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English Pages 202 [201] Year 2015
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)|