight a b u s e of language

P

is a " l o c a l P o i s s o n o p e r a t o r . "

B e f o r e w e state the final r e s u l t of this c h a p t e r w e shall need an o p e r a t o r w h i c h e x t e n d s f u n c t i o n s g i v e n in the c l o s e d u p p e r - h a l f to the w h o l e Suppose

space

of and has c o m p a c t s u p p o r t in

U.

Extend

-97f

to

in

in the o b v i o u s w a y ( i . e . , f=0 o u t s i d e U).

N e x t , in

set

(7.58)

where

is r a p i d l y d e c r e a s i n g as

and

(7.59)

Then (7. 60)

and c l e a r l y

Definition.

We shall say that

about this e x t e n s i o n o p e r a t o r ,

in extends

f.

see [33], Chapter VI,

F o r further details 3.

The f o l l o w i n g a d d i t i o n a l r e m a r k s a r e u s e f u l in the a p p l i c a t i o n s : (i)

The o p e r a t o r

(ii)

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

, w h e r e the s e c o n d e x t e n s i o n is of the s a m e type

as (7. 58), e x c e p t

is r e p l a c e d b y

s a t i s f i e s the c o n d i t i o n (7. 59) a l s o , and s i n c e any

Now s a t i s f y i n g (7. 59)

w i l l d o , w e shall not b o t h e r to i n t r o d u c e notation to d i s t i n g u i s h t h e s e extensions. (iii)

If

d e f i n e d in

is any 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 acting on f u n c t i o n s w e extend it to a c t on f u n c t i o n s w h i c h a r e in by setting

(7.61)

-98The p a r t i c u l a r e x t e n s i o n o p e r a t o r (7. 58) (i e . , the p a r t i c u l a r u s e d w i l l not he r e l e v a n t , and w i l l not b e kept t r a c k o f . T h e m a i n r e s u l t of this c h a p t e r c o n c e r n i n g a l o c a l s o l u t i o n of the inhomogeneous Dirichlet problem follows. Let

U b e a b o u n d a r y c o o r d i n a t e n e i g h b o r h o o d in M ' with c o o r d i -

nates

,

if w e identify

s u c h that

U w i t h a s u b s e t of

By c h o o s i n g a s u f f i c i e n t l y

small boundary neighborhood

has an i n v e r s e in w e have the f o l l o w i n g s o l u t i o n to the

inhomogeneous Dirichlet problem for

7.62.

Theorem

(i)

in

V.

Let

and

Let solution for

g i v e n in L e m m a 7. 2 7 and l e t

type d e f i n e d b y (7. 52).

P

E

b e the o p e r a t o r of P o i s s o n

We s e t

(7.63) w h e r e the p a s s a g e f r o m Then (7.64) (7.65) where we set (7.66)

E

to

be the l o c a l f u n d a m e n t a l

is as in ( 7 . 6 1 ) .

-99T h e a b o v e r e s u l t m a y b e v i e w e d as the " e x i s t e n c e " p a r t of the s o l u tion of the D i r i c h l e t p r o b l e m ,

in the l o c a l v e r s i o n w e a r e w o r k i n g in.

T h e t h e o r e m b e l o w l e a d s to the d e s i r e d " r e g u l a r i t y " we c h o o s e another neighborhood

7. 6 7 .

Theorem.

results.

F o r this

so that

Suppose

then in (7.68) where

and

a r e i n t e g r a l o p e r a t o r s with

kernels

and

Proof.

, respectively).

Let

(7.69) and w = u - v . theorem. by (7.63). of onal.

T h e n in

Next let

and

d e n o t e the k e r n e l o f the o p e r a t o r

Here and

P

.

Ug

w e o b s e r v e that

,

while

Here

defined

away f r o m the d i a g -

b e a s m o o t h o p e n d o m a i n in

d e c o m p o s e the b o u n d a r y of

G

F o l l o w i n g t h r o u g h the d e f i n i t i o n s

M o r e o v e r ( 7 . 6 4 ) i m p l i e s that

Next let

, by the p r e v i o u s

and ( 7 . 6 6 )

so that .

W e c a n then

into two p a r t s :

l i e s o n the h y p e r p l a n e

, and

is the r e s t of

-100the b o u n d a r y (lying in

What is c r u c i a l is that

is at a

positive distance f r o m W e apply G r e e n ' s t h e o r e m to the f o r m the m e a s u r e which

w and the r e g i o n

induced by o u r " L e v i - m e t r i c "

using

( s e e C h a p t e r 5), f o r

is s e l f - a d j o i n t . T h e r e f o r e , if

(7. 70)

Here

A

is a c o m b i n a t i o n with s m o o t h c o e f f i c i e n t s of f i r s t d e r i v a -

t i v e s of

while

B - Vw is a l i n e a r c o m b i n a t i o n of

c o e f f i c i e n t s and o f f i r s t d e r i v a t i v e s of

w.

The l e f t - h a n d side of ( 7 . 7 0 ) g i v e s v a n i s h when only.

But w = u - v

Now w h e n

nu

s o the f u n c t i o n s

on

A

and

u = 0 there, hence there then t h e i r d i s t a n c e is

and

B

are smooth.

and

(which a r e s u p p o r t e d i n "

I n t e g r a t i n g out in (7. 70) s h o w s that w o p e r a t o r s with

and

and

by the f o r m u l a s ( 7 . 6 9 ) in t e r m s of

Since both "

the r i g h t - h a n d s i d e (7. 70) n e e d s to be

evaluated on

strictly positive,

G with smooth

k e r n e l s acting on

p r o o f of the t h e o r e m .

Moreover for

can be e x p r e s s e d with

kernels.

is e x p r e s s e d in t e r m s of i n t e g r a l and

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

-101C h a p t e r 8. H e r e w e c o m p u t e the operator

P.

R e c a l l that .

P

R e d u c t i o n to the b o u n d a r y N e u m a n n b o u n d a r y v a l u e of the P o i s s o n is d e f i n e d b y (7. 52) on

W e c h a n g e the m e a n i n g of

P

s l i g h t l y keeping the s a m e

n o t a t i o n , and h o p e f u l l y not i n t r o d u c i n g a m b i g u i t y .

8. 1

Definition.

W e d e f i n e the o p e r a t o r

by

Now let

b e g i v e n by

(8.2)

The f i r s t

Neumann b o u n d a r y c o n d i t i o n is

(8. 3) A c c o r d i n g to (6. 16), (and (7. 1)) the s e c o n d c o n d i t i o n is the v a n i s h i n g of (8.4) W e t h e r e f o r e i n t r o d u c e the s h o r t - h a n d of the o p e r a t o r

as f o l l o w s .

(8. 5) for

with H e r e , w i t h a s l i g h t a b u s e of n o t a t i o n ,

stands f o r the

-102m a t r i x w h i c h is obtained by d r o p p i n g the c o l u m n of

n + l - s t r o w and the

w h i c h a r e z e r o , anyway ( s e e (6. 12)).

8.6.

Definition.

8. 8.

Theorem.

W e d e f i n e the 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

is induced by the f o l l o w i n g s y m b o l

(8.9)

where (8.10)

and

and Proof.

a r e d e f i n e d in P r o p o s i t i o n 7. 57 (ii), (iii). Using the s y m b o l of

w e shall c o m p u t e

P

g i v e n in (7. 54), (7. 55) and (7. 56)

T o s i m p l i f y m a t t e r s f i r s t we note that the

r e l e v a n t s y m b o l s a r e d i a g o n a l with the e x c e p t i o n of a c c o r d i n g to (7. 55) tion of

is d i a g o n a l .

So is

Namely,

by (7. 56) with the e x c e p -

-103and the e r r o r t e r m s

T h e s e e r r o r t e r m s w i l l b e i n c o r p o r a t e d in the e r r o r t e r m s of f o r e w e shall n e g l e c t t h e m . containing an

and

,

there-

are diagonal, except for terms

, w h i c h , a c c o r d i n g to o u r c o n v e n t i o n is c o n s i d e r e d to b e

n X n m a t r i x , and f o r the t e r m s

and

d o e s not e v e n e n t e r the c a l c u l a t i o n and error terms ( 8 . 9 ) is s t r a i g h t f o r w a r d .

, thus n e g l i g i b l e .

c o n t r i b u t e s o n l y to the Now the c a l c u l a t i o n of

N a m e l y , if w e l e a v e o f f

(p and

we obtain

-104-

Simplifying we obtain

w as (T 88.h.1ie11 sfr2o.e)lpl roow Dvseeand fsi n T i thi oeno w .r eem r eW8de.e8fd.ienfei dn e inthe Prp op s eo u s idtoi -ob dnyi f f7. geirv5ei7n(gti ii a)its ,l ( i oisipy ) em rand ab tool ,r

-105W e need the f o l l o w i n g t e c h n i c a l r e s u l t . 8.13

Lemma.

of o r d e r z e r o .

Proof.

Let

denote a c l a s s i c a l 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

Then

It s u f f i c e s to c o n s i d e r a t e r m of the f o r m ( t y p i c a l of what

o c c u r s in the c o m p o s i t i o n f o r m u l a f o r s y m b o l s ) .

This proves L e m m a 8.13. 8.14

(i)

and (ii)

Proposition.

W e have

- 1 0 6 -

Proof•

W e shall d e r i v e (i).

The d e r i v a t i o n of (ii) is s i m i l a r .

By L e m m a 8. 13 it s u f f i c e s to w o r k w i t h the " r a w s y m b o l s "

and

Then (8. 15)

Thus w e a r e l e f t w i t h c o m p u t i n g the s u m

This yields

-107Now

Therefore

Finally we consider

- 1 0 8 -

T h i s p r o v e s P r o p o s i t i o n 8. 14. W e shall state a r e s u l t , s i m i l a r to T h e o r e m 8 . 8 , w h i c h s h o w s , that

c a n b e d e f i n e d i n v a r i a n t l y , the way

is d e f i n e d .

However,

w e shall not p r o v e this s i n c e we a r e not g o i n g to m a k e u s e of the p r o p o s i t i o n 8.16

Proposition.

Let

d e n o t e the P o i s s o n o p e r a t o r

P

on

i . e . . on the o u t s i d e of Then

w h e r e the 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

Again,

and

R

has the f o l l o w i n g s y m b o l

a r e d e f i n e d in P r o p o s i t i o n 7. 57 ( i i ) , (iii).

The f o l l o w i n g is o b v i o u s f r o m what w e h a v e a l r e a d y p r o v e d ; the f o r m u l a is stated f o r f u t u r e r e f e r e n c e . 8.17

where

Lemma.

Q^

is an o p e r a t o r of P o i s s o n type of o r d e r z e r o .

We r e c a l l ( s e e (13.4) of [ 9 ] ) that the L a p l a c i a n

-109a s s o c i a t e d w i t h the b o u n d a r y C a u c h y - R i e m a n n c o m p l e x has the f o l l o w i n g f o r m (we o n l y need its r e s t r i c t i o n to (0, 1 ) - f o r m s ) .

(8.18)

T h e r e f o r e w e c a n r e w r i t e P r o p o s i t i o n 8. 14 in the f o l l o w i n g fo r m . 8.19 (i)

(ii)

Proposition.

-110C h a p t e r 9.

A parametrix for • near bM; n > 1

In this c h a p t e r w e shall c o n s t r u c t an a p p r o x i m a t e l o c a l l e f t i n v e r s e f o r the 3 - N e u m a n n p r o b l e m when r e p r e s e n t a t i o n of

n > 1,

i . e . , w e shall obtain a l o c a l

in t e r m s of

f , if

(9.1) (9.2) We shall next d e s c r i b e h e u r i s t i c a l l y this l o c a l i n v e r s e ( o r a p p r o x i m a t e " N e u m a n n o p e r a t o r " ).

In the b r i e f d e s c r i p t i o n that f o l l o w s w e s h a l l

d i s r e g a r d e r r o r t e r m s , ( w h i c h w i l l turn out to b e s m o o t h i n g o p e r a t o r s ) , and pay no attention to the h o s t of c u t - o f f f u n c t i o n s that m u s t be u s e d ( w h i c h i n t r o d u c e additional e r r o r t e r m s of s m o o t h i n g o p e r a t o r s ) .

Thus

a c c o r d i n g to T h e o r e m 7 . 6 6 w e have a p p r o x i m a t e l y (9. 3) Now the f i r s t b o u n d a r y c o n d i t i o n in (9. 2) g i v e s the d i r e c t c o n t r o l of p a r t of

[U]Q ( n a m e l y i ' ( u ) ) .

in t e r m s of

f.

We w i l l c o n t r o l the o t h e r p a r t , u^,

T o d o this apply the b o u n d a r y o p e r a t o r

b o u n d a r y c o n d i t i o n ) to (9. 3).

Since

B_ 9

indirectly

(of the s e c o n d

, we have a p p r o x i -

mately (9.4) Moreover

approximately, where

o f the L a p l a c i a n on 1 - f o r m s . has an i n v e r s e

K

is b o u n d a r y analogue

(See P r o p o s i t i o n (8. 19). ) H o w e v e r w h e n g i v e n b y an i n t e g r a l o p e r a t o r , known quite

-111e x p l i c i t l y ; thus

approximately.

(It is h e r e that the l i m i t a t i o n

n > 1 is r e q u i r e d in this c h a p t e r . ) Putting t h e s e things t o g e t h e r in ( 9 . 4 ) g i v e s that a p p r o x i m a t e l y ,

and h e n c e

to a p p r o x i m a t e i n v e r s e to (9. 1) is then (9. 5) W e now p a s s to the p r e c i s e v e r s i o n of (9. 5) W e u s e the notation of C h a p t e r s 7 and 8. b o u n d a r y n e i g h b o r h o o d s of s u c h that, if

in

Thus

are

i d e n t i f i e d w i t h s u b s e t s of IR^"^^,

has c o o r d i n a t e s

then

is identified w i t h a s u b s e t of and

a r e the c u t - o f f f u n c t i o n s of P r o p o s i t i o n 7. 57 ( i i ) , ( i i i ) .

A c c o r d i n g to T h e o r e m ( 7 . 6 7 ) , w e can w r i t e (9.3')

W e shall now r e p l a c e Poisson operators infinite o r d e r )

G and

and P.

in the a b o v e .

by the " e x a c t " G r e e n ' s and

T h i s i n t r o d u c e s s m o o t h i n g o p e r a t o r s (of F o r s i m p l i c i t y of n o t a t i o n w e shall not

k e e p t r a c k of t h e s e r e s u l t i n g e r r o r s e x p l i c i t l y , but g a t h e r t h e m t o g e t h e r at the end of the p r o o f o f L e m m a ( 9 . 1 1 ) . W e shall c o m p u t e

in t e r m s of

f.

We n o t e , that by (7. 61)

-112-

Next w e r e c a l l s o m e r e s u l t s of [ 9 ] c o n c e r n i n g a p a r a m e t r i x f o r denote admissible c o o r d i n a t e s in

(we r e f e r to C h a p t e r 4 f o r this n o t i o n ) .

D e f i n e the k e r n e l

I by

(9.6)

where

on a n e i g h b o r h o o d

of the d i a g o n a l of

We also set

element

dy

and the v o l u m e

is e x p l a i n e d in P r o p o s i t i o n 4. 37.

We d e f i n e the o p e r a t o r

(9.7) J 9.8

Proposition.

-

H

,

then

(9. 9) (9.10) where

d e n o t e s an o p e r a t o r of type 1.

16. 5 of [ 9 ] , and s h o w s that

T h i s is j u s t P r o p o s i t i o n

K is the a p p r o x i m a t e i n v e r s e f o r

-113H e r e , and in what f o l l o w s of type

m,

w i l l s y s t e m a t i c a l l y d e n o t e an o p e r a t o r

in the s e n s e of F o l l a n d - S t e i n [ 9 ],

to t h e s e o p e r a t o r s in C h a p t e r 14. o p e r a t o r s of type

m

15, 16.

F o r the p r e s e n t w e s h a l l r e c a l l that

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

m

in the " g o o d "

Thus the p a r a m e t r i c i e s c o n s t r u c t e d in [ 9 ] f o r t y p e 2 and 1, r e s p e c t i v e l y . c o m p o s e d w i t h an

9.11

Lemma.

h o o d of

in

and

directions.

w e r e o p e r a t o r s of

A l s o a v e c t o r f i e l d in the " g o o d "

S ^ g i v e s an

Suppose

W e shall r e t u r n

directions

S^

n > 1.

Let

d e f i n e d in T h e o r e m 7 . 6 2 .

be the b o u n d a r y n e i g h b o r Let '

s u c h that

Let

Then there exist

(9.12)

( R e c a l l that Proof. (9.13)

~ i n d i c a t e s an e x t e n s i o n v i a ( 7 . 6 1 ) . ) We apply

,

s u c h that

-114w h e r e we chose

so that

B y D e f i n i t i o n 8. 6 and T h e o r e m 8. 8 (9.14)

M u l t i p l y i n g (9. 13) b y

it b e c o m e s

(9.15)

A c c o r d i n g to L e m m a 8. 13 if we i n t e r c h a n g e e r r o r c o m m i t t e d is n e g l i g i b l e .

and

the

Hence (9.15) b e c o m e s

(9.16)

where

R

is 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 induced b y a s y m b o l of the

form

where

and We apply

(9.17)

w e r e d e f i n e d in P r o p o s i t i o n 7. 57 ( i i ) , ( i i i ) . to (9. 16) and u s e P r o p o s i t i o n 8. 19 (i) to obtain

-115W e apply

K on the l e f t .

A c c o r d i n g to (9. 10) w e obtain

or (9.18)

A c c o r d i n g to (9. 19) We note that

Therefore,

substituting (9.18)

into (9.19) w e obtain (9.20)

T o o b t a i n the f o r m u l a g i v e n in the s t a t e m e n t of the l e m m a w e o n l y n e e d to s i m p l i f y ( 9 . 2 0 ) .

therefore

Similarly

Namely

- 1 1 6 -

w h e r e we s e t

Collecting these f o r m u l a s we

obtain (9.21)

Since

K

is c e r t a i n l y an

w e have d e r i v e d L e m m a 9. 11.

We note that the c u t - o f f f u n c t i o n s a r e i r r e l e v a n t f r o m the point of v i e w of the e s t i m a t e s . 9. 22

Definition.

Let

We m a k e this m o r e p r e c i s e . n > 1.

Let

V

be as in L e m m a 9 . 1 1 .

that the o p e r a t o r

We

say

is of

Neumann type if it has the f o l l o w i n g f o r m (9.23)

for some

9.24

Definition.

and

Let

V

b e the b o u n d a r y n e i g h b o r h o o d g i v e n in L e m m a

-1179.11.

We say that the o p e r a t o r

is of r e m a i n d e r type if it can be w r i t t e n in the f o l l o w i n g f o r m (9.25)

with W e r e s t a t e L e m m a 9 . 1 1 as f o l l o w s . 9.26

Proposition.

Let R

Suppose

.

Let

V be as in T h e o r e m 7 . 6 2 .

Then there exist o p e r a t o r s

of Neumann type and

of r e m a i n d e r t y p e , s u c h that

whenever

i and

Notational R e m a r k It m i g h t be w e l l to r e c o r d o n c e m o r e the d e f i n i t i o n w e have u s e d in this c h a p t e r , and w h i c h we w i l l continue to u s e in f u r t h e r c h a p t e r s , that of o p e r a t o r s type) o p e r a t o r s of type of o r d e r

k.

and m;

We d e n o t e b y

namely

the ( H e i s e n b e r g - g r o u p -

is a standard 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

-118C h a p t e r 10.

The p a r a m e t r i x f o r

n e a r b M ; n=l

In this c a s e

(10. 1)

H e r e w e have the added c o m p l i c a t i o n that i.e., parametrix.

has no i n v e r s e ;

H o w e v e r , w e c a n m a k e u s e of the r e s u l t c o n c e r n i n g

the s o l v a b i l i t y of the L e w y e q u a t i o n d i s c u s s e d in C h a p t e r 3. stated s o m e w h a t i m p r e c i s e l y , We l e t

The i d e a ,

is as f o l l o w s :

d e n o t e the ( p r e s u m p t i v e ) p r o j e c t i o n on the b o u n d a r y

v a l u e s of a n t i - h o l o m o r p h i c f u n c t i o n s . (whose " p r o j e c t i o n "

T h e n on the o r t h o g o n a l c o m p l e m e n t ,

is g i v e n by

has an i n v e r s e ; n a m e l y b y

the r e s u l t s o f C h a p t e r 3 w e c a n find an i n t e g r a l o p e r a t o r a p p r o x i m a t e l y ; and so given by

s o that

has an i n v e r s e on the i m a g e of

On the o t h e r hand l e t

b e the " p r o j e c t i o n "

o p e r a t o r ( g i v e n by an o r d i n a r y 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 of o r d e r 0) c o r r e s p o n d i n g to a c o n i c n e i g h b o r h o o d of the c h a r a c t e r i s t i c v a r i e t y of The i m p o r t a n t f a c t is that the p r o j e c t i o n

a p p r o x i m a t e l y , and h e n c e

is s u b o r d i n a t e to p r o j e c t i o n

is e l l i p t i c away f r o m its c h a r a c t e r i s t i c v a r i e t y . 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 of o r d e r approximately.

It is then e a s y to s e e that

moreover Thus t h e r e e x i s t s s o that is an

-119a p p r o x i m a t e i n v e r s e to A n a l t e r n a t i v e f o r m f o r this is

We set

(10.2) and w e d e f i n e

w h e r e the c o n v o l u t i o n i s with r e s p e c t to the H e i s e n b e r g g r o u p d e n o t e the o p e r a t o r g i v e n b y c o n v o l u t i o n on (10. 3)

(10.4)

Lemma.

We set

,

Let

with the f u n c t i o n

Then

(10. 5) Proof.

T h i s is a c o n s e q u e n c e of L e m m a 3. 18, on taking c o m p l e x

conjugates. W e r e t u r n to in

and let

Let

u = u(x,y)

U be a b o u n d a r y n e i g h b o r h o o d o f

d e n o t e a d m i s s i b l e c o o r d i n a t e s in

as g i v e n in P r o p o s i t i o n 4 . 3 . Let

(10.6)

as in P r o p o s i t i o n 9. 8.

Set

-120where

on a n e i g h b o r h o o d of the d i a g o n a l of

and

and

(10. 7) R e c a l l that if

and

then

(10.8)

( s e e P r o p o s i t i o n 4 . 3).

We s e t

w h e r e we set

(10.9)

Proposition.

Proposition 9.8.

(10.10)

Let

V

be the b o u n d a r y n e i g h b o r h o o d g i v e n in

Then

(10. 11) on Proof.

The r e s u l t f o l l o w s f r o m L e m m a 1 0 . 4 m o d u l o the u s u a l

c h a n g e s n e c e s s a r y to t r a n s f e r a r g u m e n t s f r o m the H e i s e n b e r g g r o u p to a strongly p s e u d o - c o n v e x CR manifold ( e . g .

s e e the p r o o f of P r o p o s i t i o n

1 6 . 2 of [ 9 ] ). F r o m now on, to s i m p l i f y the notation w e shall d r o p the u s u b s c r i p t s from

and Let

This defines

as a l i n e a r f u n c t i o n of

-121with p a r a m e t e r s

10.12

x

Lemma.

Let

e

h

and

in

W e set

(10.13)

T

h

n

w

e

r

e

(10.14)

Proof.

C o n j u g a t i n g P r o p o s i t i o n 3 . 2 7 we get

(10.15)

W e note that

Let

(a)

the i n t e g r a l in (10.15) c o n v e r g e s a b s o l u t e l y , and (b)

-122where (10. 16)

It is e a s y to s e e that

is an i n t e g r a l o p e r a t o r with a

t h e r e f o r e it r e p r e s e n t s a o p e r a t o r . I compute in (10. 15).

.

shall

W e b e g i n b y changing the v a r i a b l e s of i n t e g r a t i o n

A c c o r d i n g to P r o p o s i t i o n 4 . 3

where

To o b t a i n w e

kernel

Therefore

w h e r e w e set (10.17) Therefore (10.18)

and the c h a n g e of v a r i a b l e s

yields

-123-

and s i n c e

,

letting

we obtain (10.14).

T h i s y i e l d s L e m m a 10.12. 10.19

Lemma.

Proof.

if

The

and

Since

w e have

Similarly

Therefore

Let

simple computation yields

a r e n o n n e g a t i v e i n t e g e r s and d e p e n d o n l y on

-124-

(10. 2 0 )

Since

If

n e a r the o r i g i n (10.20) i m p l i e s

then

T h e r e f o r e w e obtain the f o l l o w i n g , r a t h e r c r u d e , e s t i m a t e , w h i c h , theless suffices for our purposes. Namely (10. 21)Now w e can e a s i l y p r o v e L e m m a 10.19.

If

never-

then

-125Therefore

w h i c h p r o v e s L e m m a 10.19. Next w e c o n s t r u c t the i n v e r s e f o r s y m b o l of

is

D^.

R e c a l l that the p r i n c i p a l

where

the c h a r a c t e r i s t i c v a r i e t y

of

.

Therefore,

is the set

The idea of the c o n s t r u c t i o n is that in s o m e c o n i c n e i g h b o r h o o d of has an i n v e r s e b y v i r t u e of P r o p o s i t i o n 10. 9 and L e m m a 10.12; on the other hand, more precise

where (

is e l l i p t i c , h e n c e i n v e r t i b l e , away f r o m

To be

-126Let

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

is n o n s i n g u l a r

d e f i n e s a h a l f - s p a c e in

by c h o o s i n g

in

Since

Therefore,

s u f f i c i e n t l y s m a l l , the c o m p l e m e n t of

contains a c o n i c neighborhood,

where

c o n i c n e i g h b o r h o o d of the r a y

.

We note that

is the c h a r a c t e r i s t i c v a r i e t y of We choose

s o s m a l l that

is a

I

with

is e l l i p t i c o u t s i d e of the c o n i c set

for some

when

Let

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

zero for large

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

and

v a n i s h in the c o m p l e m e n t of

and

d e n o t e the 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 induced by the and

respectively.

c o n t a i n s the s u p p o r t of w i t h s y m b o l in

T h e n the c o m p l e m e n t of

in its i n t e r i o r .

and

symbols

w i t h s y m b o l in

Furthermore, can b e c o m -

bined v i a the c a l c u l u 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 ( s e e H o r m a n d e r [

] ).

10 . 22

H e n c e w e have the f o l l o w i n g r e s u l t . Lemma.

and

b o t h b e l o n g to

-127and

are both integral o p e r a t o r s with

Now w e d e f i n e the l e f t i n v e r s e , o r p a r a m e t r i x ,

kernel. , as

follows. (10.23) where

d e n o t e s the p a r a m e t r i x f o r

in the s u p p o r t of

i.e.,

(10.24)

10.2 5

Lemma.

Proof.

where we used L e m m a 10.22. f o r m of 10. 26 of

Let

N o t i c e that

c a n a l w a y s be put in the

This proves L e m m a 10.25. Lemma.

in

Suppose

n=l .

Let

d e f i n e d in T h e o r e m 7 . 6 2 .

Then there exists

b e the b o u n d a r y n e i g h b o r h o o d Let

s u c h that

s u c h that

-128(10.27)

Proof.

The p r o o f is s i m i l a r to the p r o o f of L e m m a 9- 11 and w e

w i l l not r e p e a t it. 10.28

Definition.

Let

n=l .

Let

V

be as in L e m m a 1 0 . 2 6 .

that the o p e r a t o r

We s a y

is of Neumann

type if it has the f o l l o w i n g f o r m (10.29)

with

10.30

Proposition.

P r o p o s i t i o n 9 . 2 6 holds when

We shall s e e that i n v e r s e to

10.31

n = 1.

g i v e s an a l t e r n a t i v e a p p r o x i m a t e

, in the f o l l o w i n g s e n s e ( s e e a l s o ( 1 0 . 2 5 ) ) .

Lemma

Proof. u s i n g (8.14) and (10.10).

By (10.24 h o w e v e r ,

and

-129the l a t t e r e q u a l s

b e c a u s e of ( 1 0 . 2 2 ) .

Putting t h e s e t o g e t h e r

p r o v e s the l e m m a . Remark. N

a

Tf w e u s e L e m m a (10. 31), an a p p r o x i m a t e Neumann o p e r a t o r

can b e w r i t t e n as

(10. 32) plus an e r r o r t e r m i n v o l v i n g Now the f o r m (10. 32) is the s a m e as the c o r r e s p o n d i n g f o r m ( 9 . 2 3 ) for

when

n > 1.

The e r r o r t e r m w i l l be

(10.33) w h i c h w i l l be e a s i e r to handle than (10. 32).

-130-

Part III.

The Estimates

Guide to Part III

Here we deal with the regularity properties of the solutions of the g-Neumann problem, (11 . 1) and (11.2) below.

L

2

theory The

2 L theory of Kohn allows one to write down an "abstract"

solution to the problem, in terms of a "Neumann operator" N. proceeds by using L that NfsC (M)

2

The theory

Sobolev estimates, and culminates in the assertion

if f e C ° ° ( M ) .

proofs given, in Chapter 11.

These results are reviewed, with no We then turn to estimates for

N

in other

function spaces.

Several types of operators Using the construction of approximate Neumann operators carried out in Chapters 9 and 10, the problem can be reduced to finding estimates for the following classes of operators: (i)

The restriction operator, mapping functions of

M to functions

on bM. ( i 1)

Pseudo-differential operators of the standard kind acting on

functions on bM (or M). (iii)

These are denoted generically by T.

Integral operators, related to the convolution operators on the

Heisenberg group, acting on functions on bM. ally by S.

These are denoted generic­

(iv)

P o i s s o n o p e r a t o r s , c a r r y i n g functions on bM to functions o n M .

Estimates for these operators T h e r e q u i r e d e s t i m a t e s a r e c a r r i e d o u t , f o r t h e m o s t p a r t , in C h a p t e r s 12 - 1 4 .

Many of t h e s e e s t i m a t e s a r e e i t h e r " c l a s s i c a l " o r

p r e v i o u s l y known.

N e v e r t h e l e s s w e p r e s e n t m a n y of t h e d e t a i l s h e r e ,

enough to give a c o m p l e t e view of the s u b j e c t .

B e s i d e s e s t i m a t e s of t h e

type a r i s i n g f o r (iii), t h e novel r e s u l t s a r e c e r t a i n e s t i m a t e s f o r P o i s s o n operators.

T h e f o r m e r a r e d e s c r i b e d in T h e o r e m 1 4 . 3, and t h e l a t t e r

in T h e o r e m 1 4 . 4 and L e m m a 1 5 . 34. Estimates for the ^-Neumann problem C h a p t e r 15 c o n t a i n s o u r m a i n r e s u l t s , w h i c h a r e obtained a s a n application of the above e s t i m a t e s .

T h u s t h e r e i s a l w a y s a gain of one

in a l l d i r e c t i o n s , a gain o t two in the " g o o d " d i r e c t i o n s , and a l s o a gain of two in t h e n o r m a l a n t i - h o l o m o r p h i c d i r e c t i o n . T h e equation a U = f T h e Neumann o p e r a t o r l e a d s t o a solution of

gU = f

(whenever

t h i s i s p o s s i b l e ) , w h i c h solution i s o r t h o g o n a l to h o l o m o r p h i c f u n c t i o n s . T h e e s t i m a t e s f o r t h i s solution a r e e s s e n t i a l l y c o n s e q u e n c e s of the c o r r e ­ sponding r e s u l t s f o r t h e g - N e u m a n n p r o b l e m , b u t t h e r e a r e c e r t a i n a d d i ­ t i o n a l o b s t a c l e s t h a t m u s t be s u r m o u n t e d .

T h e r e s u l t s a r e (roughly) a

gain of one-half in a l l d i r e c t i o n s , and a gain of one in t h e "good" d i r e c t i o n s The d e t a i l s a r e in C h a p t e r 1 6 .

-132C h a p t e r 11.

R e v i e w of the

theory

We shall now s u m m a r i z e the r e s u l t s of Kohn [ 2 1 ] , ( s e e a l s o the e x p o s i t i o n in [ 8 ] ), c o n c e r n i n g the tion of the

and r e g u l a r i t y t h e o r y f o r the s o l u -

Neumann p r o b l e m .

As before,

M

is an o p e n s u b - d o m a i n in a l a r g e r c o m p l e x m a n i f o l d

has a s m o o t h b o u n d a r y , b M , w h i c h is s t r o n g l y p s e u d o - c o n v e x , d e n o t e s the ( 0 , 1 ) f o r m s in

which are

d e n o t e s its c l o s u r e in the C h a p t e r 5).

We a r e c o n c e r n e d ,

up to the b o u n d a r y ;

n o r m (using the L e v i - m e t r i c of

in e f f e c t , with the p r o b l e m of s o l v i n g

(11.1) w i t h the b o u n d a r y c o n d i t i o n s (11.2) where

u

and

f

are (0,1) f o r m s .

The f o l l o w i n g t h e o r e m d e s c r i b e s the 11.3

Theorem.

and

theory.

o r i g i n a l l y d e f i n e d on t h o s e

which

s a t i s f y the b o u n d a r y c o n d i t i o n s ( 1 1 . 2 ) , has a unique e x t e n s i o n to a s e l f a d j o i n t (unbounded) o p e r a t o r (i)

H •

conditions (ii)

domain

on

, s a t i s f y i n g the f o l l o w i n g ; ,

then

u

s a t i s f i e s the b o u n d a r y

(11.2).

Let

= null s p a c e of

(the " h a r m o n i c "

space).

f i n i t e - d i m e n s i o n a l , and c o n s i s t s of e l e m e n t s that b e l o n g to Moreover (iii)

Suppose

N is d e f i n e d as

where

Then

is

-133and

.

The o p e r a t o r Remarks.

N

Then

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

is c a l l e d the Neumann o p e r a t o r . O b s e r v e that if

then

and

f

is o r t h o g o n a l to

g i v e s a w e a k s o l u t i o n of o u r o r i g i n a l p r o b l e m ((11 .1)

and

(11 . 2)) in the s e n s e that (11.4) where

is any e l e m e n t in

c o n d i t i o n s (11 . 2). e l e m e n t s of

Moreover

If a l s o

w h i c h s a t i s f i e s the b o u n d a r y u

is the unique w e a k - s o l u t i o n ,

modulo

then we g e t a s o l u t i o n to ( H . I )

and ( 1 1 . 2 ) in the u s u a l s e n s e . T a k i n g f o r g r a n t e d T h e o r e m 1 1 . 3 , o u r m a i n t a s k then w i l l b e to p r o v e the r e g u l a r i t y r e s u l t s f o r the N e u m a n n o p e r a t o r

N, r e f i n i n g c o n c l u -

s i o n s (iii) and (iv) to v a r i o u s f u n c t i o n s p a c e s . T h e study of t h e s e f u n c t i o n s p a c e s is the s u b j e c t of C h a p t e r s 12-14. W e r e t u r n to the N e u m a n n o p e r a t o r p r o p e r in C h a p t e r 15. Note:

In c a r r y i n g out the e s t i m a t e s f o r

shall a s s u m e that f

N(f)

in C h a p t e r s 15 and 16 w e

is o r t h o g o n a l to the h a r m o n i c s p a c e ; the c o m p l e m e n t

is a f i n i t e d i m e n s i o n a l s p a c e of s m o o t h f o r m s , and h e n c e this r e p r e s e n t s no l i m i t a t i o n on the v a l i d i t y o f o u r r e s u l t s .

-134C h a p t e r 12.

The B e s o v s p a c e s

W e shall c o n s i d e r the B e s o v s p a c e s , d e n o t e d by [33], Chapter V , a.

Here W e shall a l w a y s w r i t e F o r s i m p l i c i t y of notation w e have w r i t t e n m = 2n+l.

Definition

The space

is f i n i t e . b.

c o n s i s t s of all

f o r w h i c h the n o r m

(Here

ou n s w eof shall A F c hoar r a c tre raipzpaltiicoant :i o Rate a p p rneed o x i m aan t i oenq u i v a l e n t c h a r a c t e r i z a t i o n . Suppose

; then t h e r e e x i s t s a f a m i l y

s m o o t h f u n c t i o n s so that rate, while as

in

in

n o r m as

of at a d e f i n i t e

c a n be c o n t r o l l e d

appropriately,

More precisely:

that (12. See 1) L the e mbmi ba l. i o g rSuppose a p h i c a l r e m a r k s Tat hen the end of C h a p t e r 13.

so

-135-

(12.2)

C o n v e r s e l y , if f

so that ( 1 2 . 2 ) h o l d s , then

is e q u i v a l e n t with the

m

.

r o o t of the s u m of

The

n o r m of

and the two quantities

(12.2).

Proof.

Let

u(x,y)

be the P o i s s o n i n t e g r a l of

f , ( a c c o r d i n g to

Chapter V , of [33]) then w e know that

(12. 3)

by (61) and (62) of that c h a p t e r . Let

.

The s e c o n d inequality of (12. 3) p r o v e s the

s e c o n d inequality of ( 1 2 . 2 ) .

Moreover o

( T h i s is H a r d y ' s i n e q u a l i t y ) , and so the f i r s t inequality of ( 1 2 . 2 ) f o l l o w s f r o m the f i r s t inequality of ( 1 2 . 3 ) . C o n v e r s e l y s u p p o s e that ( 1 2 . 2 ) is s a t i s f i e d .

and

Write

-136-

Now take and u s e the f a c t that

Then

The finiteness

that

c.

is o b v i o u s f r o m the f a c t

The l e m m a is p r o v e d ,

The space W e c o n s i d e r the h a l f - s p a c e

u s e f u l to w r i t e here

with b o u n d a r y

It w i l l be

as c o o r d i n a t e s f o r p o i n t s in and

Sometimes we write

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

for

The s p a c e

s u c h that

w i t h the o b v i o u s n o r m (the m e a s u r e u s e d is the usual Lebesgue measure for is g i v e n b y G a g l i a r d o ' s

The c o n n e c t i o n with the B e s o v s p a c e lemma:

In the p r e v i o u s c h a p t e r s w e have u s e d p instead of y , but the p r e s e n t notation is m o r e c o n v e n i e n t f o r o u r p u r p o s e s h e r e .

-13712.4

Lemma.

Suppose F is s m o o t h in

where

.

Let

, and b e l o n g s to

f be the r e s t r i c t i o n of

(See 4. 3 in C h a p t e r VI of [ 3 3 ] ,

F

to

.

Then

T h i s is the s p e c i a l c a s e when

a = 1. )

T h e c o n v e r s e of this l e m m a is a l s o t r u e , but w e shall need a g e n e r a l i z a t i o n in t e r m s of P o i s s o n o p e r a t o r s w h i c h w e shall now c o n s i d e r . d.

Poisson operators W e c o n s i d e r m a p p i n g s of f u n c t i o n

f

in

to f u n c t i o n

F on

, g i v e n by

(12.5)

Here

p

(7. 31).

is the " s y m b o l " of the o p e r a t o r

P

as d e f i n e d in (7. 32) and

In the r e s t of this c h a p t e r and in the n e x t , we shall l i m i t o u r -

s e l v e s to o p e r a t o r s of P o i s s o n type of o r d e r

0

( u n l e s s the c o n t r a r y is

stated). 12.6

Main l e m m a .

The o p e r a t o r

T o p r o v e this w e need the f o l l o w i n g : 12.7

Lemma.

The o p e r a t o r

maps

to

-138mappir.g f u n c t i o n s on

to functions on

to the standard s y m b o l c l a s s

P r o o f of L e m m a 1 2 . 6 . in X.

, (where

belongs

maps

We a s s u m e that

has c o m p a c t support

Then by the F o u r i e r T r a n s f o r m

where

(12.8)

for each

N > 0.

However,

H o w e v e r , by the M a r c i n k i e w i c z m u l t i p l i e r t h e o r e m ( s e e [33], Chapter 4) e a c h of the o p e r a t o r s

is bounded on

with n o r m

An integration in

then p r o v e s L e m m a 1 2 . 7 . T o p r o v e the m a i n l e m m a , w e p r o v e f i r s t that

Now

F = F ( x , y ) , and f o r f i x e d y ,

is g i v e n by 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 w i t h s y m b o l in the c l a s s

u n i f o r m l y in

Lemma

12.7

Thus b y

-139and s i n c e y

p

and thus

F

has finite support in y , then an integration in

gives

Next c o n s i d e r

.

It w i l l make e s t i m a t e s e a s i e r if w e a s s u m e

(as w e w i l l f r o m now on) that

v a n i s h e s when

the e r r o r , given by s y m b o l s with c o m p a c t support in , whatever

(because clearly gives

f we start w i t h ) .

Now

has s y m b o l

and as one can o b s e r v e hence g i v e s a bounded o p e r a t o r on by L e m m a

12.7.

is again a s y m b o l of P o i s s o n type f o r e a c h fixed

y , u n i f o r m l y in y ,

Thus,

(12.9)

Now

and e a c h

b e l o n g s to the c l a s s of s y m b o l s of the P o i s s o n type

again

0.

Thus invoking

L e m m a 1 2 . 7 again g i v e s

(12.10)

Now in ( 1 2 . 9 ) and ( 1 2 . 1 0 ) let T h i s leads to

e = y , if

and integrate in y .

-140-

and so

with n o r m bounded b y

The c o n s i d e r a t i o n of

is a n a l o g o u s .

c o r r e s p o n d s to " s y m b o l " observed f o r each fixed

and y

, by L e m m a 1 2 . 1 .

In f a c t

w h i l e , as w e have a r e s y m b o l s of P o i s s o n t y p e , and h e n c e

g i v e b o u n d e d o p e r a t o r s on

c o n c l u d e s the p r o o f of the m a i n l e m m a .

u n i f o r m l y in

This

-141C h a p t e r 13.

We l e t

The s p a c e s

and

d e n o t e the standard L i p s c h i t z s p a c e s ,

d e s c r i b e d in C h a p t e r V , where

§4 of [ 3 3 ] .

Thus a bounded f u n c t i o n

when

the sup n o r m . ) F o r and w h e n

i

(Here

, as f b e l o n g s to denotes

, we require

we p r o c e e d inductively, i . e . ,

and

j= l, . . . ,m.

On on

w e d e f i n e the s p a c e w h i c h can b e e x t e n d e d to

The n o r m is the quotient n o r m . s u b s p a c e of

so as to b e l o n g to

M o r e p r e c i s e l y , let

b e the c l o s e d

c o n s i s t i n g of all

s o that a.

to c o n s i s t of all f u n c t i o n s

Define

A characterization In a n a l o g y with what w a s done f o r

t e r m s of rate of a p p r o x i m a t i o n s .

we characterize

in

We shall state the analogue of L e m m a

(12. 1) s o m e w h a t d i f f e r e n t l y . 13.1

Lemma.

Suppose

Then

write (13.2)

where

and

if and only if w e can

-142Proof.

T h i s l e m m a is not new.

P o i s s o n i n t e g r a l of

f

T o p r o v e it, l e t

u(x,y)

be the

(as in C h a p t e r 5 of [ 3 3 ] ) and u s e the f a c t that

(13. 3)

( s e e (49) and (51) of that r e f e r e n c e ) , and set

Now while

Conversely,

suppose

by the s e c o n d inequality in (13. 3). f can be w r i t t e n in the f o r m ( 1 3 . 2 ) .

F o r the s e c o n d s u m w e u s e the e s t i m a t e

Thus

F o r the f i r s t s u m u s e the e s t i m a t e

Thus

Now if

set

N

so that

p r o v i n g the l e m m a .

,

and the r e s u l t is

Then

-143T h e r e is an analogue of the l e m m a w h i c h h o l d s f o r f

Again

is to have the d e c o m p o s i t i o n (13. 2) and

(13.4)

m e a n s the n o r m of all s e c o n d d e r i v a t i v e s . ) b.

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

Suppose

is a standard p s e u d o -

differential o p e r a t o r , with (13. 5)

Lemma. Proof.

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

to i t s e l f .

By the u s u a l c o m m u t a t i o n p r o p e r t i e s of

and

c a n e a s i l y r e d u c e to the c a s e

.

, one

We shall c o n s i d e r in d e t a i l this

case. W e need a p r e l i m i n a r y l e m m a . w h i c h v a n i s h e s when Let

13.6

(13.7)

Lemma.

Suppose

is a f i x e d

function

or

d e n o t e the 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 w h o s e s y m b o l is

One has the f o l l o w i n g e s t i m a t e s :

,

for each

-144P r o o f of L e m m a 1 3 . 6 .

S u p p o s e f o r s i m p l i c i t y that

Then

Let

and it s u f f i c e s to s e e that

(13.8)

(with A independent of k)

Now s i n c e is non-vanishing only when

o b v i o u s e s t i m a t e s show that

(13.9) Now

F o r the f i r s t i n t e g r a l u s e the e s t i m a t e

(the c a s e

of ( 1 3 . 9 ) ) . F o r the s e c o n d i n t e g r a l u s e the e s t i m a t e (the e s t i m a t e with

in(13.9)).

The r e s u l t is (1 3. 8), p r o v i n g (13. 7)

The c a s e s c o r r e s p o n d i n g to

w a y , c o n c l u d i n g the p r o o f of L e m m a

a r e done in the s a m e

(13.6).

T o p r o v e L e m m a (13. 5), w e need f o r an the type ( 1 3 . 2 ) , w h e r e , h o w e v e r , the e l e m e n t s c o n t a i n e d in

a d e c o m p o s i t i o n of g

have t h e i r s p e c t r u m

-145T o do this f i x a

w h i c h is an e v e n f u n c t i o n in

for

Now d e f i n e

by-

Then since

Moreover, where

with

O b s e r v e that

is s u p p o r t e d in

since

H e n c e if

and

Also

is e v e n and

then

that (By 1 3 .the 1f0oW T )l lsheoaiw sm c aeit snhaosnow rw ugfsuf imcf iensn Now itsto sh othe sneee p athat rl sooo f p of r o vLees mand the m a by full 13.Ls5. eem tm of Sa u pi n 1pe3oq.s1u e ,aand l itthi eethe sr e f( 1 or3 ree. 4 m , )a . rks

-146T h e t e r m s c o r r e s p o n d i n g to s p e c t r u m c o n t a i n e d in

s o as to be

can be d i s r e g a r d e d s i n c e

and so

Now in the d e f i n i t i o n of take

k=0

is as r e g u l a r as w e w i s h .

( s e e the r e m a r k s p r e c e d i n g L e m m a = 1, when

has

.

13.6)

Then

(13.11) since

Q

with

and

a g r e e on the s p e c t r u m of

W e now invoke (13. 7)

, and u s e ( 1 3 . 4 ) , the r e s u l t is (13.10) and L e m m a 13. 5 is p r o v e d W e c o m e to o u r m a i n r e s u l t .

13.12 then

Main l e m m a . P

U P

is an o p e r a t o r of P o i s s o n - t y p e of o r d e r

maps

Proof.

We c o n s i d e r f i r s t the c a s e w h e n

.

Write

and if

and

We p r o v e that

(13.13)

where

v m e a n s the g r a d i e n t

and

m a t r i x of all s e c o n d p a r t i a l d e r i v a t i v e s (including the

m e a n s the .

T h i s is

0

-147p r o v e d in the s a m e w a y as in the p r o o f of L e m m a 13. 5 b e f o r e , w h e n w e o b s e r v e (as w e a l r e a d y h a v e b e f o r e ) that fixed

y

is type

u n i f o r m l y in y , and

u n i f o r m l y in y ,

for each

, a r e a l s o of type

etc.

Now (13.13)does not y e t c o n c l u d e the p r o o f of the t h e o r e m (when W e need an e x t e n s i o n o p e r a t o r , m a p p i n g f u n c t i o n s on to f u n c t i o n s on

.

W e use the m a p p i n g

where

!

if

and (13.14) as a l r e a d y d e s c r i b e d in C h a p t e r 7 ( s e e (7. 58) and the d i s c u s s i o n that f o l l o w s ) . A b a s i c p r o p e r t y of this m a p p i n g is the f a c t that (13. 15)

and

F o r the p r o o f of this s e e [ 3 3 ] , C h a p t e r VI,

§3.

Now and b y (1 3. 1 5) the as d o the

s a t i s f y the s a m e kind of i n e q u a l i t i e s ( i . e . ,

(13.13))

but now on all of

Thus by L e m m a (13. 1), (and the v a r i a n t alluded to at the end of its proof)

Since the r e s t r i c t i o n of as d e s i r e d .

h i g h e r v a l u e s of

to

T h i s p r o v e s the r e s u l t when

a we use r e c u r s i o n .

if the f i r s t p a r t i a l s of

Thus

is

F , w e get For , with

But t h e s e can be e x p r e s s e d

-148as P o i s s o n - t y p e o p e r a t o r s of the f i r s t p a r t i a l s of

f , and t h e s e a r e in

etc.

B i b l i o g r a p h i c a l c o m m e n t s f o r C h a p t e r s 12 and 13 1.

In using the r e f e r e n c e [ 3 3 ] , the r e a d e r should be w a r n e d that

throughout §5. 1 and 5. 2 of C h a p t e r V t h e r e o c c u r s a s y s t e m a t i c s l i p in sign:

The a - 1

2.

in e x p r e s s i o n s s u c h as (61), ( 6 2 ) , should be r e p l a c e d b y

T h e o n e - d i m e n s i o n a l v e r s i o n of L e m m a 1 3. 1 and (1 3. 4) is in

Z y g m u n d [ 3 6 ] , C h a p t e r III, ( 1 3 . 1 4 ) and ( 1 3 . 2 0 ) .

Note that what we c a l l

A J he c a l l s 3.

L e m m a s 1 2 . 1 , 1 2 . 4 , and 1 3. 1 c a n b e found in N i k o l ' s k i i

C h a p t e r s 4 - 6 , in c o n s i d e r a b l y m o r e g e n e r a l f o r m .

He d e n o t e s

[25], and

respectively. 4.

P r o b a b l y the o n l y n o v e l r e s u l t s c o n t a i n e d in t h e s e c h a p t e r s a r e

M a i n L e m m a s 1 2 . 6 and 1 3 . 1 2 d e a l i n g w i t h P o i s s o n o p e r a t o r s .

For

r e l a t e d e a r l i e r e s t i m a t e s s e e A g m o n , D o u g l i s , and N i r e n b e r g [O].

some

-149C h a p t e r 14. a.

The s p a c e s

on M and b M

Definitions W e c o m e c l o s e r to o u r u l t i m a t e a p p l i c a t i o n s . M

is a d o m a i n in a c o m p l e x m a n i f o l d w i t h s m o o t h b o u n d a r y

bM.

What w e have done a b o v e w i l l m a k e it e a s y to d e f i n e the s p a c e s also

and Then if

In f a c t s u p p o s e , f o r e x a m p l e .

is a l o c a l d i f f e o m o r p h i s m of

w h o s e s u p p o r t is c o n t a i n e d w h e r e

and

is r e g u l a r , then

T h i s f o l l o w s i m m e d i a t e l y f o r the c h a r a c t e r i z a t i o n g i v e n in L e m m a 12. 1. T h i s a l l o w s one to d e f i n e nate n e i g h b o r h o o d s of

in t e r m s of a finite p a t c h i n g of c o o r d i -

bM.

S i m i l a r l y one can d e f i n e the s p a c e s

f i r s t by u s i n g L e m m a 13. 1 ( s e e a l s o 1 3 . 4 ) ) , when higher

r e c u r s i v e l y by differentiation. The space

comment.

m a t t e r s to

has a standard d e f i n i t i o n w h i c h n e e d s no f u r t h e r

The s p a c e

of p a t c h i n g s :

can a l s o b e d e f i n e d .

It n e e d s two kinds

The f i r s t in t e r m s of i n t e r i o r n e i g h b o r h o o d s , w h i c h r e d u c e ; the s e c o n d in t e r m s of b o u n d a r y n e i g h b o r h o o d s

w h i c h r e d u c e m a t t e r s to b.

1; and then f o r

.

A g a i n nothing new is r e a l l y i n v o l v e d .

F o u r t y p e s of o p e r a t o r s We shall d e s c r i b e f o u r t y p e s of o p e r a t o r s in t e r m s of w h i c h o u r

e s t i m a t e s w i l l be s t a t e d . (i)

O p e r a t o r s of type I.

T h e r e is o n l y a single o p e r a t o r in this c l a s s

It is the r e s t r i c t i o n o p e r a t o r m a p p i n g f u n c t i o n s on

to f u n c t i o n s on b M

-150by (ii) on

O p e r a t o r s of type II.

bM

to f u n c t i o n s on

T h e s e a r e the o p e r a t o r s m a p p i n g f u n c t i o n s

b M , w h i c h in c o o r d i n a t e n e i g h b o r h o o d s a r e g i v e n

b y c l a s s i c a l 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 of

order

zero ,

( i . e . , with

s y m b o l in (iii) on

O p e r a t o r s of type III.

bM

to f u n c t i o n s on

These are o p e r a t o r s mapping functions

b M , w h i c h a r e g i v e n by H e i s e n b e r g - g r o u p type

k e r n e l s of type

(see [9 ] , p . 4 8 6 ) .

w i l l a l s o b e w r i t t e n as (iv) on

O p e r a t o r s of type IV.

bM

of

, and

to i n d i c a t e the

These are o p e r a t o r s mapping functions

to f u n c t i o n s on (no m a t t e r what

is a l w a y s f

is).

The o p e r a t o r s

F o r p o i n t s n e a r the b o u n d a r y

in the i n t e r i o r bM,

these

o p e r a t o r s a r e g i v e n in a p p r o p r i a t e c o o r d i n a t e n e i g h b o r h o o d s by o p e r a t o r s of P o i s s o n - t y p e ( o r d e r 0), as in (12. 5), and (7 31). c.

The e s t i m a t e s A l l m a p p i n g s w i l l be bounded in the n o r m .

14.1

T h e o r e m I.

The o p e r a t o r of type I m a p s

(a) (b)

14.2

T h e o r e m II.

O p e r a t o r s of type II m a p

Also

and

-151(a) (b)

14.3

T h e o r e m III.

O p e r a t o r s of type III m a p

(a)

when the o p e r a t o r is of type 0.

(b) w h e n the o p e r a t o r is of type 14.4

T h e o r e m IV.

1,2.

O p e r a t o r s of type IV m a p

(a) (b) P r o o f o f T h e o r e m I.

P a r t (a) f o l l o w s f r o m L e m m a 1 2 . 4 , L e m m a 12. 1

w h i c h is the c h a r a c t e r i z a t i o n in t e r m s of a p p r o x i m a t i o n s , and the l o c a l d e f i n i t i o n of

P a r t (b) n e e d s no c o m m e n t .

P r o o f o f T h e o r e m II.

P a r t (a): a c c o r d i n g to L e m m a 1 2 . 7 , an

of type II m a p s

to i t s e l f b o u n d e d l y .

Such o p e r a t o r s

operator

commute

with v e c t o r f i e l d s ( m o d u l o o p e r a t o r s again of type II) and so b y the c h a r a c t e r i z a t i o n L e m m a 1 2 . 1 , we g e t p a r t (a).

P a r t (b) is p r o v e d

similarly,

invoking L e m m a 13. 5, and the c h a r a c t e r i z a t i o n L e m m a 13. 1 (or ( 1 3 . 4 ) ) . P r o o f of T h e o r e m III. (i) [9], (ii)

P a r t (a).

E a c h o p e r a t o r of type

W e need the f o l l o w i n g f a c t s . is bounded on

to i t s e l f .

See

§15. If

is a s m o o t h b a s i s of v e c t o r f i e l d s and

S is any

-152o p e r a t o r of type

, then

, all of type

so that

(14. 5)

T h i s is ( i m p l i c i t ) in [ 9 ], p. 490; a l s o s e e R o t h s c h i l d - S t e i n [29]. Since

can be c h a r a c t e r i z e d in t e r m s of r a p i d i t y of

a p p r o x i m a t i o n , with a p p r o x i m a t i n g

function c o n t r o l l e d in

Lp norms,

p a r t (a) then f o l l o w s f r o m f a c t s (i) and ( i i ) . P a r t (b).

T h i s p a r t w e announced in [ 1 2 ] , L e m m a ( 6 . 2 ) , p a r t (b).

The

i d e a we had in mind f o r the p r o o f of the l e m m a w a s l a t e r g e n e r a l i z e d , and in a m o r e g e n e r a l f o r m a p p e a r s in R o t h s c h i l d - S t e i n [ 2 9 ] , T h e o r e m 14. T h e r e s e e m s little point in r e p e a t i n g that a r g u m e n t h e r e . P r o o f of T h e o r e m IV.

P a r t (a) and (b) a r e s i m p l y c o n s e q u e n c e s of M a i n

L e m m a 1 2 . 6 , and M a i n L e m m a 13. 3, t o g e t h e r with the r e m a r k s made.

already

-153C h a p t e r 15.

Let 11.3.

Main results

N d e n o t e the ( e x a c t ) Neumann o p e r a t o r d e s c r i b e d in T h e o r e m

O u r p u r p o s e h e r e w i l l be to p r o v e the r e g u l a r i t y of

of function s p a c e s

N

in t e r m s

and o t h e r s that w i l l be d e f i n e d b e l o w .

F o r s i m p l i c i t y of notation we a r e using

to d e n o t e not only the

p r e v i o u s l y d e f i n e d s p a c e of s c a l a r - v a l u e d f u n c t i o n s , but a l s o its analogue of (0, 1) f o r m s on

M

w h o s e c o m p o n e n t s b e l o n g to

the o t h e r s p a c e s studied in C h a p t e r 14. not lead to any c o n f u s i o n .

similarly for

But this abuse of notation should

In all o u r t h e o r e m s w e have

We shall say that a s m o o t h v e c t o r f i e l d

X

d e f i n e d on

if r e s t r i c t e d to the b o u n d a r y it p o i n t s in the " g o o d " d i r e c t i o n s ,

15.1

Theorem.

is a l l o w a b l e i.e.,

N has a unique e x t e n s i o n s o that the indicated m a p p i n g s

are bounded. (a)

(b) w h e r e p is any s e c o n d - d e g r e e p o l y n o m i a l in a l l o w a b l e v e c t o r f i e l d s .

(c) * I n any a p p r o p r i a t e c o o r d i n a t e n e i g h b o r h o o d s , the v e c t o r f i e l d s g i v e a b a s i s f o r the h o l o m o r p h i c v e c t o r f i e l d s w h i c h a r e tangential at b M . Thus and are allowable However, is the h o l o m o r p h i c v e c t o r f i e l d wliich ( n e a r b M ) has the p r o p e r t y that Re w h e r e p is the g e o d e s i c d i s t a n c e f r o m b M . f r o m b M is of no s i g n i f i c a n c e .

The e x a c t f o r m of Z^^^j away

-154We shall p r o v e t h e s e r e s u l t s by showing the b o u n d e d n e s s of the m a p p i n g s on

The r e s t then f o l l o w s b y a s i m p l e l i m i t i n g a r g u m e n t .

Now if

then

and

p r o b l e m (11 . 1) and (11 . 2).

For

15.2

-Neumann

n = 1 to e x p r e s s (in a p p r o p r i a t e

coor-

in t e r m s of o u r a p p r o x i m a t e Neumann o p e r a t o r

t o g e t h e r with the r ' e m a i n d e r estimating

s o l v e s the

u w e apply P r o p o s i t i o n 9• 2 6 , when

o r P r o p o s i t i o n 10. 30 when dinate p a t c h e s ) u

u

and

R(u).

The q u e s t i o n then b e c o m e s that of

R.

P r o o f of p a r t (a).

The f i r s t t a s k w i l l b e to p r o v e :

Proposition.

is b o u n d e d f r o m

We can w r i t e

symbolically (disregarding smooth cut-off func-

t i o n s ) in the f o r m (15. 3) where

G

is the G r e e n ' s o p e r a t o r w h i c h b y the e l l i p t i c t h e o r y is s m o o t h i n g

of o r d e r 2 in all d i r e c t i o n s ;

is a d i f f e r e n t i a l o p e r a t o r of o r d e r 1 ;

R e s t is the r e s t r i c t i o n - t o - t h e - b o u n d a r y o p e r a t o r (type I in the t e r m i n o l o g y of C h a p t e r 14); on b M ; type I V ) .

is a standard 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 of o r d e r 1

is an o p e r a t o r of type When

; and

n = 1 the d e s c r i p t i o n of

P

is the P o i s s o n o p e r a t o r , is s l i g h t l y m o r e

and w e shall d i s c u s s this c a s e s e p a r a t e l y b e l o w . T h e p r o o f of P r o p o s i t i o n 1 5 . 2 r e q u i r e s a l e m m a .

15.4

Lemma.

G

is a bounded m a p p i n g f r o m

* S e e Note on page 133.

complicated,

-155It s u f f i c e s to p r o v e this in the s e p a r a t e c o o r d i n a t e n e i g h b o r h o o d s with which we are dealing.

Then the p r o b l e m is r e d u c e d to c o r r e s p o n d i n g

p r o b l e m in

L e t us r e c a l l the e x t e n s i o n o p e r a t o r

in ( 7 . 5 8 ) ) , and a l s o in the p r o o f of L e m m a 13. 12.

(used

By (13. 15) w e know

that it is a bounded o p e r a t o r f r o m

Now ( s e e

(7.63)) (15. 5) E

is a standard 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 of o r d e r - 2 , t h e r e f o r e

by " c o m m u t i n g " d e r i v a t i v e s p a s t maps

E , w e s e e by L e m m a (12. 7) that

E

Next w e c o n s i d e r the t e r m

We o b s e r v e b y the s y m b o l i c c a l c u l u s that (15.6) and (15. 7) where

and

a r e P o i s s o n o p e r a t o r s of o r d e r

In taking a d e r i v a t i v e of o r d e r k+2 of k+1 of the d e r i v a t i v e s p a s t

P , and then p a s t

d i f f e r e n t i a l o p e r a t o r of o r d e r T h i s r e s u l t s in a f u n c t i o n in e l e m e n t in

.

0, and having

0. we can c o m m u t e

E ; turning k-1

E

into p s e u d o -

d e r i v a t i v e s act on o n

, and the r e s t r i c t i o n g i v e s an

F i n a l l y an a p p l i c a t i o n of L e m m a 1 2 . 6 c o n c l u d e s the p r o o f

of the p r e s e n t l e m m a . F o r f u r t h e r r e f e r e n c e w e r e c o r d the

analogue of the l e m m a .

-156w h i c h is p r o v e d v e r y s i m i l a r l y . (15. 8)

Lemma.

G

is a bounded m a p p i n g f r o m

W e r e t u r n to the p r o o f of P r o p o s i t i o n 1 5 . 2 ,

and now we need to

c o n c e r n o u r s e l v e s o n l y w i t h the s e c o n d t e r m of the right s i d e of (15. 3). If w e apply

k d e r i v a t i v e s w e obtain ( a f t e r u s i n g ( 1 5 . 6 ) and (15.7)) a s u m

of s u m s of the f o r m (15.9) and w h e r e

a r e e a c h P o i s s o n o p e r a t o r s of o r d e r

0.

Now b y [ 9 ], P r o p o s i t i o n 15. 14, the o p e r a t o r s o p e r a t o r s of the kind

(type :

are each

in o u r t e r m i n o l o g y ) .

B e c a u s e of

(14. 5) ( o r m o r e p r e c i s e l y its t r a n s p o s e ) we can " c o m m u t e " the o p e r a t o r s , and then c o m b i n e t h e m w i t h

The result r e p l a c e s

e a c h s u m in ( 1 5 . 9 ) b y s u m s of t e r m s of the f o r m (15. 10)

Now

, thus b y L e m m a ( 1 5 . 4 )

So in v i e w of L e m m a 1 2 . 4 the r e s u l t of applying b e l o n g s to

.

Rest is in

and and b y T h e o r e m 1 4 . 3 ( a ) ,

.

F i n a l l y by T h e o r e m 14. 4 (a),

Since this is any d e r i v a t i v e of o r d e r

k

of the

in ( 1 5 . 3 ) , and s i n c e all o u r i n c l u s i o n s a r e b o u n d e d m a p p i n g s .

P

(15.10) term

Proposition

1 5 . 2 is c o m p l e t e l y p r o v e d , when In the c a s e

n = 1 ,

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

has to

-157be r e p l a c e d b y

( s e e P r o p o s i t i o n 10. 30 and (10. 23)); but t h e s e

t e r m s can a b s o r b any d e r i v a t i v e of o r d e r

1 , giving nothing w o r s e than

and the a r g u m e n t then is c a r r i e d out as b e f o r e . Proposition 15.2.

This p r o v e s

T o handle the r e m a i n d e r t e r m ( s e e ( 9 . 2 5 ) ) we need the

following: (15.11)

Lemma.

Let

Then there exists

so that

R

T h e d i f f i c u l t y w i t h the o p e r a t o r order

is bounded f r o m ! R

is that it is s m o o t h i n g of at m o s t

One c o u l d o v e r c o m e this d i f f i c u l t y b y an i t e r a t i o n a r g u m e n t ,

but a m o r e s e r i o u s o b s t a c l e w o u l d s t i l l stand in the w a y , n a m e l y that cannot be bounded on

b e c a u s e r e s t r i c t i o n of an

h y p e r - p l a n e m a k e s no s e n s e .

R

f u n c t i o n to a

Thus w e need to s t a r t with

where

(in f a c t T h e e l e m e n t s of the f r a c t i o n a l S o b o l e v s p a c e

are defined

as r e s t r i c t i o n s of e l e m e n t s in where

is the s u b s p a c e of

v a n i s h on

c o n s i s t i n g of t h o s e f u n c t i o n s w h i c h

( F o r the f a c t s about the s p a c e s

see e . g . ,

[ 3 3 ] , C h a p t e r V . ) B e c a u s e of the p r o p e r t y (13. 15) of the e x t e n s i o n m a p this is c o n s i s t e n t w i t h the u s u a l d e f i n i t i o n w h e n

k

is an i n t e g e r .

More-

o v e r b e c a u s e of (13.15), and the standard i n t e r p o l a t i o n t h e o r e m f o r ( s e e [ 4 ] , T h e o r e m 10), it f o l l o w s that w h e n e v e r then

and the m a p p i n g is b o u n d e d , f o r any r e a l

k.

In p r o v i n g the l e m m a w e l i m i t o u r s e l v e s to the l e a s t r e g u l a r t e r m

-158of

R

w h i c h can be w r i t t e n as

(15.12) W e o b s e r v e f i r s t that: (15.13)

The m a p p i n g

Suppose

is bounded f r o m First

t h e r e f o r e s o is

Secondly,

if

as w e have a l r e a d y pointed out, and Rest

maps

( s e e [ 1 7 ] , C h a p t e r II, o r [ 3 3 ] , C h a p t e r V ) , and this is i n c l u d e d in Next an o p e r a t o r of type

maps

P r o p o s i t i o n 19. 7 and [ 2 9 ] , L e m m a 16. 1). (see [33], Chapter V, to

to

( s e e [ 9 ],

But s i n c e

§ 3 . 5 ) , L e m m a 12.6 s h o w s that

P

m a p s the r e s u l t

Thus (15.13) is p r o v e d . W e a l s o o b s e r v e that

(15.14)

The m a p p i n g

is bounded f r o m

T o p r o v e this w e c a n d i s r e g a r d the f a c t that properties,

and s i n c e it is a l s o an

to i t s e l f b o u n d e d l y . by T h e o r e m 1 4 . 4 ( a ) .

N o w if

has any s m o o t h i n g

b y T h e o r e m 14. 3 ( a ) , it m a p s then

Rest

The c a s e f o r

and so

is p r o v e d s i m i l a r l y ; (the

a r g u m e n t h e r e is v e r y m u c h the s a m e as in the p r o o f of P r o p o s i t i o n 1 5 . 2 ) . F i n a l l y an a p p l i c a t i o n of the i n t e r p o l a t i o n t h e o r e m f o r

spaces

s h o w s ( s e e [ 4 ] , T h e o r e m 10) that (15.13) w i t h (15.14) i m p l y the r e q u i r e d boundedness for

A , and h e n c e f o r

R.

L e m m a 15. 11 is t h e r e f o r e p r o v e d .

Now f o r a p p r o p r i a t e c u t - o f f f u n c t i o n s ( s e e P r o p o s i t i o n s 9 - 2 6 o r 10. 30).

w e have

Taking a s u m o v e r a finite c o v e r i n g

-159of

by c o o r d i n a t e p a t c h e s , and invoking L e m m a (15. 11) and P r o p o s i t i o n

15. 2 g i v e s (15.15)

where

and k = 0 , l , 2 , . .

Now it is w e l l known that by h o l o m o r p h i c c o n v e x i t y a r g u m e n t s one gets

, where

and

a

and

b

a r e any p o s i t i v e n u m b e r s ,

(See H i r s c h m a n n [ 1 6 ] and Caldero'n [14] . )

this it f o l l o w s that f o r any

e > 0, there exists a constant

C

From

so that S

(15.16)

I n s e r t i n g (15.16) in (15.15) (with

c h o s e n s o that

l e a d s to

(15.17)

We a r e now v e r y c l o s e to o u r g o a l , and need o n l y r e m o v e the t e r m ||uj| L

f r o m the r i g h t s i d e .

T h i s is d o n e by p r o v i n g

(15.18)

By the

t h e o r y ( s e e T h e o r e m 1 1 . 3 , (iii))

(15.19)

Next if

p

is s u f f i c i e n t l y l a r g e (p > 2n+2), then by S o b o l e v ' s t h e o r e m (see [33], Chapter V ,

give s

§2).

Thus the c a s e

k=0

of (15.17)

-160(15.20)

But if

then an e l e m e n t a r y a r g u m e n t s h o w s that

C o m b i n i n g this w i t h (15.19), the f a c t that

and

(15. 20) f i n a l l y l e a d s to

and h e n c e (15.18) if T h e n a standard i n t e r p o l a t i o n t h e o r e m , using (15.19), g i v e s (15.18) f o r .

Finally, since

N

is s e l f - a d j o i n t , a duality a r g u m e n t a l s o

p r o v e s (15.18) w h e n N

r e s t r i c t e d to

With this and (15.17) we have p r o v e d that is bounded f r o m

to

and h e n c e

has a unique bounded e x t e n s i o n , p r o v i n g p a r t (a) of T h e o r e m 15. 1. P r o o f of p a r t (b) of T h e o r e m 15. 1 . (15. 21)

Proposition.

We have f i r s t

is bounded m a p p i n g f r o m

to

The p r o o f is v e r y s i m i l a r to that of P r o p o s i t i o n 15. 2 e x c e p t now two d i f f e r e n t i a t i o n s in the " g o o d " d i r e c t i o n s ( i . e . , in t e r m s of can be a b s o r b e d in

K

giving an o p e r a t o r of type

( s e e [ 9 ] , P r o p o s i t i o n 15.14); the l a t t e r is bounded on know. Next

and

as w e a l r e a d y

-161(15.22)

T h i s f o l l o w s b e c a u s e in the nnain t e r m of

R

( s e e (15.12)) one of the d e r i v -

a t i v e s in the " g o o d " d i r e c t i o n s can be a b s o r b e d in of type

, giving an o p e r a t o r

T h e n the p r o o f of (15.14) s h o w s that (15.22) f o l l o w s .

So

P r o p o s i t i o n 9. 26 o r 10. 30 i m p l y that

H o w e v e r , b y p a r t (a) of the t h e o r e m w e have a l r e a d y p r o v e d , and ( 1 5 . 2 2 ) , we get

and p a r t (b) is a l s o p r o v e d . P r o o f of p a r t ( c ) of T h e o r e m 15. 1 •

T h i s is the d e e p e s t p a r t of the

t h e o r e m and r e q u i r e s the m o s t d e l i c a t e a n a l y s i s so f a r .

In e x p l a i n i n g

this it w i l l b e g o o d to r e v i e w s o m e of the i d e a s of the c o n s t r u c t i o n of the a p p r o x i m a t e Neumann o p e r a t o r One m a i n t a s k in C h a p t e r s 8 and 9 w a s to find the a p p r o x i m a t e l e f t i n v e r s e of the o p e r a t o r

The r e q u i r e d o p e r a t o r w a s

and w e

had in f a c t ( s e e P r o p o s i t i o n 8. 19 and (9. 10)) (15.23) w h e r e the e r r o r t e r m is of the f o r m F r o m this w e can o b t a i n a b e t t e r a p p r o x i m a t i o n to a l e f t i n v e r s e , n a m e l y f o r any i n t e g e r

m.

-162(15.24) where

N o t i c e that the f o r m s e n s e s ) , if

m

is now s m o o t h i n g of a h i g h o r d e r (in o u r v a r i o u s

is l a r g e .

Suppose we u s e operator.

instead of in K in o u r a p p r o x i m a t e Neumann

T h e n w e shall obtain an identity of the kind

is of the f o r m ( o m i t t i n g c u t - o f f f u n c t i o n s ) (15.25)

(15.26)

and

is of the f o r m

w h e r e the e r r o r is s m o o t h i n g of h i g h o r d e r . F o r the a p p l i c a t i o n s b e l o w w e s h a l l a l s o n e e d an a p p r o x i m a t i o n of h i g h d e g r e e to the r i g h t - i n v e r s e of have

where Thus (15.24') if

Now a n a l o g o u s l y to (15.23) w e

-163Hence So

Finally

(15.27) where error if

w h i c h is s m o o t h i n g of high d e g r e e

m

is l a r g e . W e c a n c o m e now to the p r o o f of p a r t ( c ) of T h e o r e m 15. 1 .

(15.28)

Proposition.

is bounded f r o m

It s u f f i c e s to c o n s i d e r the s m o o t h i n g of o r d e r

P

t e r m in ( 1 5 . 2 5 ) , the

2 in all d i r e c t i o n s . ,

where

G t e r m being

Now b y L e m m a 8.17

is an o p e r a t o r of P o i s s o n type of o r d e r

Thus

0.

term

is handled as in P r o p o s i t i o n 15. 2, and g i v e s a bounded o p e r a t o r f r o m to where

Next, P ( e r r o r . . . ) is s m o o t h i n g of high o r d e r (and s o m a p s

F i n a l l y what r e m a i n s is maps

15.29

P ( R e s t D^G)

which by our p r e v i o u s

arguments

The p r o o f of P r o p o s i t i o n 1 5 . 2 8 is c o m p l e t e .

Lemma.

maps

T h i s is now a s t r a i g h t f o r w a r d c o n s e q u e n c e of ( 1 5 . 2 6 ) , and r e q u i r e s no f u r t h e r d i s c u s s i o n . W e can now f i n i s h the p r o o f of p a r t ( c ) . P r o p o s i t i o n 15. 28 and L e m m a 15. 2 9 ,

We have b e c a u s e of (15. 2 5 ) ,

-164-

the l a s t inequality by p a r t (a) of the t h e o r e m .

Thus T h e o r e m 15. 1 is n o w

completely proved. We now g i v e the

a n a l o g u e of T h e o r e m 15. 1, p a r t s (a) and ( c ) .

A v a r i a n t of p a r t (b) in this c o n t e x t w i l l b e g i v e n b e l o w in T h e o r e m 15. 33. 15.30

Theorem. (a)

N

is bounded f r o m

(b)

Proof.

is b o u n d e d f r o m

F o l l o w i n g the a r g u m e n t s f o r the

one can show that b e c a u s e the t e r m s

inequalities

, and w h i c h o c c u r in

closely

(This and

R

map

to

b y T h e o r e m 14. 3 (b). ) Thus as b e f o r e

As a consequence (15.31) H o w e v e r , b y T h e o r e m 15. 1

M o r e o v e r if

then a c l a s s i c a l v a r i a n t of S o b o l e v ' s t h e o r e m

( s e e [ 2 5 ] , C h a p t e r 6) s h o w s that

where

.

Inserting

-165this in (15. 31) g i v e s p a r t (a) of the t h e o r e m .

P a r t (b) is p r o v e d in the

s a m e w a y as p a r t ( c ) of T h e o r e m 15. 1, using the r e f i n e d a p p r o x i m a t e Neumann o p e r a t o r

and its e r r o r t e r m

The d e t a i l s m a y b e l e f t

to the r e a d e r . A slight m o d i f i c a t i o n of the a r g u m e n t p r o v e s 15. 32

Corollary.

Suppose

Then

F o r the p r o o f w e need to o b s e r v e that if d i f f e r e n t i a l o p e r a t o r of o r d e r where Chapter V.

then

In f a c t

is the B e s s e l potential of o r d e r a ; s e e [ 3 3 ] ,

is 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 w i t h s y m b o l in the c l a s s

Thus

has o r d e r

[ 4 ], T h e o r e m 8). 14.2(b).

J^

-1

is a standard p s e u d o -

0 , and

maps

to

(see e . g .

Thus the a s s e r t i o n is p r o v e d if w e a p p e a l to T h e o r e m

Similarly,

maps

to

The r e s u l t s on L i p s c h i t z s p a c e s f o r the N e u m a n n o p e r a t o r have not b e e n c o m p l e t e l y p a r a l l e l with t h o s e f o r the S o b o l e v s p a c e s , b e c a u s e we have not s h o w n the full i m p r o v e m e n t that c o m e s about in the " g o o d d i r e c t i o n s " as in p a r t (b) of T h e o r e m 15. 1.

T h i s w e r e m e d y now.

W e shall f i r s t need s o m e d e f i n i t i o n s ; w e b e g i n by r e c a l l i n g one that w e have a l r e a d y u s e d . A smooth v e c t o r field its r e s t r i c t i o n to

bM

X

d e f i n e d on

is said to be a l l o w a b l e if

points in the " g o o d d i r e c t i o n s , "

i.e.,

if

-166W e a l s o d e f i n e the s p a c e studied in [ 9 ], p . 4 9 2 .

in a n a l o g y w i t h the s p a c e

F i r s t if

then

is an i n t e g r a l c u r v e in the function

of an a l l o w a b l e v e c t o r f i e l d ,

is in the c l a s s i c a l

In g e n e r a l , if

with

if w h e n e v e r

s p a c e , as a f u n c t i o n of

k an i n t e g e r , then

t.

if

range o v e r allowable v e c t o r f i e l d s .

15.33

Theorem.

Suppose

then

Remark.

The t e c h n i c a l d i f f i c u l t y i n v o l v e d in d e a l i n g w i t h the s p a c e s

is that an o p e r a t o r of type o p e r a t o r of type

preserve

d o e s not p r e s e r v e .

n o r d o e s an

T h i s r e q u i r e s that w e g i v e an a r g u m e n t

w h i c h is d i f f e r e n t f r o m that of p a r t (b) of T h e o r e m 15. 1 . The p r o o f w i l l r e q u i r e the f o l l o w i n g l e m m a . 1 5. 34

Lemma.

The P o i s s o n o p e r a t o r d e s c r i b e d in c o o r d i n a t e p a t c h e s

b y (7. 5 2 ) - ( 7 . 56)) m a p s

Proof.

and

W e n o t i c e f i r s t that r e s t r i c t i n g o u r s e l v e s to a suitable

coordinate patch,

where and

to

has the p r o p e r t y that

a r e all o p e r a t o r s of P o i s s o n type of o r d e r In f a c t b y (7. 55)

is bounded f r o m

has s y m b o l

0. I in o u r t e r m i n o l o g y ) .

-167with

where

d e f i n i t e 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

x.

is a p o s i t i v e

Thus

has the

kernel representation

(15. 35)

with

where

is the q u a d r a t i c f o r m c o r r e s p o n d i n g to the i n v e r s e m a t r i x ;

this is b e c a u s e of the w e l l - k n o w n identity

( s e e [ 3 3 ] , p. 61).

Hence

which proves

o u r a s s e r t i o n about

has s y m b o l of o r d e r

-1,

and so the a s s e r t i o n

c o n c e r n i n g it a l s o h o l d s . Now let

X

be an a l l o w a b l e v e c t o r f i e l d .

In an a p p r o p r i a t e

coordi-

nate n e i g h b o r h o o d w e c a n w r i t e

with

b ( x , 0) = 0, s i n c e

we also denote by (15. 6) and (15. 7), (15.36)

X

X

is tangential at the b o u n d a r y ,

(y=0).

Thus if

its r e s t r i c t i o n to the b o u n d a r y , w e have b e c a u s e of

-168and (15. 37) where

and

a r e P o i s s o n o p e r a t o r s of o r d e r

Next let

be the f u n c t i o n s d e f i n e d on ,

of L e m m a 13.5 .

f u n c t i o n in

0.

where

is d e f i n e d in the p r o o f

Then we have (as is e a s i l y v e r i f i e d ) , that

t,

, by

is an e v e n

and

(15. 38)

where

Also Now let

,

is in of the v e c t o r f i e l d

as a f u n c t i o n of X.

X

t, w h e r e

y

W e c l a i m that

is an i n t e g r a l c u r v e

T h i s can be s e e n by the d i s c u s s i o n in [ 9 ], p p . 4 9 2 ,

4 9 3 , if w e d e f i n e the " m e t r i c " w h i c h has

as in [ 9 ], p. 4 9 2 .

p in t e r m s of a n o r m a l c o o r d i n a t e

as o n e o f its b a s i s e l e m e n t s .

(local) h o m e o m o r p h i s m s given by

Next let

system

d e n o t e the

and s e t

(15. 39)

W e m a y a s s u m e (upon m u l t i p l i c a t i o n by a suitable c u t - o f f f u n c t i o n ) that

f

is s u p p o r t e d in a suitably s m a l l c o o r d i n a t e n e i g h b o r h o o d so that

(15. 39) is w e l l d e f i n e d .

However,

-169since X.

and

f

is in

along i n t e g r a l c u r v e s of

That is w e have

(15.40)

Now

(

1

5

and so

.

3

9

'

)

a

n

d

again b y (15. 38)

(15.41)

Similarly, (15.42)

W e h a v e , h o w e v e r , that

, because

T h e r e f o r e if

with

H e n c e the b o u n d e d n e s s of

s h o w s that

(by ( 1 5 . 4 0 ) ) (15.43)

Since

f

is in

then

s e e [ 9 ], T h e o r e m 2 0 . 1 .

Thus b e c a u s e of its d e f i n i t i o n (15. 39) (15.44)

Again because

(and

we have

-170-

T h e r e f o r e b y {15. 3 9 ' )

(15.45)

By (15. 36), oo The

L

b o u n d e d n e s s of

P^

gives

By L e m m a 13.12, (15.44)).

Therefore,

(15.46) is handled the s a m e w a y , using ( 1 5 . 3 7 ) and ( 1 5 . 4 5 ) .

The r e s u l t

is (15.47) The c o m b i n a t i o n of ( 1 5 . 4 3 ) , b e l o n g s to

on any i n t e g r a l c u r v e of

To summarize: However

and

( 1 5 . 4 5 ) , and (15.47) s h o w s that

If and

X , b y L e m m a 13.1 (and ( 1 3 . 4 ) ) . then

is a P o i s s o n o p e r a t o r of o r d e r

a r e o p e r a t o r s of o r d e r 0, and s i n c e w e h a v e that

w e have p r o v e d that if

if if

then

- 1 ; thus

when

Therefore

-171T h e r e s t o f L e m m a 1 3 . 4 is then p r o v e d by i n d u c t i o n , u s i n g the r e c u r s i v e definition of

and the identity (15. 36) w h e r e

X

is any a l l o w a b l e

vector field. W e p r o v e next that the a p p r o x i m a t e Neumann o p e r a t o r (We a l r e a d y know that it m a p s this f i r s t in the c a s e

L e t us d o

By the f o r m of

g i v e n in (15. 3), and s i n c e

it s u f f i c e s to c o n s i d e r the t e r m Rest

maps

, and

1 4 . 3 (b)).

Moreover

and

maps

Rest

m a p s this to

.

Now

(see T h e o r e m

maps

(For this,

s e e [ 9 ] , p . 465 and p . 4 9 2 . ) Thus an a p p l i c a t i o n of L e m m a (15. 34) s h o w s that

maps The argument for

n=l

r e q u i r e s that instead of using the a p p r o x i m a t e

Neumann o p e r a t o r g i v e n b y (10.29) and (10.23) (which has an e x t r a n e o u s t e r m in it), w e u s e the f o r m given by (10. 32). o p e r a t o r is of type g o e s as b e f o r e .

If

, and the a r g u m e n t f o r the t e r m that i n v o l v e s it

The " e r r o r t e r m "

, then

a t o r of type

is then of the f o r m

.

0, and so m a p s

So the r e s u l t f o r

However

is an o p e r -

to i t s e l f , b y [ 9 ], p p . 465 and 4 9 2 .

is p r o v e d in this c a s e a l s o .

W e a l r e a d y know (by T h e o r e m 15. 30) that o u r p r e v i o u s a r g u m e n t s show that 15. 33 is c o m p l e t e .

T h e r e the

Hence

, and the p r o o f of T h e o r e m

-172C h a p t e r 16.

Solution of

W e b e g i n by pointing out that o u r s o l u t i o n of the i s , s t r i c t l y in the i n t e r i o r of of two in the u s u a l s e n s e .

Neumann p r o b l e m

M , e l l i p t i c in the s e n s e that t h e r e is a gain

T h i s of c o u r s e f o l l o w s by the g e n e r a l

"interior

r e g u l a r i t y " of s o l u t i o n s of e l l i p t i c e q u a t i o n s , but in o u r c a s e it is a c o n s e q u e n c e of T h e o r e m s 15. 1, p a r t (b), and T h e o r e m 15. 33, s i n c e a l l o w a b l e v e c t o r f i e l d s a r e not r e s t r i c t e d away f r o m the b o u n d a r y . A n o t h e r f a c t is that the " n o r m a l "

c o m p o n e n t of the s o l u t i o n b e h a v e s

in an e l l i p t i c w a y e v e n up to the b o u n d a r y . follows.

Let

T h i s can be m a d e p r e c i s e as

be a s m o o t h ( 1 , 0 ) f o r m w h i c h n e a r the b o u n d a r y is

g i v e n by

.

(What it is away f r o m the b o u n d a r y is i r r e l e v a n t . )

In t e r m s of it we have the d e c o m p o s i t i o n , n e a r the b o u n d a r y , f o r any (0,1) fo r m

u,

where

is the c o m p o n e n t of

, and

is o r t h o g o n a l to

In g e n e r a l w e put The D i r i c h l e t b o u n d a r y c o n d i t i o n of . ( 1 The 1 . 2 ) mis r ei ns p 16.1 T h e o r e m app g o n s i b l e f o r the mf oalpl os w i n g r e g u l a r i t y p r o p e r t y . and

to

Proof.

, ji

to

It a l s o m a p s

It is s u f f i c i e n t to p r o v e that

p r o p e r t i e s as t h o s e c l a i m e d f o r and

,

and that

( s i n c e w e a l r e a d y know that

to

has the s a m e b o u n d e d n e s s maps

to or

-1 73uε

under our hypotheses).

16.2 Lemma.

On the s u b s p a c e of

W e s h a l l need the following o b s e r v a t i o n

g , s u c h that y(g) = 0 , y(P) i s a

P o i s s o n o p e r a t o r of o r d e r -1 . In f a c t the m a i n t e r m of P i s a d i a g o n a l o p e r a t o r .

The non-diagonal

p a r t of P h a s s y m b o l of o r d e r -1 ( s e e (7. 55)). T h u s the a r g u m e n t s w e h a v e a l r e a d y p r e s e n t e d in d e t a i l a b o v e s h o w t h a t tj(N ) and a

y( R ) h a v e the r e q u i r e d r e g u l a r i t y p r o p e r t i e s and the

theorem is proved. W e c o m e now to the p r o b l e m ( 1 6 .3)

au = f

where f (16.4)

is a given (0,1) form which satisfies Qf = O

in the w e a k s e n s e , i . e . ,

'Ii ( ( g ) φ , f) = 0 , f o r any φ e

CO

" (M), which has

c o m p a c t s u p p o r t in M . 2 — Whenever f e L (M), Harmonic forms , i.e.,

f s a t i s f i e s (16.4) and f

(f, φ) = 0,

is orthogonal to the

φ eV , then b y the f o r m a l i s m of the

g - p r o b l e m ( s e e e . g . [ 8 ] , p . 52), w e know that (16. 5)

U = JJN( f )

i s the unique w e a k s o l u t i o n of (16. 3) w i t h the p r o p e r t y that (16. 6)

(U, F ) = 0

2 — for all holomorphic function F which are in L (M). Observe that by the regularity theorems already proved for

N,

-174-

16.7 Then

Theorem.

Suppose

f

Is a ( 0 , 1 ) f o r m w i t h

and

U, g i v e n b y (16. 5 ) , is the u n i q u e ( w e a k ) s o l u t i o n of

which

s a t i s f i e s (16. 6 ) , f o r a l l h o l o m o r p h i c f u n c t i o n s , solution

U

morphic

F

when

with

then the

of (16. 3) is d e t e r m i n e d u n i q u e l y b y m

(a)

•

(U, F ) = 0

f o r all h o l o -

where

is any a l l o w a b l e

Moreover

U, and

ji

X

vector field.

Proof.

W e s h a l l r e s t r i c t o u r a t t e n t i o n to

s u f f i c i e n t l y s m a l l o p e n s e t in

U.

Thus

U

is a

, as w e h a v e d o n e s y s t e m a t i c a l l y a b o v e

and m a k e o u r e s t i m a t e s f o r s u p p o r t in

where

etc. , where

and h a s c o m p a c t

w U l m e a n that

w h e n b y an a p p r o p r i a t e c o o r d i n a t e s y s t e m w e h a v e i d e n t i f i e d

J with

a n e i g h b o r h o o d o f the o r i g i n in In

U w e c h o o s e an o r t h o n o r m a l f r a m e

and i t s d u a l f r a m e

.

T h e n w e k n o w ( s e e C h a p t e r 4 ) that

(16.8)

* See a l s o the d i s c u s s i o n in C h a p t e r 17.

-175where

N o t i c e that

, and so p a r t (a) f o l l o w s f r o m T h e o r e m (16. 1),

and T h e o r e m 15. 1, p a r t ( b ) , s i n c e the

Now if

X

a r e a l l o w a b l e v e c t o r f i e l d s if

is any a l l o w a b l e v e c t o r f i e l d , w e a l r e a d y know (by

T h e o r e m 1 5 . 1 , p a r t s (a) and ( b ) ) , that We can i n t r o d u c e new c o o r d i n a t e s ,

so that

and

locally.

, and a f t e r m u l t i p l i c a -

tion w i t h suitable c u t - o f f f u n c t i o n s w e have (16.9)

W e c l a i m that as a c o n s e q u e n c e of (16. 9) we have

In f a c t , l e t in (7. 58).

F be equal to the e x t e n s i o n of

to all of

T h e n b e c a u s e of (16. 9), and the c o m m u t a t i v i t y of

, as with

w e get

(16.10)

W e c l a i m that as a r e s u l t

(16.11)

Now l e t

b e the o p e r a t o r w h i c h is g i v e n by m u l t i p l i c a t i o n on

the F o u r i e r t r a n s f o r m s i d e by the f u n c t i o n

-176We know ( s e e [ 4 ]) that h e n c e on if

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

f o r all r e a l

y

with p o l y n o m i a l g r o w t h in y .

then so is

p o l y n o m i a l l y in y .

(and

, with n o r m g r o w i n g at m o s t

Next

, with n o r m again

p o l y n o m i a l l y g r o w i n g at w o r s t in y ,

since

Thus by the

c o n v e x i t y a r g u m e n t in [ 4 ], w e have

.

However,

w h e r e the o p e r a t o r s a r e bounded on on

Thus

(and h e n c e

, by the M a r c i n k i e w i c z m u l t i p l i e r t h e o r e m .

(A can

b e taken to be m u l t i p l i c a t i o n on the F o u r i e r t r a n s f o r m side by a s m o o t h f u n c t i o n of c o m p a c t s u p p o r t . s m o o t h and w h i c h f o r l a r g e

B

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

Thus

Going b a c k to the d e f i n i t i o n s w e s e e that e a c h b e l o n g s l o c a l l y to

, and so p a r t (b) of the t h e o r e m is a l s o p r o v e d .

The f a c t that T h e o r e m 15. 33.

if

follows directly f r o m

The f a c t that

w e have p r e v i o u s l y d o n e . an o p e r a t o r of type

In f a c t the

is a l r e a d y i m p l i c i t in what P

t e r m in

( s e e (15. 3)) i n v o l v e s

If w e apply an a l l o w a b l e v e c t o r f i e l d to this

t e r m w e get a s i m i l a r t e r m , but with

r e p l a c e d by

need u s e T h e o r e m 14. 3, p a r t (b) to c o m p l e t e the a r g u m e n t .

P

We then only The d e t a i l s

a r e so s i m i l a r to p r e v i o u s a r g u m e n t s that they m a y b e l e f t to the r e a d e r .

Chapter 17.

Concluding Remarks

In this chapter we point out some further results in order to round out the picture we have presented above.

We give only an indication of

the proofs, since the reader who has followed us this far should have no difficulty in filling out the required details. a.

The domain of • The first question we pose is that of giving a characterization of

those u, in terms of regularity conditions on u and boundary conditions, so that u belongs to the self-adjoint extension •

of • (described in

C h a p t e r 11) i . e . , w h e n i s u = N ( f ) , f e l ? ( M ) ; o r m o r e g e n e r a l l y w h e n i s u = N(f), where

f e lF (M)? k

Observe f i r s t that by T h e o r e m 1 5. 1 , if

P — fe L^(M), then

p(Z, Z)u 6lP (M) k

(17. 1) where ρ

is any polynomial of second degree in the allowable vector fields

Also (17.2) where

Z

n+1

is a holomorphic vector field which near the boundary equals

Incidentally, conclusion (a) of Theorem 15. 1 , namely that (17. 3) is a consequence of (17.1) and (17.2).

-178In f a c t , l e t

be a b a s i s of the h o l o m o r p h i c v e c t o r

f i e l d s (in an a p p r o p r i a t e c o o r d i n a t e n e i g h b o r h o o d of the b o u n d a r y ) . the d e r i v a t i v e s of of

Then

u along the r e a l and i m a g i n a r y p a r t s of

u b e l o n g to

because,

a r e a l l o w a b l e v e c t o r f i e l d s , when

M o r e o v e r b y the c o m m u t a t i o n r e l a t i o n ( L e m m a 6. 2 0 ) , again b e l o n g s to

F i n a l l y b e c a u s e of ( 1 7 . 2 )

Tu , and so

O b s e r v e a l s o that if (17. 1) and (17.2) a r e s a t i s f i e d (then b e c a u s e of (17. 3)) and the e x p l i c i t f o r m of

g i v e n in ( 6 . 2 3 ) , w e have

(17.4) Finally since

it f o l l o w s that its r e s t r i c t i o n

to the b o u n d a r y is a w e l l - d e f i n e d e l e m e n t in Thus

( s e e T h e o r e m 14.1)).

is w e l l - d e f i n e d and it can be p r o v e d by a s i m p l e l i m i t i n g

a r g u m e n t that (17. 5)

Similarly since

then in v i e w of ( 6 . 1 6 ) ,

and again (17.6) The c o n s i d e r a t i o n s a b o v e can b e s u m m a r i z e d as f o l l o w s . 17.7

Theorem.

Suppose

if and only if

u

u

is g i v e n on

M.

Then

u = N(f), where

s a t i s f i e s the r e g u l a r i t y c o n d i t i o n s (17. 1) and

(17. 2 ) , t o g e t h e r w i t h the b o u n d a r y c o n d i t i o n s (17. 5) and (17. 6).

-179b.

Solution of

when

f

is bounded

A c c o r d i n g to T h e o r e m ( 1 6 . 7 ) , the m a p p i n g solves

(which

w h e n e v e r this is p o s s i b l e ) is bounded f r o m In k e e p i n g w i t h the r e s u l t s w e have p r o v e d f o r

to a l s o show that it is bounded f r o m

we wish In o r d e r to state

the r e s u l t it is c o n v e n i e n t to give the s p a c e w e f i r s t g i v e the a l l o w a b l e v e c t o r f i e l d s on finite c o v e r i n g of

to

a norm. a norm.

T o d o this

We consider a

b y c o o r d i n a t e p a t c h e s , and in e a c h s u c h p a t c h w e

express

T h e n the n o r m of

, is the s u p r e m u m

of the d e r i v a t i v e s of o r d e r not g r e a t e r than one of the

taken o v e r

the v a r i o u s c o o r d i n a t e p a t c h e s . F i n a l l y the

n o r m of

all f u n c t i o n s

u

is the s u p r e m u m of the

where

n o r m s of

r a n g e s o v e r all s e g m e n t s of i n t e g r a l

c u r v e s of a l l o w a b l e v e c t o r f i e l d s of n o r m 17.8 (17.9)

Theorem.

The m a p p i n g

so that

defined f o r those

and

satisfies

W e w i l l now r e v i e w the b a c k g r o u n d of this t h e o r e m . In the c a s e w h e n an i n t e g r a l o p e r a t o r

is a s u b - d o m a i n of so that

Henkin [ 1 3 ] c o n s t r u c t e d when

and f o r w h i c h

=i=In o u r p r e v i o u s d i s c u s s i o n w e did not g i v e t h e s p a c e s n o r m s . H o w e v e r , if w e d e f i n e t h e i r n o r m ( s i m i l a r l y to the norm described b e l o w ) then T h e o r e m 15. 33, and T h e o r e m 16. 7 ( c ) , can be r e s t a t e d in t e r m s of b o u n d e d n e s s in t h e s e n o r m s .

-180-

Henkin and Romanov [14] proved the estimate (17.10)

Il H ( f ) Il

< A | | f (I

1/2

.

L 00

The latter followed e a r l i e r results of Grauert and Lieb [10 Kerzman [20], and others about similar operators. of us introduced the

Γ

J,

A little later one

spaces in [34] ; it was also asserted that an

estimate like (17.9), but slightly weaker, held for

H(f).

Actuallythe

proof we had in mind only showed something even weaker (namely that H(f)

is in

A

-L-S

in the "good directions").

together with other "sharp" results for

The details of that proof,

H(f), appear in the thesis of

S. Krantz [23]. To prove Theorem 17. 8 we shall use still another solution of the problem

§U = f, the one which is studied by Phong in his forthcoming

dissertation [28]· 3Φ(ί) = f

This solution,

f—>φ(ί), is the one characterized by

with φ(ί) orthogonal to holomorphic functions, the orthogonality

being in terms of integration taken on the boundary. of Kohn, one can give a simple expression for φ(ί) of the corresponding

Following a suggestion in terms of solutions

^ problem

In fact we shall see that the estimate (17. 9) holds not only for the solution $N(f), but also for

H(f), and

φ(ί).

H(f) was also proved recently by Henkin,

The fact that it holds for but by different methods.

We f i r s t sketch the proof of Theorem (17. 9) when η > 2. * The definitions given in [34] for Γ The Γ

OL

Let C

have undergone a notational change.

spaces used here correspond to the Γ .

our spaces Γ are called Γ ,, in Krantz on a/2 ,a **Personal communication.

r

OU

23 •

spaces in [34].

Also

-181d e n o t e the C a u c h y - S z e g o p r o j e c t i o n o p e r a t o r , w h i c h b y the f o r m a l i s m f o r the b o u n d a r y c o m p l e x ( s e e e . g .

[ 8 ], C h a p t e r V) c a n b e w r i t t e n

(17. 11) where

G^

is the " N e u m a n n o p e r a t o r " f o r the

r e g u l a r i t y p r o p e r t i e s of

complex.

it f o l l o w s that if

F r o m the

then

and h e n c e the h o l o m o r p h i c f u n c t i o n Now s e t

where

and T h e n the f u n c t i o n

satisfies

on b M to h o l o m o r p h i c f u n c t i o n s . (17.12) and if

and

w

is o r t h o g o n a l

F r o m this it f o l l o w s that

then

(17.13) where

is the " r e s t r i c t i o n " of the ( 0 . 1 ) f r o m , and s i n c e

f to

bM.

Now

is the s o l u t i o n of (17. 13) o r t h o g -

onal to h o l o m o r p h i c f u n c t i o n s , it f o l l o w s f r o m [ 9 ] ,

17 that

(17.14)

Similarly since (except for smoother t e r m s ) of

fj^ b y o p e r a t o r s of type

is e x p r e s s e d in t e r m s

, w e a l s o have b y T h e o r e m 14. 3, (b)

(17. 15) N o w b y an a n a l o g u e of what w a s d o n e in C h a p t e r 7 f o r the D i r i c h l e t

-182problem for

where

and

w e have

a r e a p p r o p r i a t e G r e e n and P o i s s o n o p e r a t o r s .

Since

w e get b y e l l i p t i c e s t i m a t e s (of the type w e h a v e

a l r e a d y m a d e ) that (17.16) Next analogue of L e m m a (15. 34) t o g e t h e r with (17.14) and (17.15) s h o w s that

Putting t h e s e t o g e t h e r g i v e s (17.17)

O u r t h e o r e m w i l l then be p r o v e d if we c a n show that the h o l o m o r p h i c function

:

satisfies

(17.18)

T h i s is a c o n s e q u e n c e of the f o l l o w i n g g e n e r a l f a c t : 17.19

Lemma.

Suppose

Then

F

is h o l o m o r p h i c in M

and b e l o n g s to

and

T h i s r e s u l t is a c t u a l l y true in m u c h m o r e g e n e r a l s e t t i n g . no p s e u d o - c o n v e x i t y h y p o t h e s e s on the b o u n d a r y need b e m a d e . t h e r e is an a n a l o g u e w h e r e

and

a r e r e p l a c e d by

First, Secondly, and

-1 83respectively for

0 < a < »·

T h a t w a s a n n o u n c e d b y o n e of u s i n [ 3 4 ] ;

since details have not yet appeared we shall give the proof in the case that is needed, namely a = 1/2·

The general case can be proved similarly

Everything is based on the following two simple observations. Suppose

P i s a p o i n t i n M a n d t h e d i s t a n c e of

P from bM is

a n y c o m p l e x o n e - d i m e n s i o n a l d i s c , c e n t e r e d a t P of r a d i u s

δ·

Then

6 lies in M.

H o w e v e r , if t h e o n e - d i m e n s i o n a l d i s c l i e s a l o n g a " g o o d d i r e c t i o n " t h e n e v e n if i t h a d a m u c h l o n g e r r a d i u s i t w o u l d s t i l l l i e i n M . there exists positive constants

c^

a n d c ^ , s o t h a t if

X

To be precise,

is any allowable

v e c t o r f i e l d (of n o r m < 1 , a s d e f i n e d a b o v e ) , t h e n (17.20) radius

The one-disc, centered at P, in the direction c^g

1/2

(C X p , a n d o f

, lies in M.

Moreover this disc is at a distance at least c^6 from bM. ( 1 7 . 2 0 ) i s a d i r e c t c o n s e q u e n c e of t h e d e f i n i t i o n o f a n a l l o w a b l e v e c t o r f i e l d , si n c e

X

'bM

1,0 0,1 eT φ T... .. (bM) (bM)

T h e s e c o n d o b s e r v a t i o n i s a n e a s y c o n s e q u e n c e of C a u c h y ' s i n t e g r a l formula.

We suppose that

f

i s a h o l o m o r p h i c f u n c t i o n of o n e c o m p l e x

variable defined in the disc D (17.21)

If

If I < 1

(17.21')

If

||fl|

r

of r a d i u s

in D , then

|f'(0)|