Beijing Lectures in Harmonic Analysis

BEIJING LECTURES IN HARMONIC ANALYSIS EDITED BY E. M. STEIN PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1986 ...

Author: Elias M. Stein

66 downloads 1169 Views 14MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

BEIJING LECTURES IN HARMONIC ANALYSIS

EDITED BY

E. M. STEIN

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1986

Copyright © 1986 by Princeton University Press "LL RIGHTS RESERVED

The AMals of Mathematics Studies are edited by William Browder, Robert P. Langlands, John Milnor, and Elias M. Stein Corresponding editors: Stefan Hildebrandt, H. Blaine Lawson, Louis Nirenberg, and David Vogan Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding materials are chosen for strength and durability. Paperbacks, while satisfactory for personal collections, are not usually suitable for library rebinding ISBN 0-691-08418-1 (cloth) ISBN 0-691-08419-X (paper) Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

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

TABLE OF CONTENTS

vti

PREFACE NON-LINEAR HARMONIC ANALYSIS, OPERATOR THEORY AND P.D.D. by R. R. Coifman and Yves Meyer MULTIPARAMETER FOURIER ANALYSIS by Robert Fefferman

3 47

ELLIPTIC BOUNDARY VALUE PROBLEMS ON LIPSCHITZ DOMAINS by Carlos E. Kenig

131

INTEGRAL FORMULAS IN COMPLEX ANALYSIS by Steven G. Krantz

185

VECTOR FIELDS AND NONISOTROPIC METRICS by Alexander Nagel

241

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS by E. M. Stein

307

AVERAGES AND SINGULAR INTEGRALS OVER LOWER DIMENSIONAL SETS by Stephen Wainger

357

INDEX

423

v

PREFACE In September 1984 a summer school in analysis was held at Peking University. The subjects dea It with were topics of current interest in the closely interrelated areas of Fourier analysis, pseudo-differential and singular integral operators, partial differential equations, real-variable theory, and several complex variables. Entitled the "Summer Symposium of Analysis in China," the conference was organized around seven series of expository lectures whose purpose was to give both an introduction of the basic material as well as a description of the most recent results in these areas. Our objective was to facilitate further scientific exchanges between the mathematicians of our two countries and to bring the students of the summer school to the level of current research in those important fields. On behalf of all the visiting lecturers I would like to acknowledge our great appreciation to the organizing committee of the conference: Professors M. T. Cheng and D. G. Deng of Peking University, S. Kung of the University of Science and Technology of China, S. L. Wang of Hangzhou University, and R. Long of the Institute of Mathematics of the Academia Sinica. Their efforts helped to make this a most fruitful and enjoyable meeting. E. M. STEIN

vii

NON-LINEAR HARMONIC ANALYSIS, OPERATOR THEORY AND P.D.E.

R. R. Coifman and Yves Meyer Our purpose is to describe a certain number c:i results involving the study c:i non-linear analytic dependence of some functionals arising naturally in P.D.E. or operator theory. To be more specific we will consider functionals i.e., functions defined on a Banach space of functions (usually on Rn ) with values in another Banach space of functions or operators. Such a functional F:B 1 -• B 2 is said to be real analytic around 0 in B 1 if we can expand it in a power series around 0 i.e. 00

F(f) =

l: i\k(f) k=O

where i\k(f) is a "homogeneous polynomial" of degree k in f. This means that there is a k multilinear function

(linear in each argument) such that i\k(f) = i\k(f, f, ···f) and

(1)

for some constant C. (This last estimate guarantees the convergence of the series in the ball

I f II 81 < ~ ·)

3

4

R. R. COIFMAN AND YVES MEYER

Certain facts can be easily verified. In particular if F is analytic it can be extended to a ball in B~ (the complexification of B 1 ) and the extension is holomorphic from B~ to B~ i.e., F(f +zg) is a holomorphic (vector valued) function d

zl

\z\ < 1,

C,

Vf, g sufficiently small.

The converse is also true. Any such holomorphic function can be expanded in a power series, (where Ak is

t!

x the kth Frechet differen-

tial at 0 ). We will concentrate our attention on very concrete functionals arising in connection with differential equations or complex analysis, and would like to prove that they depend analytically on certain functional parameters. As you know there are two ways to proceed.

1. Expand in a power series and show that one has estimates (1).

2. Extend the functional to the complexification as "formally holo1 morphic" and prove some boundedness estimates. Let L denote a differential operator like a(x)

d:

xtR,

a(z)

~

ztC

the coefficients a(x) (or aij(x) ) will be assumed to belong to some Banach space B 1 of functions (for example L"" ). It is natural to ask when such objects as:

or more generally, ¢(L) (where ¢: C .... C ), can be defined as a bounded operator (say on L 2 or some Soboleff space), and a functional calculus developed i.e., ¢ 1 (L)¢ 2 (L) "' ¢ 1 ¢ 2(L) ·

NON-LINEAR HARMONIC ANALYSIS

5

Many questions arise: a) Does F(a) = cf>(L) viewed as an operator valued function of a depend analytically on a ? This is equivalent to asking whether we can consider complex valued coefficients in L and still have estimates on cf>(L). b) What is the largest domain of coefficients a for which we have estimates for cp(L) ? This question is the same as asking what is the

largest 8 1 for which (1) holds, and what is the domain of holomorphy of F(a) in this space. The answer to question a) will require first that we understand methods for expanding functionals in a power series, and second, that the nature of the multilinear operators Ak be sufficiently well understood to provide estimates (1). As for question b) we will see that the largest spaces possible for the coefficients involve rough coefficients and leads us to work with coefficients in L 00 , B.M.O. and other "exotic spaces." We now start with a fundamental example related to the Cauchy integral. We let L =-1-M a 1-ra idx If we define h(x)

=

with llalloo ll') 2

The first term is dominated by

For the second we observe that

lrl Integrating in y we get

which shows that second term is bounded by Cllbll,.,. We have thus shown the necessity of the condition T(l)

l

BMO. Before stating the theorem

precisely we would like to reformulate it somewhat.


Let ¢

E

C 0(R 1) with JtPdx = 1 and r/1

=

¢(x). Let ¢f(u) =

} ¢ { ut) and similarly for r/Jf(u). We claim that under the preceding assumptions on T we have

In fact, assume for simplicity, that rp is supported in (-1,1). Since Jr/ldu

=

0, if we assume lx-yl > 3t

(where we used the fact that jy-zj < t, !x-u!< t, lx-y\ > 3t and the hypothesis layK(x,y)j ~ lx-yl- 2

).

If lx-yl < 3t we use the antisymmetry of k(x,y) to write

but

lr/lf(u)¢f(z)-¢f(u)r/l~(z)\ ~

lu-:1 and the fact that lu-x!< t, t

lu-zl

< t, lx-yl < 3t and IK(z,u)l ~ -

1-

lz-ul

imply

Combining these estimates proves our claim.

11

12

R. R. COIFMAN AND YVES MEYER We can now state

THEOREM (G. David, J. L. Journe) [7]. Let T:~ .... ~- such that for some

e>O

II:S!. t

1 =pt(u-v) 1 1 + ~U-VI +E t

and Il
*

1

l+e

=pt(u-v).

1 + I!:!.::Y I t

Then the necessary and sufficient condition for T to extend to a bounded operator on L 2 into L 2 is that T(l) and T*(1) be in B.M.O. We would like to make a few comments concerning the conditions *. We have just seen that if T(f) = p.v.

I

k(x,y)f(y)dy

where 1°

lk(x,y)J < - - 1 1x-y 1

and

for some e > 0. II

s T.

13


(This last condition followed from k(x,y) = -k(x,y) and 1°, 2°) then the conditions

*

are verified. It can be shown that if

*

is valid then T

can be represented as a limit of integral operators whose kernels satisfy the conditions 1°, 2°, 3°. We will refer to 1°, 2° as standard estimates, and to 3° as the weak cancellation (or boundedness) property. This last condition is independent of 1°, 2°, and can be proved in a variety of ways, as we shall see. To see how the theorem can be applied to reduce the degree of nonlinearity a of a "polynomial" and to obtain estimates consider

We check by a simple integration by parts that

If we make the induction hypothesis that Cn_ 1 (a,f) is bounded on L 2

it would follow by the preceding remarks that Cn_ 1 (a,f) maps L"" to B.M.O. from which we deduce that Cn(a,l)

t

B.M.O. and that 3C > 0

such that IICn(a,f)li

2

L

~ cnllalln

L 00

llfll

2 •

L

We will see later that this reduction of en to Cn-I is not a trick but can be accomplished in general to reduce the study of n+l multilinear operators to an n-multilinear. Proof of the theorem. For simplicity of notation we'll denote

We start with the fundamental theorem of calculus writing for f t

C~(R)

14


J

1 /e

Tf

=

lim P 2TP 2 f t-+ o t t

=

-lim

e-+0

t

~ (P 2TP 2 f) !!!. at t t t

e

We'll study only the first term (since the second is its adjoint). We start by observing that t

£P[g (O

=

t:

·P..)- 1 , in fact, L - 1f

=J

i f(l-ta)fdv(w) . 11(z-w)

We now observe that if h(z) is such

a h = 1-ta az

and h(z) - z fL oo

then

from which we find (L-{) f = xL!. f and (L-(r 1 = xL - 1 !_ f )(

)(

.

or more precisely

(L-()-1f =

~

~ and

~J-1- eil(h(z)-h(w)}(+(z-w >(I ~- f(w)dw z-w

E

uw

(L-(rlf =feil(h(z)-h(w))(+(z-w)£1 ah f(w)dw

aw

32


= Jk(h(z)-h(w),z-w)!

f(w)dw

where k(u,v) =

f cf>(()eiu~+V(d~ c

is the Laplace transform of cf>. This gives us an explicit kernel realization of the calculus. It can be shown independently that

Since

~

=

l+a, for this to be bounded we must have

~

£

L 00 i.e.,

a*.!...£ L 00 • In other words, in order to have a functional calculus in z2

1~

t

(consisting of bounded operators in L 2(C 1)) it is necessary to

assume that a

£

L oo and the 2-dimensional Hilbert transform (or Ahlfors

Beurling transform) of a is also bounded. This version d H 00 is quite interesting and should be better understood.

§5. Until now we have only shown that small perturbations of certain operators are bounded, we would like to describe an extension method due to G. David.


We proved earlier that the commutators

f(y) d C k(a, f) =J(A(x)-A(y)\k x y x-y Y

J

A'(x) = a(x)

satisfy

(This is A. P. Calderon's theorem [1].) If we recombine these terms in a series we obtain, for example, that

I

f(y) dy x-y+iA(x)- iA(y)

is a bounded operator on L 2 if IIA'IIoo < ~ and that

It was proved in [4] by a careful analysis of the functional calculus interpretation of the Cauchy integral that in fact,

This more precise result implied a much stronger statement. THEOREM

1 (4].

Let ¢ ( C 11 (R). A real valued IIA'IIoo < 00 then

33

34

R. R.

COIFMAN AND YVES MEYER

It is our purpose to describe a direct real variable method to obtain this result. The main idea, due to G. David, is that on each interval a Lipschitz function with a given norm is, in fact, equal (modulo an additive constant) on a substantial subset to a Lipschitz function with a fraction of the norm. By iteration this permits estimates for large norms in terms of smaller ones.

-

To get results like Theorem 1 it is enough to show that for A real i.t'A(x) -A(y) IIT,;(OII2

=

Je

X y

~ 0 such that for each interval I there is a constant C(I)

(dependin~

continuously on I ) for which lxd: lb(x)-C(I)I

Then b

f

~al >~III.

BMO and llbiiBMO ~ cNa.

The main idea to estimate the BMO norm of T(l) is to replace inside eachinterval I, TA byanoperator TA &action of I and

AI

where

AI~A

onalarge

I

has a smaller Lipschitz norm, and then compare

TA (1) to TA (1). This is achieved via the following lemma, the first of I

which is the rising sun lemma (or the one-dimensional version of the Calderon-Zygmund decomposition). LEMMA

1. Let A be such that C-M < A '(x) < C+M

then for each I there exists a function AI and a constant CI such that AI= A on a set

E lEI>} III

and

Proof. We can assume C a) m~A')~M.

=

M i.e., 0 M and we construct A 11 as above.

A 11 = 2M(x-a) -A(x) except on a set of me as

~ ~ II I

A(x) = 2M(x-a)- A 11 =AI A'(x)= 2M- A{= AI

4

2

,

21

2

2M - - M - - M < 2M - A 1 0,

c

> 0 such that

lor each I there exists K 1(x,y) satisfyin8 standard estimates uniformly in I with

Batisfying

and there is a subset

E ~I

with

Vx cE, Vy cE,

lEI > 8111

such that

K1(x,y) = K(x,y).

38


Then T maps L 00 to BMO with IITII

If we let u(M)

=

sup

\\A:\Ioo- 0 and we conJecture

clearly the case whenever mf

x,y

that this condition gives the domain of holomorphy. [Before discussing other examples we urge the reader to identify the space of holomorphy for F(a)

= sgn ( 1:a :x) . [Caution: It is not enough to consider all Ak

separately.)

44


A more interesting example arises if we reconsider the Cauchy integral on rectifiable curves, which we chose to parametrize by arc length (and not as graphs). We write z(s)

=

f 05

ei(a(t))dt and consider the L 2 operator valued

functional

J

00

C(a)f = 21 . p.v. 771

-

f(t)z'(t) dt = ~ A (a)f z(s)-z(t) ._, k

00

here again one can define the norm

lila Ill

=

IlA 1(a) I L 2 ,L 2

and one finds that this norm is equivalent to the BMO norm of a. On the other hand it is easy to show that if

llallsMO 0.

Then there exist disjoint cubes Qk such that

1 (1) aa. IQ'I o

For these Q' we stop the bisection process. For the

rest, we continue until we first arrive at a cube Q' such that - -1- f. ,f >a, 1Q'l Q at which point we stop. The cubes Q' at which we stop are then our Qk. By construction

_l_

f.

IQkl ok

f >a. Let Qk be the cube containing Qk which was bisected

to produce Qk. Then IO'kl = 2°1Qkl and since we did not stop at Q'k,

_1_ ff1 f < a. It follows that IO'kl k -

49

MULTIPARAMETER FOURIER ANALYSIS

proving (1). Notice that (3) follows from (1) because \Qk\

~

k fQ

f so k

summing on k, we have \UQk\

:::~I

J cs~ J Qk

Finally, (2) follows, since for each x

c.

Rn

I

UQk, x belongs to a sequence

of cubes Ck whose diameters converge to zero and such that Cl

f.

k ck

f:::; a. It follows from Lebesques theorem on differentiation of

integrals that f(x)

Sa a.e. for such x.

We all know how important the maximal operator of Hardy-Littlewood is in the subject of Fourier analysis. This is the operator given by

Mf(x) = s u p1- r>O jB(x;r)j

I

\f(t)! dt .

B(x;r)

Going along with this we also define Msf(x) =

sup xeQ dyadic cube

-1 \Qj

J

if(t)l dt .

Q

(Recall that a dyadic interval of R 1 is one of the form [j2k,(j+1)2k] j,k e Z and a dyadic cube is a product of dyadic intervals of equal length; recall also the basic property of dyadic cubes-if Q 1 ,Q 2 are dyadic either Q 1 nQ 2

=0,

Q 1 ~Q 2 or Q 2 ~Q 1 .) Thenthefollowingsimple

lemma sheds some light on the relationship of the Calderon-Zygmund lemma to the Maximal Operator.

50

ROBERT FEFFERMAN

2: 0

LEMMA. Let f

L 1(Rn) and a

t

> 0. Let Qk be Calderon-Zygmund

cubes as above, and let Q'k denote the double of Qk. Then there exist positive numbers c and C such that (1) UQk cal (2) uQ'k ,2(Mf >Cal

(3) FurthermJXe, if Qk are dyadic, then UQk 2 (M8f >Cal.

Proof. (1) Let x X (

Qk

c B(x;r)

t

and

-

Mf(x)

Qk. Then there exists a ball B(x;r) such that \B(x;r)\ <en. Then IQkl -

>

1 - jB(x;r)j

I

jfj > llQkl ) _1_ - \IB(x;r)l IQkl

I

B(x;r)

jfj >....!_ a -en .

Qk

(2) Let x I UQk. Let r > 0. Then we estimate

~ ajB(x;r)J

1-

I,

J

ops10

Qi

:S a\B(x;r)j -t 2na

I,

f

JQjl .

oinsl0 Key point: if Qj

n

B(x ;r)-:/ rJ then Q j ~ B(x;l Or) so that

ops#0 That is, Mf(x)

'S C 0 a.

B(x,r)


51

The use of the Calderon-Zygmund lemma is apparent in the CalderonZygmund Theorem on Singular Integrals: Let X and N be Banach spaces and let B(X,N) denote the bounded operators from X to N. Let K: Rn x Rn/lx=yl-. B(X,N) satisfy (1)

lhl 5 :;g for ihl lx-yjn

IK(x,yth)--K(x,y)! 8 (X,N) S

< Ix-y I 2

and for some 5 > 0.

(2) if Tf(x) = J

Rn

Po> 1 IITfll

K(x,y)f(y)dy for f

S Cllfll

P L

0 (N)

£

LP(X), and suppose for some

P L

0 (X)

Then, for T we have !lxl \Tf(x)IN >ali S ~ lifll 1 L

(X)

and IIT£11 P L

Proof. Let

(N)

:S Cplif!l

L

P

(X)

for 1 < p 0 and f E L 1(X). Set _l_ IQkl g(x)

=

f dt if

X (

Qk ,

Qk

f(x) and b(x) = f (x)- g(x). Then

As for Tb(x), suppose

I

xI UQk.

if

X

I Qk.

52

ROBERT FEFFERMAN

Tbk(x)

=

J

K(x,y)bk(y)dy.

Qk

Let Yk be the center of Qk; then

I

K(x,yk)bk(y)dy = K(x,yk)

f

Qk

bk{y)dy = 0

Qk

so Tbk(x)

=

J

IK(x,y)-K(x,yk)lbk(y)dy

Qk

and

summing over k we have

Thus

From this weak (1,1) estimate, interpolate to get the LP result. We now quote some important examples: 1. Classical Calderon-Zygmund Convolution Operators. Here Tf = f * K where K(x) is a complex-valued function satisfying

53

MULTIPARAMETER FOURIER ANA LYSIS

\K(x)\ ~ C! lxln ;

(a)

J

({3)

K(x)dx = 0 for all 0 < p 1 < p2 ;

Pt~lx\~2 and

5

\K(x+h)-K(x)\ < C ~ lxln;.o

(y)

The Riesz transforms Rl

=

\hi 2, 00

cp(x) = 0. Then define K:Rn ++ L 00((0, oo); dt) by K(x){t) = ¢t(x) = t -ncp(x/t). Then I'VxK(x)(t)i

=

lc lxl/2, \lcp(x/t)

=

< CIIV¢11 -

L oo(Rn)

1 lx jn+l

0 ).

Again IK(x+h)-K{x)l L

< C _lh_l if lhl < !. 2 lxl , lxln+l

oo -

andwealsohave llf*KII L oo(L oo) ~Cil£11 L oo since ih¢t(x)l~ll¢11 1 11£11. oo Then Mf(x) "-' If* K(x)l

Loo

so M is bounded on LP(R~, p > 1 and

weak 1-1. 4. The Estimates for Pointwise Convergence of Singular Integrals on L 1(Rn). Suppose that K(x) is a classical Calderon-Zygmund kernel and let Ke{x) existence a.e. of

=

K(x) · XjxJ>e(x), for e > 0. We are interested in the

lim h Ke(x) for f e ~o

f

L 1 (Rn). In order to know this, it


55

is enough to show that T*f(x) =sup If* Ke(x)l satisfies the weak type e>O estimate i!T*f(x) >ali~~

fRn

If\. It turns out that by using the Hardy

Littlewocxl maximal operator it is not difficult to prove T*f(x) ~ C{M(Tf)(x) + Mf(x)l which immediately gives the boundedness of T* on LP(Rn) for p > 1 . However, it fails to give the weak type inequality for functions on L 1 (Rn). This inequality follows easily from the observation that T* is a singular integral. Let

and set Kix) = K(x)[1-~(~)] . Then \Kix)-K(x)\

~ \x~n )(~\xl 9 e (x)

so that T*f(x) ~sup If* ~e(x)\ + Mf(x), and so we need only show that e>o sup if* Kel is weak type (1,1). In crder to do this let H:Rn-+ L ""((O,oo);de) e>O be given by H(x)(e) = K~x). Then \H(x)-H(x+h)l ~ 9!1._ and H is • L oo \xln+l bounded from L 2 -+ L 2 (L j , so H is weak 1-1 .

5. The Maximal Function as a Littlewood-Paley..Stein Function. Let f

E

L 2 (Rn), f(x)'~ 0 for all x. Use the Calderon-Zygmund decomposition

with

a=

cubes

Cj, j

f

Qt where

Z for some C > 0 sufficiently large, to get (dyadic)

~ f . f ""' Cj.

decomposition,

IQLI Q~

Define f j as in the Calderon-Zygmund

f(x)

if x

i

U Q~ k

and

~.f = {. 1 -f·, J J+ J

then observe that:

56

ROBERT FEFFERMAN

(1) !1/ lives on ~

QL

Qt.

and has mean value 0 on each

(2) Llif is constant on every Q ~ for i < j. +oo (3) f.-> 0 as j-> J

-oo

and f. ->f as j ... +oo so f J

From (1) and (2) it is clear that the -

!lf!IL2(Rn)

=

(~ llt1lll~2)

Lll

=

~ j=-00

!1.£. J

are orthogonal so that

1/2

l/2 =

li

( ; lt1/(x)j 2)

IIL2 ·

2) 112 1·s . 11 y, ob serve t hat t h e square f unction ('"'j".f(x)j F ma .._ ~ j J

essentially just the dyadic maximal function. In fact, if then x

E

Qi

for some k and Llj(f Xx) ""'

ci « M0 (f)(x)

ci. It follows that

Before finishing this section, we shall need estimates near L 1 for the maximal function.

If Q denotes the unit cube in Rn then for k a positive integer

J

J

Q

Q

Mf(log+ Mf)k-ldx < oo if and only if

The proof runs as follows': If f

llx!Mf(x)

E

L(log+ L)k then

>all ~ ~

J jf(x)j>a/2

and so

!£!(log+ !f!)kdx < oo.

f(x)dx


IIMfll L(log

J 00

(log a)k- 1

I

~

L)

k-1

57

:s

lf(x)ldxda :S

lf(x)l>a/2

1

~ llfll

L(log L)

k.

Conversely, (Stein) Calderon-Zygmund decompose Rn at height a > 0. We have

J

f(x)dx :S

M0 f(x)>ca

J

f(x)dx

~

UQk

This yields

J

\f(x)l(log+M0f(x))kdx

~ Joo ~ '

I

\f(x)\dx(log a)k-lda

M0 f(x)>Cna

Qk

I

1

0.

The nature of the real variable theory involved does not

seem to depend at all on the dimension n. In marked contrast, it turns out that a study of the analogous operators commuting with a multiparameter family of dilations reveals that the number of parameters is enormously important, and changes in the number ci parameters drastically change the results. Let us begin by giving the most basic example, which dates back to Jessen, Marcinkiewicz, and Zygmund. We are referring to a maximal operator on Rn which commutes with the full n-parameter group of dilations (x 1 ,x 2,···,xn) ... (8 1 x 1 ,B 2x 2 ,···,onxn)• where oi>O isarbitrary. This

is the "strong maximal operator," M(n), defined by

M(n)f(x) = sup _!_ xfR

IRI

J

jf(t)j dt

R

where R is a rectangle in Rn whose sides are parallel to the axes. Unlike the case of the Hardy-Littlewood operator, M(n) does not satisfy

For instance when n

=

2 and when

then for lxtl. lx 2 1 >28,

and

fa = s- 2 X(jx 1 j -1- or (}' - ((}'- (})

(J' - (J

->c

(}'

and this in our case of (Jk = 2-k is valid with c = 1/2. (If (Jk = (1- e)k, c =e.) Then summing over j in (2.5) we have

lSI1J~ ~ S

~j

before R

x~. 2: J

R1J~ ~ I I

R

~ before

in other words

and by the boundedness of M( 2 ) we see that

1

x~. 2: 2 c R

J


67

by (2.5). We have controlled m by M< 2 > here in just exactly the way M(n) was previously controlled by M(n- 1 ). And while m is a 3-parameter maximal operator, M( 2 ) is a 2-parameter one. Finally, we should note that, as before, this covering lemma implies the maximal theorem for m as claimed. In fact, suppose we have shown that given IRk I there exists IRk I a subsequence so that (1) /URkj

(2}

~ c/URk/

II~XR lip·~ c/URki 11P', (here !.+1_=1). ~

p

k

p'

Then

llmf> all ~ (:C

1/fll ~p /;

•

In fact we have, by definition {Rkl so that lmf> al ~ URk and - 1- f. /f/ >a for all k. By the covering lemma, select the class IRk!. IRk/ Rk Then /Rk/

~} f~

lfl and so k

and the estimate on

II mf >all

follows by a division of both sides by

IURk/1/p: Our last topic for this lecture will be the so-called Zygmund Conjecture. I believe it was Zygmund who was the first to realize the difference between the one-parameter and several parameter harmonic analysis. Particularly, he remarked that in differentiation theory a "big picture" was evolving. He considered n functions ¢ 1 ,¢ 2 , ···,¢n of the positive real variable t, with each ¢i(t) increasing and the family of rectangles

68

ROBERT FEFFERMAN

lRt!t>_o given by Rt =

ll [- cpi(t), cpi(t)J. 2 2

Form a maximal operator M

i=l

defined by

M(f)(x) = sup -11 I t>O Rt

J

if(x+y)l dy .

Rt

Then M is of weak type 1-1 , just as in the special case of the HardyLittlewood operator where cpi(t) = t. Zygmund noticed that the proof of this was virtually the same as the Hardy-Littlewood theorem. All one had to do was to prove a Vitali-type covering lemma for Rt's using the fact that if IJ,l is the class of all translates of the Rt and if R, S R n S -f.

!

IJ,l and

0 and if R corresponds to a bigger value of t than does S,

then S ~ R, the S-f old dilation of R. Next, he considered the collection of rectangles Rs,t, s,t > 0 where R t = s, where

cp

[-~ ~] 2'2

x

[-!_2'2t.]

X [-

cp(s,t) cp(s,t)J 2 , 2

is a function increasing in each variable septrately, fixing the

other variable. In other words, Zygmund next conjectured that since

m

is a 2-parameter family of rectangles in R 3 , the corresponding maximal operator, which we shall call Mz should behave like the model 2-parameter M( 2) in R2 :

operator

llMzf(x) >a, lxl < 111

~~

ilfi!L(log L>a for some R f m containing X, IRI R then there exists a dyadic R 1 whose R1 (double) contains x such that - 1- 1 f. lfl >~ .) Now let R 1 = R and, given R 1 , ···,Rk, select Rk+l IRt Rl "' as follows: Let Rk+l be the first R on the list of Rk so that

J exp(I x'R' ) S. C'. lx19 k E'1.=it- U Rk. Then

We claim that the Rk satisfy Rk be

J

R' 1,···,R'N

andlet

k>j

J

exp(f X'R') = Jexp(f X'R' )dx + k=l

Uflj

k

~N

k=l

To see this let the

k

J ~N-1

1

e x j i x'R') +···+Jexp(x'R') \

k=l

n

and

so we have

Now let us show that jUR) > cjURjl· Let R be an unselected rectangle. Then

1

~1

70

ROBERT FEFFERMAN

where the sum extends only over those chosen

Rk

which precede R .

Let us slice R with a hyperplane in the X l'x2 direction. Call

s.Sj

the

""' Then slices of R and. Rj.

J

I~ I exp(~ xs-)dxidx ~C. 2

s

(Again we sum only over those

Sj

which appear before S .) Now, each

Rj

appearing before R has the property that its x 3 (or z ) side length exceeds that of R. It follows that each corresponding Sj has ei~r its XI

or x2 side length longer than that of

s.

Call those

sj

havfng

longer xI side length than xI length of S of type I, and the otiher of type II. Put S = I xJ. Then

so it follows that

on UR j ; hence

URj+x, [exp(Ix~ 1/2 implies

> 'YI·

(For, in general if Q is arbitrary of side 8 and lEI> l/2IQI, consider w(8x) on Q/8 and multiply w(8x) by the right constant p to have

f1 18 pw(8x) = 1 .

Then pw(8x) on E/8 would have [pw(8x)](E/8) > [pw(8x)](Q/8)

'Y1

< •. w(E) >

1J ·)

w(Q)

But by the AP condition,

c(

and

£w} £ ,-1

(

5

~p-1

w- 1 /(p-I'

(

5

so

J

w>£

E

-c

~p-1

~ w-l/(p-l~

0.

for all cubes Q with

)

0.

(() From (e) it is immediate (see also (y)) that w f AP implies w fA q for some q (TJ)

0

< p.

If f is a locally integrable function in some LP space and

1 ). To prove this let f f LP(Rn) be given and a c (0,1). Let Q be a cube centered at

x,

Q

and

its double. Then write f = xQ'f + XcQ'f =

f 1 + f 0 • We must show that

and

~~I

f Q

As for the first inequality,

M(f 0 )a dx

~ CM(f )a(x) .

75


(This is an immediate consequence of the weak type estimate for M on L 1 .) This shows that

As for the second inequality, choose a cube C centered at

l~l

f

if 0 1 dx

x such that

~ ~ M(fo>OO.

c Now, if x

t:
then any cube C' centered at x which intersects c~ is

containeq inside a cube of comparable volume centered at

- 1IC'I

J

J

c'

c

if 0 I < A - 1 IC I

x

so that

It 0 Idx •

Wehaveproventhatforall XlQ, M(f 0 Xx):s_A

I~Jfclf 0 Jdx

sothat

Let us begin to discuss the weight theory by showing that the HardyLittlewood maximal operator is bounded on LP(w) if and only if w lAP, 1 < p < oo [12]. In the first place if f

=

w- 1 /(P- 1 ) XQ and if M is

bounded on LP(w) one sees right away that

or

76

ROBERT FEFFERMAN

~

( _1_

IQI

IQI

f

p-1

qr( f. w and proves the lemma.

• E

R

Proof of the Theorem. We prove a covering lemma, namely, if !Rkl are

rectangles with sides parallel to the axes, there exists !Rkl so that w(UR kL :S Cw(URk) and

To prove this order Rk by decreasing xn side length, and then select a rectangle R when

where the union is taken over all those k such that Rk precedes R and

R'knR=0. Now if we slice R, an unselected rectangle, with a hyperplane perpendicular to the xn direction we have

so that M(n-l w

>(x U(R'k>d ) > .,

on URk. By induction, w(URk) :S Cw(U(Rk)d).

"' have disjoint parts property with respect to dx and so they Now the Rk have this property with respect to w dx also. It follows that

Therefore

MULTIPARAMETER FOURIER ANALYSIS Now to estimate

II! X""

II

,

Kk LP(w)

81

, let us slice the R'k with a hyperplane

perpendicular to xn, calling the slices sk. Then [S'k-

u S'J.I > 2!.

IS'k I

jk Lp ( w)

~ Cw(USk) 1 /p'. Raising this to the pth

power and integrating in xn we have

REMARK. Given this covering lemma, cover the set IM~)(f) > al by Rk such that W

(Rl ) k

f.

fw >a. Then we need only estimate w(URk) of

Rk

the covering lemma. But

82

ROBERT FEFFERMAN

The maximal operator is weak type (p,p), p > 1 and we are finished by interpolation. One application of this theorem is that with it, one can obtain weighted norm inequalities for multi-parameter maximal operators which cannot be handled directly through iteration. We give the following example.

9t denotes the family of rectangles with side lengths of the form s, t, and s·t in R 3 , where s and t > 0 are arbitrary. (Suppose Suppose

the sides are also parallel to the axes.) Define the corresponding maximal opera tor M by Mf(x) = sup -11 xcRc.!R

Rl

J

lfldt.

R

Then it is natural to ask for which weights w do we have

The answer is AP(!R) where this class is defined in the obvious way [14]:

for all R c 9t. To prove this result we need the following. LEMMA.

If w c /IP( !R) then w satisfies a reverse Holder inequality.

Proof. Since w is uniformly

/IP in the x 1 variable, w will satisfy a

reverse Holder inequality uniformly in that variable:


83

( C independent of p ). Fix I, an interval of the x 1 -axis, and define a measure in the x 2 , x 3 plane whose Radon-Nikodym derivative W(p) is defined by

We claim that W satisfies an A00 condition uniformly (in I ) relative to the class of rectangles whose side lengths are t , II It in the x 2 ,x 3 plane. Then let S be such a rectangle in the x 2 ,x 3 plane and E C S such that lEI/IS! ~ 1/2. Then

Since w c A00(9l),

ffExl w ~ (1-q)

ffR w and so

fE

W(p)dp :S C(1-q) f5 W(p)dp,

and by taking 8 small enough, C will be so close to 1 that C(1-q) < 1 and W is uniformly A00 on the collection of all such S. Therefore since S is just !-parameter (just a linear change of variable in one of the x 2 or x 3 variable away from squares) we have that W satisfies a reverse Holder inequality: (For 8' some value < o)

Jw 1 ~dp)

1/(1~')

( _!_ lSI This shows that

and so

w

< c _!_

-

IS!

Jw. s

84

ROBERT FEFFERMAN

(

..!.. IRI

J

1 /(1-i-S')

W1 -tS')

< C'...!..

-

IRI

J

R

w

'

R

dJl.

R

Now on to the theorem. Because w

l

AP(9t) and w- 1 /(p-1)

l

AP,(gt)

satisfies a reverse Holder inequality, w £AP-e(9t) for some e > 0. It follows that Mf(x) ~ CMw(fP-f) 1 /(p-e) , and it just remains to show that Mw is bounded on LP(w). A quick review of the proof that M~), n ~ 3, is bounded on LP(w) reveals that all we really used was that w satisfy an A00 condition in the x 1 and x 2 variables as well as a doubling condition in the x 3 variable: w((R)d) ~ Cw(R). All of these are satisfied by our w here, and this concludes the proof since Mw(f)

:S ~)(f).

Now we wish to relate some of our results on multi-parameter maximal functions to the theory of multiplier operators. We shall work in R 2 , and consider the following basic question: For which sets S ~ R 2 is x_ 8 (() a multiplier on LP(R 2 ) for some p For Xs to be a multiplier of course means that, if for f

l

1-2 ?

c;(R 2 ) we set

Tf (() = x_ 8 (() f(() then we have the a priori estimate

In his celebrated theorem, Charles Fefferman showed that if S is a nice open set in R 2 whose boundary has some curvature then x_ 8 (() will only be an L 2 multiplier [15]. The other nice regions left are those whose boundaries are comprised of polygonal segments. If S is a convex polygon then Xs will obviously be an LP multiplier for all p, 1 02 > 03 ... >on> on-1 _, 0 be a given sequence of angles 0 and let

85


4 Figure 2

Ro

be the polygonal region pictured above. Then we shall define

Te

by

Consider as well the maximal operator M0 defined by

Mff.(x) =

sup x(R(s 0

where

a0

-11 1 R

J

If I dt

R

is the family of all rectangles in R 2 which are oriented in

one of the directions Ok, but whose side lengths are arbitrary. We claim

86

ROBERT FEFFERMAN

that the behavior of havior of

Mo

To

on LP(R 2 ) for p > 2 is linked with the be-

on L(p12>((p/2)' is the exponent dual to p/2 ). More pre-

To

cisely, suppose that

is bounded on LP(R 2 ) and we assume the

weakest possible estimate on M0 , namely llMoxE

Mo

is of weak type on L
'. Conversely, if

To

L
'(R 2 ) then

Mo

>HI ~ CIEI.

Then

is bounded on

is bounded on Lq(R 2 ), for p'< q < p [16].

To prove this assume first that

To

is bounded on LP(R 2). Then

the first step is to notice that this implies that

This is proven by observing that if we dilate the region Ro by a huge factor p to get

R~

and translate

R~

properly (by rk) then R:•rk

looks like a half plane with boundary line making an angle of ok with the positive x axis. Then if rk(t) are the Rademacher functions, and Tp,rkfA

0

and

=X

p,rk

Ro

f

then

87


Taking the limit as p

oo,

4

\1 (~1Tkfk12)

we have

1~

IILP

~ell (~ifk12)

1~

IILP,

where Tk is the Hilbert transform in the direction Ok. The next step is to use the above inequality to prove a covering lemma for rectangles in

mo.

Let IRkl be a sequence of such rectangles.

""' 1, R ~ "" Select R given that R 2 ,· .. , Rk-l have been chlSen provided IR n [j~k

Rj ll < ~ JR J.

Then if R is unselected

Mo<Xu'R'/ ;: - ~

on R

so that

Let Ek

=

Rk-

u

~J.

j 1/100. Applying Tk =Hilbert transform in the direction perpendicular to ek to Tkfk we see that Tk(Tt/k) > 1/100 on all of R~. Repeating twice more we get

I

I x~) I(~

IILP ~ c I(~ lx~k 12 )

1/2

This shows that

Me

1/2

IILP ~ Cllxu~k\ILP ·

is of weak type (p/2)'.

Conversely, assume that

Me

is bounded on L(PI 2 >'. Define

Sk=l(o:((1'( 2 )\2k~( 1 01 whose values lie in Rn+l: F(x,t) = (u 0 (x,t),·· ·,un(x,t)) where the ui{x,t) satisfy the "Generalized Cauchy-Riemann equations," n

au.

I~ (x,t) i=O

=0

(t

=xo)

1

and

au.1

ax. J

auj

- "'L"" for all i,j. = ax1

These Stein-Weiss analytic functions are then said to be HP(R~+ 1 ) if and only if sup t>O

(J

1 IF(x,t)IPdx) /p = JIFII

[17].

HP(Rn)

Rn

Again, these functions have an interpretation in terms of singular integrals, since if a Stein-Weiss analytic function F(x,t) is sufficiently "nice" on R~+l, then the boundary values ui(x) satisfy ui{x) = Ri [u 0 ]{x) where Ri is the ith Riesz transform given by Ri(f Xx) = f

cnxi * --. lxln+l

1

In particular we may consider an H (Rn+

+1

) function {by

identifying functions in R~+l with their boundary values) as a function f with real values in L 1(R 0 ) each of whose Riesz transforms Rif also belong to L 1 (Rn). An interesting feature of HP spaces is that they are intimately connected to differentiation theory as well as singular integrals. To discuss this, let us make some well-known observations. For a harmonic function u(x,t) which is continuous on R~+l and bounded there, u is given as an average of its boundary values acc01ding to the Poisson integra 1:

91


Let f(x) = l(y,t)\ \x-y\ < tl. Then since convolving with Pt at a point x can be dominated by an appropriate linear combination of averages of f over balls centered at x of different radii, it follows that if u*(x)

=

sup

\u(y ,t)\ , then u*(x)

'S cMf(x)

.

(y,t)lf{x)

Unfortunately, if u(x,t) is harmonic, for p :S 1, u

=

P[fJ, and

fan \u(x,t)\Pdx ~ C then the domination u* ~ CMf is not useful, since M is not bounded on LP , and it is not true in general that u*(x) for a,.e.

F

say

X l i

< oo

Rn. On the other hand, suppose F is Stein-Weiss analytic,

H 1 (R~+l).

Then a beautiful computation shows that if 1 >a > 0 is close enough to 1

{a~ n~l) then 6(\F\a)?: 0 so that \Fia is subharmonic.

H

s(x,t) is subharmonic and has boundary values h(x) then s is dominated by the averages of h, i.e., s(x,t) ~ P[h](x,t) . Applying this to G = \Fia (which has we see that G*

'S M(h)

Ll/a so that M(h)

l

for some h

t

JG 1 /a(x,t)dx

~ C for all t > 0)

Ll/a. Now M is bounded on

Ll/a and so G* (Llla. It follows that F*

l

L1 •

Just as for a random f ( L 1(Rn) we do not necessarily have Rif ( L 1 (R 0 ) (singular integrals do not preserve L 1 ) it is also not true that for an arbitrary L 1 function f that for u = P[f]. u* then u* (

L 1(Rn).

l

L 1 . But if f

Thus the nontangential maximal function F*(x) ~

sup

\F(y ,t)\

l

L 1(Rn)

(y,t)lf(x)

if and only if the analytic function f

l

H 1 (R~+l).

l

H1 (R!+ 1 )

92

ROBERT FEFFERMAN

We know so far that we can characterize HP functions in terms of singular integrals and maximal functions. There is another characterization which is of great importance. To discuss it, let us return to HP functions in R ~ as complex analytic functions, F

=

u + iv. It is an

interesting question as to whether the maximal function characterization of HP can be reformulated entirely in terms of u. That is, is it true that F* c LP if and only if u* c LP ? In fact, this is true, and the best way to see this is by introducing a special singular integral, the LusinLittlewood-Paley-Stein area integral,

s 2(u)(x) =

JJ

l'i7ui 2 (y.t)dtdy

l(x)

which we already considered in the first lecture. As we shall see later, for a harmonic function u(x,t), IIS{u)ll

LP

~ llu*!l

LP

for all p

> 0 [18].

The importance of S here is that the area integral is invariant under the Hilbert transform, i.e., S(u)

=S(v),

since l'i7vl = l'i7ul.

When we combine the last two results, we immediately see that

It is interesting to note that the first proof of !IS{u)ll

""'!iu*ll P, LP L 1 ~ p > 0 was obtained by Burkholder, Gundy, and Silverstein [19] by using probabilistic arguments involving Brownian motion. Nowadays direct real variable proofs of this exist as we shall see later on. To summarize, we can view functions f in HP spaces by looking at their harmonic extensions u to R~+l and requiring that u* or S{u) belong to LP(Rn).

93


It tums out that there is another important idea which is very useful

concerning HP spaces and their real variable theory. So far, we have spoken of HP functions only in connection with certain differential equations. Thus, if we wanted to know whether or not f take u

= P[f] which of course satisfies

~u =

l

HP we could

0.

This is not necessary. If f is a function and f/J < c;(Rn) with

fR

¢n =1, then we may form f* (x) =

sup

If *f/Jt(y)l, ¢t(x) =

(t,y)E r(x)

en rp(x/t) and if 1/J < c;(Rn) and J.P = 0 we may form s3.(f Xx) ==

is suitably non-trivial (say radial, non-zero)

rr

J,

'I'

if * .Pt(y)l 2 dy dt tn+l

[18] .

rex)

Then C. Fefferman and E. M. Stein have shown that 11£11 P n "" llf*ll P n "" IIS,,.(f)il P n for 0 H (R ) L (R ) 'I' L (R )

< p < oo

•

Thus, it is possible to think of HP spaces without any reference to particular approximate identities like Pt{x) which relate to differential equations. In addition to understanding the various characterizations of HP spaces, another important aspect is that of duality of H 1 with BMO, which we shall now discuss. A function rp(x), locally integrable on Rn is said to belong to the class BMO of functions of bounded mean oscillation provided

~~I

J

lr/J{x)- r/Jo I dx

'S

M

for all cubes Q in Rn ,

Q

f. f/J. IQI o

where r/Jo = _!_

The BMO functions are really functions defined

modulo constants and

II

llaMo is defined to be sup

l~l J0

lr/J-r/Jol·

ROBERT FEFFERMAN

94

According to a celebrated theorem of C. Fefferman and Stein, BMO is the dual of H 1 (18]. This result's original proof involves knowing that singular integrals map L"" to BMO and also a characterization of BMO functions in terms of their Poisson integrals which we now describe. Suppose p. :2: 0 is a measure in R~+l and Q S.: Rn is a cube. Let S(Q)

=

{y,t)ly £Q, 0 < t <side length (Q)I. Then we say that p. is a

Carleson measure-on R~+l iff p.(S(Q)) ~ CIQI. The basic property that characterizes Carleson measures is

for all functions u on R~+l. In connection with this type d measure there is the characterization of functions in BMO(Rn) in terms of their Poisson integrals. A function cp(x) on Rn with Poisson integral u(x,t) is in BMO if and only if the associated measure

is a Carleson measure. C. Fefferman proved this and used it to prove that every function in BMO acts continuously on H 1 :

These are the basic facts of HP spaces that will concern us here and which we shall later generalize to product spaces. Let us now prove that for a harmonic function u(x ,t)

95


We begin with the estimate llu*llp

:S Cpi!S(u)llp, and to do this we

shall show that

liu*>Cal[

~ C~2 ~

J

S2(u)(x)dx +

liS(u)>al~.

J

S(u}all we have

~ J~ ..-

Hu. n

1 Au .(a)da

p -

f

0

j .•- f.~ j t 1

0

.S(u)(JJ)d {3 + •• ,.

j

J 00

aP- 3dad{3 +

aP-l>..S(u)(a)da .

0

00

Assuming p < 2 as we clearly may, this is

~

I

00

f3P-l>..S(uif3)d{3

-v

1\S(u)l\~.

0

To prove the estimate on llu* >Call, we set the notation that

E: = ~M(xE) > l.

f

H,

and then claim

S 2 (u)(x)dx:::: c JJI'Vu(y.t)\ 2 t dtdy where

ln fact, if (y ,t) dR then \B(y,t) n!S(u)>all
al

R

S(u):Sa

!R

t

\B(y,t)l .

.

96

ROBERT FEFFERMAN

Then

I

I ( I iVul 2(y,t)t 1 -ndyd~

S2 (u)(x)dx "

xd S(u~a

S(u}Sa

(4.1) -=

Jf

I

dx

T(x)

\'Vu\ 2 (y,t)t 1 -nJixj(y,t)d'{.i),x/lS(u)>allldydt.

Rn+l

+

But fa (y,t)

f

R,

\lx\(y,t)£r{x),xi!S(u)>all\-= \B(y,t)nciS(u)>al\ ~ ~ \B(y,t)\ and (4.1) is

as claimed.

II. The next step is to write \'Vu \2 to 9t :

ffA(u 2)(y,t) t dydt = JJ' R

Now

t

=

A(u 2 ), and apply Green's theorem

J~an

t- u 2

aR

~do. an

~ c > 0 for some c so the above gives

Since, for purposes of all estimates we may assume that u is rather nice, we may assume u('Vu)t vanishes at t = 0, so

97


J

u(Vu)t do=

dR

J

u(Vu) t do

dR

where aR is the part of dR above IS(u) > al. It is not hard to see that !Vult~ a on dR so that

Putting all of our estimates together, we see that

I

u 2du

~J

I

t(u~a

dR

S2 (u)(x)dx +a'IS(u)

III. Next we wish to define a function

f

by

f(

x)

=

>

.,,l

)

u(x, r(x)) where

(x, r(x)) t: dR defines the function r. We claim that in the region R

lui~ Plr] +Ca.

(4.2)

This is done by harmonic majorization. It is enough to show this on dR, and this in turn is just saying that for any point p t: c1R, jU(p)l is dominated by the average over a relative ball on dR of u +Ca. This follows from the estimate

IVult~ Ca

on dR. Anyway, from (4.2) we

have, for xi IS(u) > al, u*(x) ~ CP[f]*(x) + Ca, so that finally llu* > C'all

~ IM((x) >all~ ~ llfll~ a

< _g_ a2

I S(u)~

This completes the proof.

S 2 (u)(x)dx + c llS(u)

>all .

98

ROBERT FEFFERMAN

The proof that \\u*\\p ~ Cp\IS(u)\\p which we ;ust gave has been lifted from Charles Fefferman and E. M. Stein's Acta paper [18]. To prove the reverse inequality we want to go via a different route, and we shall follow Merryfield here [20]. We prove the following lemma. In the next lecture we show how this lemma proves \lu*l\p ~ Cp\IS(u)l\p. LEMMA.

Let f(x) and g(x)

f

L 2 (Rn),

and suppose

rp

f

and u ~ P[£]. Then

for some

rp

f

c;(Rn) with

f rp

== 0 (

rp

real-valuecl).

Proof.

-2

Jf Rn+l

+

u(x,t)

~ u(x,t)(g*cJ>t) 2(x,t)dtdx

c;(R") radial


where

I

=

JJ

u(tV(g*cf>t)) t -l 12 · Vu(g*cf>t) t 1 12dt dx

Rn+l

+

and

We see that

but

99

100

ROBERT FEFFERMAN

But

so

Putting this together gives

5.

More on HP spaces At this point we wish to discuss the theory of multi-parameter HP

spaces and BMO. We saw, in the last lecture, that HP(Rn) could be defined either by maximal functions or by Littlewood-Paley-Stein theory. All of these spaces, HP and BMO were invariant under the usual dilations on R 0

,

x-+ 8x, and this is hardly a surprise, since they can be

defined by the maximal functions and singular integrals which are


101

invariant under these dilations. Here we shall define HP and BMO spaces which are invariant under the dilations (in R 2 ) (x 1 ,x 2 ) -. (8 1x 1 ,8 2 x 2 ), 8 1 ,8 2 > 0. For convenience we shall work in R 2 but all of this could just as we 11 be carried out in Rn x Rm, n, m > 1 . We shall call our HP and BMO spaces "product HP and BMO" and denote them by HP(R~ x R~) and BMO(R! x R~) so that we reserve HP(R 2 ) for the one-parameter space ci functions on R 2 • Let (x 1 ,x 2)

f

R 2 and denote by r(x) the set

Let u(x,t) be a function in R~ x R~, x

f

R2 , t

f

R+ x R+' which is

biharmonic, i.e., harmonic in each half plane separately. Then the nontangential maximal function and area integral ci u are defined by u*(xl'x 2) =

iu(y 1 ,t 1 ,y 2 ,t 2)1

sup (y,t)tf'<x 1 ,x 2 )

and

S2 (u)(x) =

Jf

1";;\ \72u(y.t)l 2dyldy2dtldt2.

f'(x)

More generally, if ¢

f

C~(R 2 ) and if

then for a function f on R 2 f*(x) =

sup (y, t) f f'<x)

If* ¢t t (x)l 1' 2

102

ROBERT FEFFERMAN

then

Given f(x 1 , x 2 ), we define its hi-Poisson integral by u(x 1 ,t 1 ,x 2 ,t 2 ) = P[f](x,t) = f * Pt t (xl'x 2), where the hi-Poisson (or just Poisson for 1' 2

short) kernel is defined by Pt t (x 1 ,x 2 ) 1, 2

=

P(x~\

t1 1t; 1 P(x 1)

t2)

tl

and

where P is the !-dimensional Poisson kernel. Then, of course, P[f] is bi-harmonic in R! x R!. In analogy with the 1-parameter case, we define f c HP(R!xR!) if and only if u* c LP(R 2) where u = P[fl. It is not hard to see, just as in the single parameter case that for any ¢ c c;(R 2), \lf*ll

L

P "'

llu* II

LP

f¢

=

1,

for p > 0 ,

so that we may use any approximate identity which is sufficiently nice to define product HP spaces. In terms rf area integrals, we also have, for 1/J(x 1 ,x 2) suitably nontrivial, say ¢

even in x 1' x 2 and not = 0,

To complete the chain of equivalences, we would like to know that

IIS(u)\1

L

P "'

\lu*\1

L

P.

103

MUL TIPARAMETER FOURIER ANALYSIS

In fact, this is true, but is not obvious, and so we in tend to present the proof here. The proofs are by iteration, but they are not of the same totally straightforward nature as the iteration in the jessen-MarcinkiewiczZygmund theorem. Often, this is the case in the analysis of product domains, namely, the proof is by iteration, but this requires a different way o{

looking at the one-parameter case than one is used to.

Proof that of S{u) ( LP, then, u*

l

LP (Gundy-Stein) [21 ]. We can

assume that u -= P[f]. Then since the 2-parameter area integra 1 is invariant under taking Hilbert transforms in each variable separately, we see that we may write

where ~2

f++

is supported in

g1, ( 2 > 0

and

f+-

is supported in ( > 0,

< 0, etc., and we have S(f±±) ( LP. By reflection we may assume f

is analytic (i.e., P[f]

is hi-analytic) and then show that f*

l

LP. But

for u = [f] a hi-analytic function, we know that for a > 0, ju(x 1 ,t 1 ,x 2 ,t 2 ) Ia is subharmonic in each half plane (xi,ti) l separately; this implies

Ri

that

lfa 0. In order to simplify things a little, we shall take a modified definition of u* in what follows, namely

u*(x) =

sup (y,t)fr

10

lu(y,t)l

10(x)

where

This is an irrelevant change, since a trivial computation shows that ·llu*llp is, for a larger aperture, no more than a constant times llu*llp for a smaller aperture, the constant depending only on the apertures involved.

In (5.3), take ¢(x) Let us estimate

=

1 for all lxl < 1/3, and g(x) =X

u

*< x )a)all < 2 ~ 0 jR(y,t)ll and where R(y;t) is

the rectangle in R 2 with sides parallel to the axes and with side lengths 2t1'2t 2 centered at y. Notice that if !R(y;t)nlu*>all < 1 ~ 0 !R(y;t)l then g * t(Y)

=

=

Ptg(y) > c for some c > 0. It follows that

i + ii + iii + iv. Consider i: If "' Qt(gXy)

*I 0

then u*(x)

~a

for some x

But then ju(y ,t) I ~ a so i is less than or equal to

f

R(y; t).


ii is less than or equal to

Again

(iii) is similar to (ii). Finally, (iv) is less than or equal to

So we have

I lu*;Sal

S 2(u)(x)dx

~c

a 2 !1u*>all +

J u*(x);Sa

u* 2 (x)dx

109

110

ROBERT FEFFERMAN

and we have seen before that this implies that !IS(u)il

L

P

:S. Cpllu*ll

LP

, 2 > p > 0.

The next topic that we shall consider is that of duality of H 1 and BMO in the product setting. In the classical case there were four results which expressed this duality. 1) The characterization of Carleson measures 11 for which the Poisson transform f ... P[f] is bounded from LP(dx) to LP(dl1) , p > 1 . 2) The characterization of functions in BMO(R 1) by a condition on their Poisson integrals in terms of Carleson measures. 3) The characterization of functions in the dual of H 1 by the BMO condition. 4) The atomic decomposition of H 1 . Let us try to guess what the analogous theory should look like in product spaces. For simplicity we consider the dual of H 1 (R~xR~). Then what should an element of BMO(R!xR~) look like? We might look at tensor products of functions in BMO(R 1 ) to get a feel for the answer. So, for example if ¢ 1 and ¢ 2 are in BMO(R 1 ) then ¢ 1 (x 1 )¢ 2 (x 2) might be our model. Of course, this function ¢(xl'x 2) satisfies

(5.4)

~~I

I

l¢(x 1 ,x 2 )- c 1 (x 1 )-c 2 (x 2 )1 2 dx 2dx 2 :S. C

R

for the appropriate choice of functions c 1 and c 2 of the x 1 ,x 2 variable. A Carleson measure in R~ x R~ would be a non-negative measure 11 for which (5.5)

where P is the hi-Poisson integral. The obvious guess is that 11

111


satisfies (5.5) if and only if IL(S(R)) ~ CIRI for all rectangles R

C

R2

with sides parallel to the axes, where the Carleson region S(R) is defined by S(I xj) == S(I) xSQ) for R =I xj. In terms of these Carleson measures, it is not hard to show that ¢ satisfies (5 .4) if and only if its hi-Poisson integral u satisfies

And finally, all of this in some sense is equivalent to asserting that every f

f

H 1 (R~ xR~) can be written as I A.kak where I IA.kl ~ Cilfll

H

1

and

ak(x 1 ,x 2 ) are "atoms," i.e., ak is supported in a rectangle Rk == Ikxjk such that

J

ak(x 1 ,x 2 )dx 1

0

for all x 2

ak(x 1 ,x 2 )dx 2 = 0

for all x 1

=

Ik

I Jk

and

In 1974 [22], L. Carleson showed that 11(S(R)) ~ CIRI was not sufficient to guarantee the inequality

J

ifiPdx.

R2

From here it is not difficult to produce examples of functions ¢(xl'x 2 ) which satisfy

112

ROBERT FEFFERMAN

R

where C 1 ,C 2 dependon R, yet ¢/LP(R 2 ) forany p>2. Therefore, this condition is not strong enough to force ¢ to belong to the dual of H1 . In other words the simple picture of the structure of H 1 (R~xR~) and BMO(R~ xR~) suggested above as the obvious guess is completely wrong.

Rather one considers the role of rectangles to be played instead by arbitrary open sets. Although this may seem a bit frightening at first glance, it turns out, and this is of course the final test of the theory, that nearly all the classica 1 theory of HP and BMO can easily be carried out using the approach suggested here. By way of introduction, we shall prove that for any function ¢

f

H 1 (R~xR;)*, if u = P[¢] we have a Carleson condition with respect

to open sets satisfied by the appropriate measure. To describe this result, we !!Jake the following definition ([23], [24], and [25 ]). Let n ~ R 2 be an arbitrary open set, and let R(y; t) be the rectangle in R 2 centered at (yl'y 2 ) = y and with side lengths 2t 1 and 2t 2 • Then S(fl) the Carleson region above n is defined as

U S(R) = l(y,t) ( (R;) 2 R(y; t) ~ n!.

S(Q) =

1

RCfl

Then we say that 11 ~ 0 in (R;) 2 is a Carleson measure if and only if ll(S(fl))o:;Cifll for every open set n~R 2 ·f (H 1 (R~xR~)* if and only if for u = P[f],

In fact, this follows immediately from the inequality (5.3). To see this, notice that lv't V2 ul is invariant under the Hilbert transform Hx.(i = 1,2) 1

so that if we prove this when f when f is of the form

f

L 00(R 2 ), we will have proven it also

113


A function a(x) on R 2 will be in H 1 (R~xR!) if and only if a Hx a, Hx a , and Hx Hx a 1

2

1

2

l

l

L 1 (R2),

L1 .

In fact, if a iH 1 (R~xR;) then S(a)iL 1(R 2) hencesoare S(H

x1

a),

S(Hx a) and S(Hx Hx a); therefore Hx.a,Hx Hx a ( L 1 • Conversely, 2

1

2

1

1

2

if a,Hxia, and Hx 1Hx 2a iL 1 (R 2 ) thenwecanform F++,F+_,F_+, and F __ iL 1 (R 2) suchthat a =~F±± and reflections ofthe F±± are boundary values of hi-analytic functions. A bianalytic function F with (distinguished) boundary values in L 1(R 2) has F* l L 1 by a subharmonicity argument applied to IF Ia, a< 1. So a*

i

H 1 (R~xR;>*.

Define a map from

i

L 1 and a

H 1 (R;xR~) ..P__, ED

i

H 1 . Let

L 1 (R 2)i by

i=1

Then ll~fll

1 "'-' 11£11 1 . ~ is obviously one to one, so t?- 1 = ~ exists eL H and is bounded on lm(~). The map o ~ extends, by Hahn-Banach to

an element of the dual e L 1 = e L 00 • Then

Thus every element of (H 1 )* is of the form

114

ROBERT FEFFERMAN

So it suffices to show that if f

But in (5.3), take g

L 00(R 2) with u

t

"'x 0 (x 1 ,x 2),

[-1 ,1] then ~t g{x) = 1 if (x, t)

l

=

P[f] then

and notice that if

f ¢>

"=

1, supp if>(x)

~

S(O) . This is because for such {x, t),

R(x; t) ~ 0 and g * cf>t(x) = fR2 if>t(x -u)du

=

1. It follows from (5.3) that

Jf117,17,u! t,t,dxdt s cnfll~fflii,.l'l dtt dx 2

S(0)

6. Duality of H 1 and BMO and the atomic decomposition

In this lecture we shall consider in greater detail the spaces HP(R~xR~) and BMO(R~xR!), which we discussed briefly in section 5. There we saw that in product spaces, the most obvious guesses at characterizations of HP atoms of BMO failed. In order to circumvent these difficulties we must take a slightly different approach than we are used to in the classical !-parameter case. In what follows we shall be working with functions in HP(R! xR~) or

BMO(R~xR!> only. The theory for R!xR!

X"'X

R! or for R~tl x R~+l

is quite similar and only requires minor changes. Now let !R be the family of all rectangles with sides parallel to the axes and Sid be the subfamily of 9l whose sides are dyadic intervals. If f( x l'x 2) is a sufficiently nice function on R 2 , and «/!

l

C 00{R 1),

ifJ is even, ifJ i 0 real valued and supp{«/1) ~ [-1 ,1], and «/! has a large

115


number of moments vanishing, then for

I ~~(e-),2d~/e00

=

1

0

we have

In fact, taking Fourier transforms of both sides, for the right-hand side we have

We can use this representation to decompose the function f as follows: R

t

!Rd. Set ~l(R) = l(y,t)

f

R;xR;\ y

f

R, f 1 < t 1 ~ 2fi where

fi , i = 1,2 is the side length of R in the xi direction I. Since

R~ xR.!

=

U

W(R), if we define

Rf!Jld

fR(xl,x2)

=

ryf{y,t)I/Jt t (xl -yl'x2-y2)dy

J"

1 2

/!

1 2

~(R)

where f(y,t)

=

hY,t(y), then f =

I

fR, and each fR is supported in

Rt!Rd

"' the double of R and has the property that R

ROBERT FEFFERMAN

116

"' I

JfR(x 11 x 2)dx 2 = 0

for all x 1

"'

J

where ~ = Tx]. It will be convenient to define a norm I IR on functions supported on a rectangle R 1 as follows.

where N is a large integer. With these preliminaries we can pass to a theorem characterizing (H 1)* in a number of useful ways. THEOREM

[25]. For a function on R 2 the following are equivalent:

(1) ¢ t H 1 (R~xR~)*.

(2) ¢ = g 1 + Hx (g 2) + Hx (g 3) + Hx Hx (g 4) for some g 11 g 2,g 3 and 1

2

1

2

g 4 in L ""(R 2).

(3) If u = P[¢] in R! x R~ 1 then

If

lvl \72ul 2(ylt)tlt2dt

~ Clilll

for all open sets

n s; R 2 .

S({l)

(4) If ¢(y 1 t) = ¢

If S(il)

* 1/Jt(y),

l¢(y,t)l 2dy dtt

then

~ CIUII

for all open sets

n c R2.


117

l: cRbR where bR(xl'x 2) are

(5) ¢J can be written in the form

RfRd

supported in

~, \bR\R ~ 1

and

. R2 . In

I c~ ~ C\01

R~U

for all n

Proof. To begin with, we proved in section 5 that (1) -=9 (2).

~ (1), since if f

trivial that (2)

f

and since f

f

open

It is also

H 1(R!xR!),

f

f(x)Hx Hx (g)(x)dx =fHx Hx (fXx)g(x)dx 1

H1

,

2

1

2

Hx Hx (f) ( L 1 • 1

2

Next, we recall that (2)

=--~

(3) was also proven in the preceding

section. Now we claim that (3) or (4') implies (1). We show that (4) implies (1), \

the other proof being similar.

We,~o

this via the atomic decomposition of

H 1 which we shall describe here only enough to derive our implication. We shall present the decomposition of H1 in greater detail later. Let f ( H 1 (R~xR~). Then S.p(f) ( L 1 (R 2 ) (this follows by vector iteration, just as in the argument that S(f) ( L 1 implies f ( L 1 ). Consider the sets nk = IS.p(f) ak(x 1 ,x 2) =

> 2k1, k ( Z. Set ~

fR(x).

RdRd

\RnUkl > 1/2\R\ \Rnnk+ 11 < 112\R\ Then, as we shall elaborate later on ~k(x 1 ,x 2 ) H 1(R +2 xR +2 ) atom where

is an

118

Then f

ROBERT FEFFERMAN

=

~ ,\kak where .\k

=

2klflkl, and by the strong maximal theorem

lflkl ~ Cl!lkl so that

~ ,\k ~ CIISr/,(OIILl ~ C'II£11Hl . Now consider ¢(x 1 ,x 2) satisfying (4). Then it will be enough to show that

f

ak(x1,x 2)¢(x1,x2)dx < c

R2

and then simply sum over k. But

(6.1)

where the sum is taken over

Then (6.1) becomes

¢(x)

*

dx. J..ryf(y,t)lfrt(x -y) ~yt 1 2 2

lukl

2{(R)

=J

oo, for each N

> 0.

so

Joo ~~(~)1 2 ~~
(x 2) where qt(i) is odd, C 00 and decreasing at oo like lxii-N (depending on how many moments of if! vanish). Now suppose we choose 7J(x) on R 1 sothat supp(7])~(1/4,4), 71£C 00{R 1), 71 evenand 00

I

k=-oo

11( \)

11(;1{).

2

= 1 . Let 71 0 (x) = I

k 0, let 71k(x) =

Set I/Jk,j(x 1 ,x 2 ) = qt{l)(x 1)71k(x 1)· '1'< 2 >(x 2)71k(x 2). Then

122

ROBERT FEFFERMAN

(a) supp('l'k,j)

s; 4R(O; 2k,2j)

(b) 'l'k,j is odd in each variable separately (c) 'l'kj is c;(R 2 ) and

By Minkowski's inequality, we have

Now, to estimate

we use the same argument as that given·above, except that now supp('l'k,j) ~ 4R(O; 2k,2j) and not the unit square. If (y,t) R(y; t)

Thus

s; {}

£

S(O), then

and the support of ('l'k,j)t( · -y) will be contained in

123


By the strong maximal theorem ll)kj I :S C(k+j)2k+j_; and it is easy to see that

J0oo f0

00

1Wkj1 2

d~

decreases like a large power of T(k+j) as

k,j ... ""· So our desired estimate follows from (6.2). So far we have proven the equivalence of (1), (2), (3) and (4). We shall not go into the details of the equivalence of (5) except to say the proof is given in the Annals paper of Chang-Fefferman [25]. Rather, let us point out a beautiful application of the equivalence of (5) with the other definitions of BMO which occurs already in the one-parameter setting. This is the theorem of A. Uchiyama [26], which tells us which families of multipliers homogeneous on Rn of degree 0 determine H1 (Rn). He showed that for multiplier operators I, Kl' K2 .-··, Km with multipliers

1, Oi(O that f, Kif E L 1(Rn) implies f

l

H1(Rn) if and only if the Oi

separate antipodal points of sn- 1 , i.e., if and only if for every ~ l sn- 1 , there exists i such that Oi(~) i Oi(-4). The way Uchiyama proves this is to show that the dual statement is true, namely, every ¢ EBMO(Rn) can be written as

This depends on a simple lemma. LEMMA.

If

()i

are as above, then given f

there exist functions g 0,···, gm

l

f

L 2 , and a vector

II l

cm+ 1 ,

L 2 so that

To prove the formula (6.4), Uchiyama decomposes ¢=I C 1 ¢ 1 as in our (5), and applies the lemma to get functions g 1(x) such that Kg 1(x) C1 ¢ 1(x) for which g(x) is perpendicular to the correct when modified ooly slightly to

g1 ,

11,

has the property that I

=

and the result,

g1 l

L"". For

the details see Uchiyama's recent paper in Acta [26]. Now, finally we wish to discuss the atomic decomposition of H1(R,!xR.;>

in greater detail. There are interesting applications of this

124

ROBERT FEFFERMAN

decomposition besides duality with BMO(R~xR~) which was presented above. We shall be content with one more application here which sheds a good deal of light on the nature of these atoms. Namely, we intend to give a second proof, directly by real variables, that on R~ x R~ if S.p(f) c L 1(R 2) then f* c L 1(R 2) [27). Suppose S.p(f) c L 1(R 2). Then in our discussion of duality we defined atoms

To simplify this notation we define w = Ok and A(x) = 2klfik\ak(x). Let

~ 1 and ~ 2 c C;'(R 1) with ~i(x) ~ 0,

f ~i

= 1 , and supp(~i) C [-1, +1).

Set

Define W = M< 2 ><xw) > - 1- • We need to estimate A * ~t t (x) for 1010 1' 2 x I (;). To do this let us make the following definitions. If R is a rectangle then R 1 , R 2 will be its sides, so that R = R 1 xR 2 . Let

~

A:(x)=

A~(x)

fR(x),

=

J

Rc9lw \R 1 \=2j

Then to estimate A

* ~t

t (x), since supp(~t( · -x)) s; R(x; t) = S, in

1' 2

the definition of A we need only consider those f R for which For any such rectangle R dRk, since R C w,

where (;) denotes again

M<x

w

)>-

1- • Then 1010

Rn S I

0.


A* cf>t t (x) 12

=

I,

. 2 3;js1 lt t (x)

I,

t-

12

. 2 1 ;js 2 lt t (x)

fR * cf>t t (x)

m

REv

R'ns .f. 0.

125

IR21

IS 2 I t t (x) we use the following trivial lemma. 1

1 2

LEMMA. On R 1 suppose that c/>(x)

f

C 00 and is supported in an interval

{}, Suppose a{x) is supported on disjoint subintervals of {}, Ik whose lengths are all ~ yj{}j. Assume that a(x) has N vanishing moments over each Ik. Then

f

a{x)c/>(x)dx

~ Cllc/>(N+l)lloo(yj{}j)N+ 1 Jla{x)jdx.

{}

We estimate A~ * cf>t t (x) using the fact that for each fixed x 2 , 1 1 2 A~(· ,x 2 ) has N vanishing moments over disjoint x 1 intervals over 1

.

length 2 · 21. (Actually, we sould have to break up A~ into 3 pieces to 1

insure this, but we spare the reader this trivial complication.) It follows from the lemma that

Convolving in the x 2 variable, we have

lA~

* cf>t t (x)l;:; C 1 1' 2

21. I N +1 51

1

~ lSI

f 'S

IA~Idx'. 1

126

ROBERT FEFFERMAN

For this we get

By symmetry

and also

Thus

To sum up our findings, we have seen that if x I 6> then

127


(6.5)

lA

* cPt

t (x)l ~ C 1 2

I ~ Sncui)N/4~1 IS I IS I S' ~

""

J

J(y)dy

where

To finish the proof, we need another lemma: LEMMA.

Let g(x)

I fR(x) where B is a collection of dyadic

=

RdB rectan~les.

Then

Proof. Let llhll

J

g(x)h(x)dx

2

2 =

L (R )

=f I,

1 . Then

LJf(y,t)l/lt(x-y)dy

R£~

1 2 ~(R)

=

I

RdB

:~

ryf(y,t)h(y,t)dy

J.,

l"CR)

td~ 1 2

· h(x)dx

128

ROBERT FEFFERMAN

Now, notice that, by the lemma,

IIJII~ ~

If

\f(y,t)\ 2 dy

t~tt 2 ~

Rf~R)

J S~(f)(x)dx ~

C ·2 2 k·lwl.

fik;flk+l

The same estima.te holds for \lA II~ . Then

Also away from

W

so

~ Clwi 112 iwl 112 2k = C2klwl. It follows that IIA *11 1

~ C2klwl and also llakll 1 ~C.

ROBERT FEFFERMAN DEPARTMENT OF MATHEMATICS UNIVERSITY OF CHICAGO CHICAGO, ILLINOIS 60637

BIBLIOGRAPHY [1]

B. Jessen, J. Marcinkiewicz and A. Zygmund, Notes on the Differentiability of Multiple Integrals, Fund. Math. 24, 1935.

[2]

E. M. Stein and S. Wainger, Problems in Harmonic Analysis Related to Curvature, Bull. AMS. 84,1978.


129

[3]

A. Cordoba and R. Fefferman, A Geometric Proof of the Strong Maximal Theorem, Annals of Math., 102,1975.

[4]

J. 0. Stromberg, Weak Estimates on Maximal Functions with Rectangles in Certain Directions, Arkiv fur Math., 15, 1977.

(5]

A. Cordoba and R. Fefferman, On Differentiation of Integrals, Proc. Nat. Acad. of Sci., 74, 1977.

[6]

A. Nagel, E. M. Stein, and S. Wainger, Differentiation in Lacunary Directions, Proc. Nat. Acad. Sci., 75, 1978.

[7]

A. Cordoba, Maximal Functions, Covering Lemmas, and Fourier Multipliers, Proc. Symp. in Pure Math., 35, Part I, 1979.

[8]

F. Soria, Examples and Counterexamples to a Conjecture in the Theory of Differentiation of Integrals, to appear in Annals of Math.

[9]

B. Muckenhoupt, Weighted Norm Inequalities for the Hardy Maximal Function, Trans. of the AMS, 165, 1972.

[10] R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted Norm Inequalities for the Conjugate Function and Hilbert Transform, Trans. AMS, 176, 1973. [11] R. R. Coifman and C. Fefferman, Weighted Norm Inequalities for Maximal Functions and Singular Integrals, Studia Math., 51, 1974.

[12] M. Christ and R. Fefferman, A Note on Weighted Norm Inequalities for the Hardy-Littlewood Maximal Operator, Proceedings of the AMS, 84, 1983. [13] R. Fefferman, Strong Differentiation with Respect to Measures, Amer. Jour. of Math., 103, 1981.

[14]

, Some Weighted Norm Inequalities for Cordoba's Maximal Function, to appear in Amer. Jour. of Math.

[15] C. Fefferman, The Multiplier Problem for the Ball, Annals of Math., 94, 1971. [16] A. Cordoba and R. Fefferman, On the Equivalence between the Boundedness of Certain Classes of Maximal and Multiplier Operators in Fourier Analysis, Proc. Nat. Acad. Sci., 74, No. 2,1977. [17] E. M. Stein and G. Weiss, On the Theory of HP Spaces, Acta. Math., 103, 1960. [18] C. Fefferman and E. M. Stein, HP Spaces of Several Variables, Acta Math., 129, 1972. [19] D. Burkholder, R. Gundy, and M. Silverstein, A Maximal Function Characterization of the Class HP, Trans. AMS, 157, 1971.

130

ROBERT FEFFERMAN

[20] K. Merryfield, Ph.D. Thesis: HP Spaces in Poly-Half Spaces, University of Chicago, 1980. [21] R. Gundy and E. M. Stein, HP Theory for the Polydisk, Proc. Nat. Acad. Sci., 76, 1979. [22] L. Carleson, A Counterexample for Measures Bounded on HP for the Bi-Disc, Mittag-Leffler Report No.7, 1974. [23] S. Y. Chang, Carles on Measure on the Bi-Disc, Annals of Math., 109, 1979. [24] R. Fefferman, Functions on Bounded Mean Oscillation on the Bi-Disc, Annals of Math., 10, 1979. [25] S. Y. Chang and R. Fefferman, A Continuous Version of the Duality of H 1 and BMO on the Bi-Disc, Annals of Math., 1980. [26] A. Uchiyama, A Constructive Proof of the Fefferman-Stein Decomposition of BMO(Rn), Acta. Math., 148, 1982.

ELLIPTIC BOUNDARY VALUE PROBLEMS ON LIPSCHITZ DOMAINS Carlos E. Kenig*

Dedicated to the memory of jack P. Burke PREFACE This paper is an outgrowth of a series of lectures I presented at the Summer Symposium of Analysis in China (SSAC), held at Peking University in September, 1984. The material in the introduction and parts (a) and (b) of Section 1 comes from the expository article 'Boundary value problems on Lipschitz domains' ([19]), which I wrote jointly with D. S. Jerison in 1980. The rest of the paper can be considered as a sequel to that article. Some of the material in part (b) of Section 2, and all d. Section 3 comes from the recent expository article "Recent progress on boundary value problems on Lipschitz domains'' ([23] ). The results explained in Section 2, (b) and Section 3 are unpublished. Full details will appear elsewhere in several joint papers. Acknowled~ements.

I would like to thank Peking University, and the

organizing committee of the SSAC, Professors M. T. Cheng, S. L. Wang, S. Kung, D. G. Deng and R. Long for their invitation to participate in the SSAC, and for their warm hospitality during my visit to China. I would also like to thank Professor E. M. Stein for his many efforts to make the SSAC a success. Thanks are also due to Mr. You Zhong and Mr. Wang Wengshen for taking careful notes of my lectures. Finally, I would like to thank B. Dahlberg, E. Fabes, D. Jerison and G. Verchota for the many discussions and fruitful collaborations that we

*Supported in part by

the NSF.

131

132

CARLOS E. KENIG

have had throughout the years, which resulted in the work explained in this paper.

Introduction A harmonic function u is a twice continuously differentiable function on an open subset of Rn, n ~ 2, satisfying the Laplace equation n

L\u

=

l:

2

au

j=l ax~

= 0. Harmonic function arise in many problems in mathe-

J

matical physics. For example, the function measuring gravitational or electrical potential in free space is harmonic. A steady state temperature distribution in a homogeneous medium also satisfies the Laplace equation. Moreover, the Laplace equation is the simplest, and thus the prototype, of the elliptic equations, or systems of equations. These in turn also have many applications to mathematical physics and geometry. A first step in the understanding of this more general situation is the study of the Laplacian. This will be illustrated very clearly later on. Initially we will be concerned with the two basic boundary value problems for the Laplace equation, the Dirichlet and Neumann problems. Let D be a bounded, smooth domain in Rn and let f be a smooth (i.e. C"") function on aD, the boundary of D. The classical Dirichlet problem is to find and describe a function u that is harmonic in D,

continuous in

D, and equals f on aD. This corresponds to the problem

of finding the temperature inside a body D when one knows the temperature f on iJD. The classical Neumann problem is to find and describe a function u that is harmonic in D' belongs to C 1 (D), and satisfies

~

=

f on

ao'

where

~ represents the normal derivative of u on

ao.

This corresponds to the problem of finding the temperature inside D when one knows the heat flow f through the boundary surface

ao.

Our main purpose here is to describe results on the boundary behavior of u in the case of smooth domains, and to study in detail the extension

ELLIPTIC BOUNDARY VALUE PROBLEMS

133

of these results to the case of minimally smooth domains, where we allow corners and edges, i.e. Lipschitz domains. This class of domains is important from the point of view of applications, and also from the mathematical point of view. Their importance resides in the fact that this is a dilation invariant class of domains with some smoothness. They have the borderline amount of regularity necessary for the validity of the results we are going to expound on. In a smooth domain, the method of layer potentials, (which we are going to develop soon) yields the existence of a solution u to the Dirichlet problem with boundary data f ( ck,a(aD), and the bound

k

=

0,1,2,···

O(y')l

s Mjx'-y'j

and DnB = IX=(x',xn):xn>c/>(x')lnB.

H for each Q the function cf> can be chosen in C 1 (Rn- 1), then D is ca lied a C 1 domain. If in addition,

llc/>

satisfies a Holder condition

of order a, lllc/>(x')-llcf>(y')i :S·Cix'-y'la, we call D a C 1•a domain. Notice that the cone re ~ I (x ',xn): xn < -M lx'll satisfies re n B

c co.

Similarly, ri ~ {(x ',xn): xn >Mix'il satisfies ri n BCD. Thus, Lipschitz domains satisfy the interior and exterior cone condition. The function ¢ satisfying the Lipschitz condition i¢(x ') -- cf>(y') I ~ Mjx'-y'l is differentiable almost everywhere and 17¢

l

L ""(Rn- 1 ), llllc/>lloo :SM.

136

CARLOS E. KENIG

Surface measure o is defined for each Borel subset E C aD n B by

o(E) =

J

(1 + \'Vcp(x')\ 2 ) 1 12 dx',

E*

where E* =lx': (x',cp(x'))lEI. The unit outer normal to !D given in the coordinate system by ('i]cp(x '), -1 )/(1 + \'Vcp(x ')\ 2 / 12 exists for almost every x'. The unit normal at Q will be denoted by NQ. It exists for almost every Q

(a>,

with respect to do. In order to motivate the use of the method of layer potentials, we need to recall some formulas from advanced calculus, and some definitions. We will start with the derivation corresponding to the Dirichlet problem. We first recall the fundamental solution F(X) to Laplace's equation in Rn: ~F = 5, where

1

n >2

(n-2) wn \X \n-2 F(X) =

lrr log \X\

n=2

where wn is the surface area of the unit sphere in Rn. F(X) is the electrical potential in free space induced by a unit charge at the origin. It provides a formula for a solution w to the equation tlw = 1/J , with

0

1/J c C (Rn),

w(X) = F *1/J(X) =

f

F(X-Y)I{I(Y)dY.

Rn It will be convenient to put F(X,Y) ~ F(X-Y). Notice that tl.yF(X,Y) =

8(X-Y). The fundamental solution in a bounded domain is known as the Green function G(X,Y). It is the function on

DxD

continuous for X ~ Y

and satisfying tlyG(X,Y) = 8(X-Y), X c D; G(X,Y) = 0, X £ D, Y £!D.

137


G(X,Y) as a function of Y is the potential induced by a unit charge at X that is grounded to zero potential on aD. The Green function can be obtained if one knows how to solve the Dirichlet problem. In fact, let ux(Y) be the harmonic function with boundary values ux(Y)\an = F(X,Y)\ao. Then, G(X,Y) = F(X,Y)-ux(Y).

(1)

On the other hand, if we know G(X,Y), we can formally write down the solution to the Dirichlet problem. In fact, u(X) =

=

J

J

D

D

u(Y)o(X,Y)dY =

f [u(Y)~yG(X,Y)

u(Y)AyG(X,Y)dY =

-Au(Y)· G(X,Y)]dY

=

D

J[u(Q)

~Q (X,Q)- ~Q (Q)G(X,Q~ du(Q)

=

ao

=

I

u(Q)

a;Q G(X,Q)du(Q)'

ao

where the fourth equality follows from Green's formula. Thus, we have derived the formula

(2)

u(X) =

J

f(Q)

:;Q

(X,Q)du(Q)

an

for the harmonic function u with boundary values f. The problem with

138

CARLOS E. KENIG

formula (2) is of course that we don't know G(X,Q). Because of formulas (1) and (2), C. Neumann proposed the formula

w(X) =

J :Q J f(Q)

(X,Q)da(Q) =

1

f(Q)

= (un

<X-Q,NQ> IX--Q I" da(Q)

av as a first approximation to the solution of the Dirichlet problem, L\u in D,

=

0

ulao =f.

w(X) is known as the double layer potential of f. First of all w is harmonic in D. Also, one can show that as X .... Q t CD, w(X) ....

t

f(Q) +

Kf(Q), where K is the operator on CD given by

Kf(Q)

=

J-n

J ao

I 'I"P f(P)da(P) . P-Q

If Kf were zero, we would be done, and in some sense it is true that Kf

is small compared to

~ f, when the domain D is smooth. In fact, CD

has dimension n-1 , while it is easy to see that if CD is C 00 ,

Thus, the operator K: C(CD) .... C(I , and

~ Cpllfll

What do we do when

.

LP(da)

ao

is merely C 1 , or even merely Lipschitz?

As I mentioned before, the LP boundedness of K is even in doubt. In 1977, A. P. Calderon ([1]) showed that for any C 1 ·domain, K: LP(OD) .... LP(OD), 1 < p < oo is a bounded operator. Shortly afterwards, Fabes, Jodeit and Riviere ([11]) showed that K is in fact compact in this case. They were thus able to extend the theorems above to the case of C 1 domains. Before going on to the main subject matter of this paper, i.e. the method of layer potentials on Lipschitz domains, I want to discuss another important method for the Dirichlet problem for Laplace's equation. (b) The method of harmonic measure Another way of studying the Dirichlet problem for Laplace's equation is in terms of the notion of harmonic measure. Let D be a bounded Lipschitz domain in Rn. As we mentioned before, then D satisfies the exterior cone condition, and so, by a classical result ci. Zaremba and Lebesgue, we can solve the classical Dirichlet problem for !l in D, for any f tC(aD). Given X mapping f

1-->

t

D, the maximum principle shows that the

u(X) defines a positive continuous linear functional on

C(OD). Therefore, by the Riesz representation theorem, there is a unique positive Borel probability measure wx on

u(X) =

J

f(Q)dwx(Q)

an

an

such that

142

CARLOS E. KENIG

wx is called the harmonic measure for D, evaluated at X. For example, harmonic measure for the unit ball B , evaluated at the origin is a constant multiple of surface measure: w 0 = u/o(aB). (This follows from the mean value property of harmonic functions.) For different X, the measures wx are mutually absolutely continuous (a simple consequence of Harnack's principle). We fix X 0

X

t

D, and denote w = w 0 • The importance of

harmonic measure to the boundary behavior of harmonic functions on Lipschitz domains can be illustrated by the following theorem of Hunt and Wheeden (1967): If u is a positive harmonic function in a Lipschitz domain D, then u has finite non-tangential limits almcst everywhere with respect to w. Conversely, given any set E C aD, with w(E) = 0, there is a positive harmonic function u in D with lim u(X) X

-->

Q, for every Q

t

= + oo as

E. Despite its advantages, harmonic measure has

some inherent difficulties. First, it is hard to calculate it explicitly. Second, it is tied up to the maximum principle, positivity, and the Harnack principle, and so it is not useful for the Neumann problem, or for the Dirichlet problem for systems of equations. In general, harmonic measure may be very different from surface measure. H D is a C 1 •a domain, then harmonic measure and surface measure are essentially identical in that each is a bounded multiple of the other. This can be proved by the classical method of layer potentials. Along the same lines, as we saw before, one can use layer potentials to solve the Dirichlet and Neumann problems with boundary data in LP. On C 1 domains, it is no longer true that harmonic measure is a bounded multiple of surface measure, or vice versa. Moreover, as was explained before, the applicability of the method of layer potentials is not obvious. The situation for general Lipschitz domains is even less obvious. In 1977, B. E. J. Dahlberg ((4]) proved that on a C 1 or even a Lipschitz domain, harmonic measure and s•uface measure are mutually absolutely continuous. Using a quantitative version of mutual absolute continuity, and the theory of weighted norm inequalities, he proved ([5])


143

that in a Lipschitz domain D one can solve the Dirichlet problem as in the theorem above with f

£

L 2 (ao, da). In fact, he showed that given a

Lipschitz domain D, there exists e = e(D) such that this can be done for f

£

LP(ao, da), 2-e 'S p 'S

oo.

Also, simple examples to be presented

later show that given p < 2, we can find a Lipschitz domain D for which this cannot be done in LP. By establishing further properties for harmonic measure on C 1 domains, he was able to show the results above in the range 1

< p < oo for C 1 domains. (The best possible regularity

result for harmonic measure on C 1 domains is due to D. Jerison and C. Kenig (1981): if k =

~~, then log k £ VMO(dD) .)

A shortcoming of Dahlberg's method of proof, as was explained before, is that, by studying harmonic measure, it relied on positivity and the Harnack principle. This made the method inapplicable to the Neumann problem, or to systems of equations. Also the method does not provide useful representation formulas for the solution. (c) The method of layer potentials revisited In 1979, D. Jerison and C. Kenig [16], [17] were able to give a simplified proof of Dahlberg's results, using an integral identity that goes back to Rellich ([30]). However, the method still relied on positivity. Shortly afterwards, D. Jerison and C. Kenig ([18]) were also able to treat the Neumann problem on Lipschitz domains, with L 2 (aD,da) data and optimal estimates. To do so, they combined the Rellich type formulas with Dahlberg's results on the Dirichlet problem. Thus, it still relied on positivity, and dealt only with the L 2 case, leaving the corresponding LP theory open. In 1981, R. Coifman, A. Mcintosh andY. Meyer [2] established the boundedness of the Cauchy integral on any Lipschitz curve, opening the door to the applicability of the method of layer potentials to Lipschitz domains. This method is very flexible, does not relie on positivity, and does not in principle differentiate between a single equation or a system of equations. The difficulty then becomes the solvability of the integral

144

CARLOS E. KENIG

equations, since unlike in the C 1 case, the Fredholm theory is not applicable, because on a Lipschitz domain operators like the operator K in part (a) are not compact, as simple examples show. For the case of the Laplace equation, with L 2 (aD, do) data, this difficulty was overcome by G. C. Verchota ([33]) in 1982, in his doctoral dissertation. He made the key observation that the Rellich identities mentioned before are the appropriate substitutes to compactness, in the case of Lipschitz domains. Thus, Verchota was able to recover the L 2 results of Dahlberg [S] and of Jerison and Kenig [18], for Laplace's equation on a Lipschitz domain, but using the method of layer potentials. These results of Verchota's will be explained in the first part of Section 2. In 1984, B. Dahlberg and C. Kenig ([16]) were able to show that given a Lipschitz domain DC Rn, there exists e = e(D) > 0 such that one can solve the Neumann problem for Laplace's equation with data in LP(OD, do), 1 < p::; 2 +e. Easy examples (to be presented later) show that this range of p's is optimal. Moreover, they showed that the solution can be obtained by the method of layer potentials, and that Dahlberg's solution of the LP Dirichlet problem can also be obtained by the method of layer potentials. They also obtained end point estimates for the Hardy

space H 1(aD,do), which generalize the results for n [21 ], and for

C1

=

2 in [20] and

domains in [12]. The key idea in this work is that one

can estimate the regularity of the so-(x)){x)) or simply by x. Nx or NQ will denote the unit normal to ao =A at Q = (x, c/>{x)). If u is a function defined on Rn\A and Q (

ao,

lim

u ±(Q) will denote

u{X) or

X-->Q

xtri lim

u(X), respectively. If u is a function defined on D, N(u)(Q) =

X->Q

xtre(Q) sup \u(X)\. xtri We wish to solve the problems (D)

~Au = 0 { u\ao

=

Au= 0 in D

in D f

t

(N) {

L 2 (aD, du)

~I

ao

= f

t:

L 2(aD,du)

The results here are THEOREM 2.1.1. There exists a unique u such that N(u)

£

L 2(aD, da),

solving (D), where the boundary values are taken non-tangentially a. e .. Moreover, the solution u has the form

u(X) =;

n

for some g c L 2(aD,da).

J ao

<X-Q,NQ> I In g{Q)da(Q) , Q-X

147


THEOREM

that N(V'u)

2.1.2. There exists a unique f

u

tendin~

to 0 at

oo,

such

L 2 (aD,da), solving (N) in the sense that No·Vu(X) ... f(Q)

as X ... Q non-tangentially a.e. . Moreover, the solution u has the form

u(X)

=

f

2)

/

"'n n-

1 IX-Q ln-2

g(Q)da(Q) ,

dD for some g

E

L 2 (ao, da).

In order to prove the above theorems, we introduce the double and single layer potentials

J

<X-Q,N >

Kg(X) = Jn

IX-Qjl?

g(Q)da(Q)

dD and

Sg(X) = -

l

cun n

2)

f

1

IX-Qin-2

g(Q)da(Q) .

dD If Q = (x, ¢(x)), X = (z ,y), then

Kg(z,y) =

J n

Sg(z,y)

=-

J Rn-1

1

cu {n-2)

n

THEOREM 2.1.3. a) If g

l

y-¢(x)-(z-x). V¢(x) g(x)dx llx-zl2 + [¢(x)-¢(z)]2]n/2

I

Jl + IV¢(x)l2

Rn- 1

llx-zl 2 + l¢(x)-y]2] 2

LP(di), da), 1 < p < oo,

n-2 -

g(x)dx.

then N('i7Sg), N(Kg)

also belong to LP(ao, da) and their norms are bounded by

Cll&ll

LP(dD,da)

.

148

{b)

CARLOS E. KENIG

lim

...!..

e-+0

e

a. e. and \\Kg\\

1 . 11m (;) e _.. 0 n

cf>(z) -~(x)-{z-x) · V~(x) g(x)dx = Kg(z) exists [\x-z\2 +[~{x)-¢{z)]2]n-2

LP(dD,do)

J -

~ C \\gl\

Lp(dD,do)

1

, do) adjoint of

=

+} g{z)- K*g{z),

where K* is the

K.

The proof of Theorem 2.1.3 a) follows by well-known techniques from the deep theorem of Coifman, Mclntos h and Meyer ([2 ]). THEOREM

([2]). Let (}: R _.. R be even, and CDC>. Let A,B: Rn- 1 ... R

be Lipschitz. Let K(z,x) = A(z)-A(x) \z-xln operator

M*g(z) =sup E-+0

I

ofB(z)-B(x)l. [ lz-x\ J

I

K(z,x)g{x)dx\

\t-x\>e

is bounded on LP(Rn- 1 ), 1 < p < oo, with

Then, the maximal

•

149


\I 'VAll""~ M, ll\7811"" ~ M.

where C = C(M,O,n,p), and

The proof of (b), (c) follow from the theorem above, together with the following simple lemmas. LEMMA.

If f c C':)'(Rn-l), then

lim _!_ e .. own

I

¢(z)-¢(x)-(z-x). \7¢(x) f(x)dx = [jx-zl2+[¢(x)-¢(z)]2]n/2

lz-x\>e

=-~_!_I ~ wn 1

zk-xk A

Iz-x 1n-1

[¢(z)-¢(x)~ lx-zl

L(x)dx' axk

where A(O) = 0, A'(t) = (1 +t 2)-n/ 2 , and where the equality holds at every z at which ¢ is differentiable, i.e. for a.e.z. LEMMA.

If a c Rn- 1 , a

.J ¢(a), f r C';;'(Rn- 1) and A is as in the

previous lellUlla, then

_!_f a-¢(a)- {a-x). 'V¢(x) wn

=

f(x)dx =

[Ja-xl2+[¢(x)-a]2]n/2

_21 sign (a-¢(a))f(a)- w1 n

f~

xk-ak A

~ lx-ajn-1

{¢~x) -j\ 1!

\

x--a

J uxk

(x)dx .

Moreover, the integral on the right-hand side of the equality is a continuous function of {a,a) c Rn. It is easy to see that (at least the existence part) of Theorems 2.1.1 and 2.1.2 will follow immediately if we can show that {~I+

K)

and

150

t

CARLOS E. KENIG

are invertible on L 2 (aD,do). This is the result of

I+K*

G. Verchota ([331). THEOREM 2.1.5.

(±~I+ K)

,

(± ~ I+ K*)

are invertible on L 2(aD, do).

In order to prove this theorem, it suffices to show that (±

~

I+ K*)

are invertible. In order to do so, we show that if f f L 2(aD, do), I (2!. I+ K*\ fll 2 ~ 11(-12 I- K*) fll 2 , where the constants } L (liD,do) L (ao,do) of equivalence depend only on the Lipschitz constant M. Let us take this for granted, and show, for example, that ~ I+K* is invertible. To do this, note first that if T

=

~ I+ K* , IITfll L 2 ~ C llfll L 2 , where C

depends only on the Lipschitz constant M. For 0::; t operator Tt =

~ I+

K:, where K:

~

1, consider the

is the operator corresponding to the

aT

domain defined by t¢. Then, T 0 = ~ I, T1 = T, and (/ : LP(Rn-l) .... LP(Rn- 1), 1 < p < oo with bound independent of t, by the theorem of Coifman-Mclntosh-Meyer. Moreover, for each t, independent of t. The invertibility of T now follows from the continuity method: LEMMA 2.1.6. Suppose that Tt: L 2 (Rn- 1) .... L 2 (Rn- 1) satisfy (a) liT tfll

L

2

> C 1llfll 2 L

(b) IITtf-Tsfll 2 ~C 2 \t-s\\fl\ 2 , O::;t,sSl. L L 1 2 2 (c) T0 : L (Rn-l) .... L (Rn- ) is invertible. Then, T1 is invertible. The proof of 2.1.6 is very simple. We are 1 I+ K*) fll 2 thus reduced to proving (2 .1. 7) II (-2

L (ao,da)

~ II (-21 I- K*) fll L 2 (ao,d [)

In order to prove (2.1.7), we will use the following formula, which goes back to Rellich [30] (also see [28], £29], [27}). LEMMA 2.1.8. Assume that u fLip (D), L\u

=

0 zn D , and u and its

151


derivatives are suitably small at

Then, if en is the unit vector in

oo.

the direction of the y-axis,

fl\7Ul 2 du=2

ao Proof. Observe that div (en1Vul 2) = div

-t; V'u t

V'u · V'u +

=

i·

t

div V'u

=

J~·~do.

ao

!Vui 2 =

2fy \lU · V'u,

~ V'u · Vu.

while

Stokes' theorem now

gives the lemma. We will now deduce a few consequences of the Rellich identity. Recall that Nx

=

(-V'cp(x),l)/Vl + IV'cp(x)! 2 , so that

COROLLARY

1 (l+M2)

11 ::; :S

2

1.

2.1.9. Let u be as in 2.1.8, and let T 1(x), T2(x),

Tn_ 1 (x) be an orthogonal basis lor the tangent plane to aD at (X, cp(X)). n-1

Let IY'tu(x)l 2 = j~1 I 1 2 • Then,

J (~Y ao

da:S

c

J ao

1vtul 2 du.

152

CARLOS E. KENIG

COROLLARY 2.1.10. Let u be as in 2.1.8. Then,

Proof.

fao

IV'ufda::;

2(fao IV'ul 2 da) 112 Van 1~1 2 daf 12 ,

by 2.18,

and the corollary follows.

lnordertoprove2.1.7,let u~Sg. Becauseof2.1.3c, V'tu iscon tinuous across the boundary, whileby2.1.4,

{~}± ={+~

1-K*} g.

We

D, to obtain 1.1.7. This finishes the proof

now apply 2.1.11 in D and of 2.1.1 and 2.1.2.

We now turn our attention to L 2 regularity in the Dirichlet problem. DEFINITION 2 .1.12. f

f

L~(A), 1

< p < oo, if f(x, ifJ(x)) has a distribu-

tional gradient in LP(Rn- 1). It is easy to check that if F is any extension to Rn of f, then V'xF(x,ifJ(x)) is well defined, and belongs to LP(A). We call this V'tf. The norm in Lf(A) will be IIV'tfiiLP(A). THEOREM 2.1.13. The single layer potential S maps L 2 (A) into Lf(A) boundedly, and has a bounded inverse. Proof. The boundedness follows from 2.1.3a). Because of the L 2 -Neumann theory, and 2.1.11, IIV'tS(£)11 2 L

> cji~. C!ifll 2 A . The un L ( ) L ( )

-1 . Fix now a p > 2, and choose {3 so close to 2TT that p TT/{3-p < -1. Then, N('i7w) iLP(aO{:J). If {} I- K*) were invertible in LP(aO{:J), then, since would have that w(z)

=s((} 1-K*)-l(~))(z)

~

€

L 00 (d~), we

has a non-tangential

maximal function in LP(dQ{:J). By the L 2 -uniqueness in the Neumann problem, w-w is constant in 0{3, but this is a contradiction. This shows that given p > 2, we can find a Lipschitz domain so that {} I-- K*) is not invertible in LP. The example can also be used to

154

CARLOS E. KENIG

show that } I+ K is not always invertible in L q , when q < 2. In fact, fix q < 2, and let p satisfy

p~-p 0 such that, tiven

THEOREM

f t LP(Cl), da), 2- e ~ p

with N(u)

t

< oo, there exists a unique u harmonic in D,

LP(aD, da) such that u converges non-tangentially almost

everywhere to f. Moreover, the solution u has the form

Jn

u(X) =

J

<X-Q,NQ> IX-Qin g(Q)da(Q) '

ao for

some g t:

LP(aD,da).

THEOREM 2.2.2.

f

l

There exists e = e(M) > 0, such that, given

LP(aD, da), 2-e ~ p < oo, there exists a unique u harmonic in D,

tending to 0 at

oo,

with N(V'u)

t:

LP(aD, da) , such that NQ V'u(X) con-

verges non-tangentially a.e. to f(Q). Moreover, u has the form

u(X) =

for some g

t

(1 2) wn n-

f ao

g(Q)da(Q) , 1 IX-Q ln-2

LP(aD,da).

THEOREM 2.2.3.

There exists e = e(M) > 0 such that given f

t:

1 < p < 2 + e, there exists a harmonic function u, with IIN(V'u)l! -

CI!V'tfll

LP(

A, and such that V'tu )

=

for some g

£

(1 2) cun n-

LP(aD,da).

J ao

L

P

~)

S

V'tf (a.e.) non-tangentially on A·u

is unique (modulo constants). Moreover, u has the form

u(X) =-

Li(A),

1 2 g(Q)da(Q), IX-Qin-

156

CARLOS E. KENIG The case p - 2 of the above theorems was discussed in part (a). The

first part of 2.2.1 (i.e. without the representation formula), is due to

B. Dahlberg (1977) ([5]). Theorem 2.2.3 was first proved by G. Verchota (1982) ([331). The representation formula in 2.2.1, Theorem 2.2.2, and the proof that we are going to present of 2 .2 .3 are due to B. Dahlberg and

C. Kenig (1984) ([6]). Just like in part (a), 2.2.1, 2.2.2, and 2.2.3 follow from. THEOREM 2.2.4. There exists

e = e(M) > 0 such that (±}I- K*} is

invertible in LP(OO, do), 1 0

harmonicity of u, and

OD~

•

2=

~=lim f e{t/J-1]~,

e

aop

E-->0

faoe 0~ ~.

OD~\ao;

p

1• '

=

aop

since,bythe

0. Let ODpe 1 = l(x,y) c iJDPE : y > c/>(x) + e I, •

Then, lim

f

e [t/J-1]

aop = fao t/Ja- fan

E--> 0

lim Jaoe [t/J-11 ~ = fao [t/J-1]a e ... o p, 2 t/1=0 on suppa. Moreover, J0 _B\7u\7t/I=J0

~

= lim

a=

E-+

0

f

ao~,l

fiKJ t/Ja

{t/J-1] ~ +

= 0, since

\lu·\lt/1*, where c/l*=c/Jo(/),

by our construction of B. The last term is also 0 by the same argument, and so a = 0. We now show that u (and hence

u ) is bounded.

We will

assume that n ~ 4 for simplicity. Since lla II 2 A < C, we know that L ( )

u(X)-= en

fa

£( Q)

D IX-Qin-2

l(x,y):y>¢(x)+1l,

da(Q), with 1\fll 2 A O. For R?:Ro= diamB*, set b(R)= f N('Vu) 2 , where, AR=I(x,¢(x)):R..\l!d..\+Ce

f."" ..\e- 1 (J. 0

f.

lm>..\1

m 2 d..\< -

\ m2 )d..\.

h>~

162

CARLOS E. KENIG

By a well-known inequality (see [14] for example), lEAl~ Callm >All. Thus, fm 2 +e ~ Ce f 0oo A1 +e llm>AIIdA+Ce f000 Ae-l(Jh>A m2)dA ~ Ce fm 2+E+ Cfm 2 he. Ifwenowchoose e0 sothat Ce 0 A,h~l-

that the same argument gives the estimate !IN(VU)llp ~ Cil'Vtullp, 2 < p < 2 + e , and the LP theory is thus completed. §3. Systems of equations on Lipschitz domains (a) The systems of elastostatics.

I"n this part we will sketch the extension of the L 2 results for the Laplace equation to the systems of linear elastostatics on Lipschitz domains. These results are joint work of B. Dahlberg, C. Kenig and G. Verchota, and will be discussed in detail in a forthcoming paper ([8]). Here we will describe some of the main ideas in that work. For simplicity here we restrict our attention to domains D above the graph of a

cp: R 2 ... R.

Lipschitz function Let ..\ > 0, p.

~

0 be constants (Lame moduli). We will seek to

... =

solve the following boundary value problems, where u

i

(ul'u 2 ,u 3)

~\{ + (..\ + p.) 'V div \: = 0 in D

(3.1.1)

...

ulao

f

.... =

f

2

£

L (il>,da)

~t; + (..\ + p.) 'V di v 1: =

0 in D

(3.1.2) { ..\(div \:)N + p.IV\:+(Vi:)tlNlao =

f

£

L 2 (il>,da).

164

CARLOS E. KENIG

(3.1.1) corresponds to lmowing the displacement vector

ii

on the

boundary of D, while (3.1.2) corresponds to knowing the surface stresses on the boundary of D. We seek to solve (3.1.1) and (3.1.2) by the method of layer potentials. In order to do so, we introduce the Kelvin matrix of fundamental solutions (see [24] for example), r(X) = (rij(X)), where A 0 ii

c xi xi TXi + 4rr·IXI3 , and

rii(X) = 4"

1 [1 A = 2 p:

1 1 1 [1 1 ] +_2il.+XJ ' c = ~ p: -z;;x

.

We will also introduce the stress operator T, where Tti' = ~ (div i:) N +

pi \l~ + vi: t I N . The double layer potential of a denc;ity g(Q) is then given by i:(X) = Kg(X) =

fao

IT(Q)r(X-Q)Itg(Q)da(Q), where the operator T is applied

to each column of the rna trix

r.

The single layer potential of a density g(Q) is

ti'(X) = Sg(X) =

J

r(X-Q) · g(Q)da(Q).

ao Our main results here parallel those of Section 2, part a). They are THEOREM

3.1.3. (a) There exists a unique solution of problem 3.1.1 in

L 2 (00, da). Moreover, the solution U' has the form

D, with N( ti')

f

ti'(X) = Kg(X),

g

f

L 2 (00, da).

(b) There exists a unique solution of (3.1.2) in D, which is 0 at infinity,

with N(\]u) Sg(X),

f

L 2 (aD, da).

Moreover the solution ~ has the form ~(X)=

gfL 2 (aD,da). 2

~

(c) If the data f in 3.1.1 belongs to L 1 (00, da), then we can solve

(3.1.1), with N(\7~)

f

L 2 (cD,da). Moreover, we can take ~

~

~

u(X) = Sg(x), g

f

2

L (00, da).

The proof of Theorem 3.1.3 starts out following the pattern we used to prove 2.1.1, 2.1.2 and 2.1.14. We first show, as in Theorem 2.1.3. that the following lemma holds:

165


LEMMA

3.1.4. Let Kg, Sg be defined as above, so that they both solve

j.LAii + (A+p.)'V div ii (a)

(b)

=

0 in R 3\CID. Then:

IINll L P 'I+ p.I \]u.... + \]u->t I) 2 do

t II 2 do .

ao

ao

Lemma 3.1.7 is proved by first doing so in the case when the Lipschitz constant is small, and then passing to the general case by using the ideas of G. David ([9] ). Lemma 3.1.8 is proved by observing that if ~ is any row of the matrix A.(div i:)I+p.I\Ji:+\Jirtl, then ~ is a solutioo of the Stokes system

/'t.~ = \]p in D { (S) div ~ = 0 in D

~lao= 7( L 2 (aD,do) This is checked directly by using the system of equations

p.!'t.\I + (A+p.)\7 div II= 0. One then invo~s the following Theorem of E. Fabes, C. Kenig and G. Verchota, whose proof will be presented in the next section. THEOREM

3.1.9. Given

f ( L 2(0D, do),

there exists a unique solution

(;,p) to system (S) with p tending to 0 at oo, and N(~) ( L 2(aD,do).

Moreover 1\N(;)II 2

L ¢(x)l, and so that T¢ is a bi-Lipschitzian inapping. Also, it is clear that T¢ is smooth for (x,y) with y > 0, and T¢(x,O) = (x, ¢(x)). We will den~te by A¢ the point T¢(0,1). Lemma 3.1. 7 is an easy consequence of LEMMA 3.1.11. Given M > 0 and ¢ with l\IV' ¢111 ~ M, there exists a constant C = C(M) such that lor all functions t; in D¢, which are Lipschitz in 5¢, which satisfy ~t; +(A+ p.)\7 div t; vii(A,J.) = V'ii(A,J.)t' we have IIN,J.(\i'ii)\1 2 'fJ

~tl vii+ vtrt Ill

'fJ

2

L

'fJ

O, e>O, a€(0,0.1), ifProposition (M,e) holds, then Proposition ((1-a)M,l.le) holds.

We postpone the proof of Proposition 3.1.13, and show first how Proposition 3.1.12 and Proposition 3.1.13 yield Lemma 3.1.11.

Proof of Lemma 3.1.11. We will show that Proposition (M,e) holds for any M,e. Fix M,e, and choose R so large that if e(10M) is as in Proposition R

3.1.12, then (l.l)Re(lOM)? e. Pick now aJ· > 0 so that

.n

(1-aJ·)=l/10.

J=l

Then, since Proposition (10M, e(lOM)) holds, by Proposition 3.1.12, applying Proposition 3.1.13 R times we see that Proposition (M,e) holds. We will not sketch the proof of Proposition 3.1.13. We first note that it suffices to show that

where

Ncp

is the nontangential maximal operator with a wider opening of

the non-tangential region. This follows because of classical arguments relating non-tangential maximal functions with different openings (see [14]

174

CARLOS E. KENIG

for example). Pick now ¢ with 111\7¢-i Ill~ 1·le, llli Ill :S (1-a)M. We will choose N¢ as follows: Since ao¢\B¢ is smooth, it is easy to see that we can find a finite number of coordinate charts (i.e. rotates, translates and dilates of

D.p ),

which are entirely contained in D¢ ,

such that their bottoms B.p are contained in aD¢, such that T.p((x,O): lllxlll < 1/2) cover (/JJ¢• and such that the o/'s involved satisfy

IIIVI/1111

S.

(t- f) M, and there exist

fl.p

:S (x,O): 1\\x\\1 < ~ )

such that llli.plll

l.lle . The non-tangential region defining N¢, on T.p ( is defined as follows: let F C {<x,O): \llxlll < ~} sider the cone on R~. y = l<x,y)

f

R~: b \xI

be a closed set. Con-

< y I, where b is a small

constant. Consider now the domain DF on R~, given by DF =

U ((x,O)+y). Then DF is the domain above the graph of a xfF

Lipschitz function (), for which IIIV Olil s_ Cb, for some absolute constant C (independent of F ). It is also easy to see that we can take now b so small (depending only on M and e )

~hat

domain above the graph of a Lipschitz function

if!,

which satisfies

IIIV~III S.

(1-

to) M,

tangential region defining N¢, for Q

Tl/1

~x,O):

if!

with

IIIVI/1-a".plll f

T .p(f~,_F) is the ~

:S l.llle.

lllxlll
0, consider the open-set Et ~ lm > tl. We now produce a Whitney type decomposition of Et into a family of disjoint sets IUj I with the property that each Uj is contained in T.p (<x,O): 1\\xl\l < for a coordinate chart cube in

D.p,

t)

each Uj contains T.p(lj), where Ij is a

11\x \\1 < ~ , and is contained in T.p(lj), where 1j is a fixed

multiple of Ij. Finally, we can also assume that there exists a constant 'Tlo suchthatif diam(Uj)S.Tlo• thereexistsapoint Qj in ao¢, with

dist (Qj ,Uj) ~ diam Uj, such that m(Q j) s_ t. Let now f3 > 1 be given. We claim that there exists

o> 0

so small that if Ej

=

Uj

n lm > {3t, iii S. otl

175


then a(Ej)

s (1-17M)a(Uj)

where 17M> 0. Assume the claim for the time

being. Then

J-

Joo ta(Ef!t)dt ~ f 2{3

ta(E,ldt - 2{3 2

0

J

2

J 00

ta(Ej)dt + 2{3 2

ta(m>8t)dt S

~ 2{3 2 (1-17M)

0

0

ta(Uj n E{3t)dt S

0

0

00

J 00

J 00

ta(Uj)dt +

0

J

Thus, if

we chocse {3 > 1 , but so that {3 2 ·(1-17M) < 1 , the desired result follows. It remains to establish the claim. We argue by contradiction. Suppose not, then a(Ej) >(1-17M) a(Uj). Let Ej

=

Tt(Ej). If 17M is chosen suffi-

ciently small, we can guarantee that IEj nijl? ·99~1- Let now Fj = Ej n Ij, and construct now the Lipschitz function as

i~ the definition of N¢.

lll'iJ«/J-a«/1111 S l.llh.

Thus,

~ :2: «/!,

=

lxdj

:f=~l.

We now apply Lemma 3.1.10"'to

then !FjnFjl ?C a(Uj), with

and such that there exists af, with lll'iJf-aclll S

corresponding to it,

lll'iJ;III S

at a time, to find a Lipschitz function f, with f Fj

«/!

t l.llle <E.

ill acill S

~

(~ -f0 ) M, «/!,

one variable

«/! on Ij, such that if III'Vf\11

s

(1

-f0)M,

(1 - laO) M so that

We can also arrange the truncation of our non-

tangential regions in such a way that on the appropriate rotate, translate and dilate of Df (which of course is contained in the corresponding coordinate chart associated to IA(div u)I t

11! 'iJu + 'iJut II S 8t.

D«/1,

which is contained in D¢ ),

To lighten the exposition, we will still

denote by Df the translate, rotate and dilate of Df. Note that Proposition (M,E) applies to it. We divide the sets Uj into two types. Type I

176

CARLOS E. KENIG

are those with diam Uj

~

TJo, and type II those for which diam Uj s_ TJo·

We first deal with the Uj of type I. In this case, Df has diameter of the order of 1. Because of the solvability of problem 3.1.2 for balls, and our normalization, we see that on a ball BCD¢, diam B A¢ (B, we have JB !Vlti 2

s_c

~

1,

JB \Adiv lti+,JVlt+vtttl\ 2 . Joining Af

to A¢ by a finite number of balls, and using interior regularity results for the system ~\i + (A+p.) 'V div lt = 0, we see that IV~(Af)l

S. C8t,

for

some absolute constant C. Then

J

s_ C a(Uj)8 2t 2 + c

Nl(A(div

lt)I+p.IV~+Vltt1) 2 da,

aof by (M,e). Thelastquantityisalsoboundedby Ca(Uj)8 2t 2 , whichis a contradiction for small 8. Now, assume that Uj is of type II. Note that in this case

th~e

exists Qj caD¢, with dist(Qj,Uj) ~ diam Uj, and such that !Vu(X)i

S.

t

for all X in the nontangential region associated to Qj. Because of this, it is easy to see, using the arguments we used to bound ivii'(Af)l in case I, that for all X in a neighborhood of Af and also on the top part of Df, we have that !Vu(X)\ m(Q) Nt
1 , if 8 is small enough, we see that we must have

Q)? m(Q).

Hence, N1 (vU -[IIU(Af); vU' -2 vir tlJ

faof Nf \'Vu

2

~

.... [Vir(A ) - virt(Aff])2 Nf Vu f 2 da ~

da ~ Ca(Uj)8 2t 2 , a contradiction if 8

is small. This finishes the proof of Proposition 3.1.13, and hence of Lemma 3.1.11. (b) The Stokes system of linear hydrostatics

In this part I will sketch the proof of the L 2 results for the Stokes system of hydrostatics. These results are joint work of E. Fabes, C. Kenig and G. Verchota ([13]). We will keep using the notation introduced in part (a).

II =

We seek a vector valued function

(u 1 , u 2 ,u 3 ) and a scalar valued

function p satisfying

(3.2.1)

~

flit = Vp in D div it= 0 in D .... .... 2 u lao ~- f ( L (00, da)

in the non-tangential sense.

THEOREM 3.2.2 (also Theorem 3.1.9). Given f ( L 2(aJ, da), there

exists a unique solution ( J,p) to (3.2 .1 ), with p tending to 0 at

oo,

and N(J)£L 2 (CD,da). Moreover, ~(X)=Kg{X), with g£L 2(aJ,da). In order to sketch the proof of 3.2.2 we introduce the matrix f'(X) of fundamental solutions (see the book of Ladyzhenskaya [25] ), f'(X) =

a..

x.x.

" lXI .

" \X\3 x.

(f 1.,.(X)), where f 1.J.(X) =-81 ~ + -81 -

.

vector q(X) = q 1 (X)), where q 1(X)

1

_J , and its corresponding pressure

1 -. = --

477\X\ 3

Our solution of (3 .2 .2) will

..

be given in the form of a double layer potential, u(X) =

fan

K.g(X) . =

IH'(Q)r(X-Q)Ig(Q)da(Q), where (H'(Q)f'(X-Q))if = oijqe(X-Q)nj{Q) +

178

CARLOS E. KENIG

ar.f act" (X-Q)nj(Q).

We will also use the single layer potential ;(X)=

J

In the same way as one establishes 3.1.4, one has: LEMMA

3.2.3. Let Kg, Sg be defined as above, with

Then, they both solve Ali= 'Vp in D, and

o-.

Also

(a)

!INII L 2 ( an,du) ~ Cll gI L 2 ( an,du) .

(b)

(Kg)±(P) = ±} g(P) - p. v.

(c)

IIN(\.lSg)\1 L 2

•

3.2.8. Let u,p be as m 3.2.5. Then,

Jl~t da ~

an

J ac

~

lvtt: 12 da +

l

J

/ns

an

~r da,

where the constants of equivalence depend only on M. Proof. 3.2.5 clearly implies, by Schwartz's inequality, that C

fan

~~~ 2da.

fan

iVI:i 2 daS:

Moreover, arguing as in the second part of the Remark 2

after 3.1.5, we see that 3.2.5 shows that

Ian

IVII 12 da S: C

Ian

IVtli 12 da

Ilac

+I

p ns hf

tx:

da

By Corollary 3.2.7, the right-hand side is bounded by

3.2.8 follows now, using 3.2.7 once more. ~

To prove 3.2.4, let u continuous across

aD.

~

=

~

S(g). By d) in 3.2.3, \ltu and ns

~s

ax.

are

J Using this fact, 3.2.3 e) and Corollary 3.2.8,

3.2.4 follows. In closing we would like to point out another boundary value problem for the Stokes system, which is of physical significance, the so-called slip boundary condition

181


(3.2. 9)

This problem is very similar to (3.1.2). Using the techniques introduced in part (a), together with the observation that if ~ir = V'p, div

ir = 0

in

D, the same is true fa each row v of the matrix [\7~+ vtrt - p·l], we have obtained THEOREM 3.2.10. Given f c L 2 (dD,du) there exists a unique solution

( ~.p) to (3.2.9), which tends to 0 at ->

Moreover, u(X)

=

-+

•

-+

S(g )(X), wzth g

f

oo,

and with N(Vir) c L 2 (dD, du).

2

L (dO, du).

DEPARTMENT OF MATHEMATICS UNIVERSITY OF CHICAGO CHICAGO, ILLINOIS 60637 REFERENCES [1]

A. P. Calderon, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sc. U.S.A. 74 (1977), 1324-1327.

[2]

R. R. Coifman, A. Mcintosh and Y. Meyer, L'integrale de Cauchy definit un operateur borne sur L 2 pour les courbes lipschitziennes, Annals of Math. 116(1982), 361-387.

[3]

R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. AMS 83 (1977), 569-645.

[4]

B.E.J. Dahlberg, On estimates of harmonic measure, Arch. Rational Mech. and Anal. 65 (1977), 272-288. - - - - · On the Poisson integral for Lipschitz and C 1 domains, Studia Math. 66 (1979), 13-24.

[5] [6]

B.E.J. Dahlberg and C. E. Kenig, Hardy spaces and the LP Neumann problem for Laplace's equation in a Lipschitz domain, to appear, Annals of Math.

[7]

----,Area integral estimates for higher order boundary value problems on Lipschitz domains, to appear.

[8]

D.E.J. Dahlberg, C.E. Kenig and G. C. Verchota, Boundary value problems for the systems of elastostatics on a Lipschitz domain, in preparation.

182 [9] [10]

CARLOS E. KENIG

G. David, Operateurs integraux singuliers sur certaines courbes du plan complex, Ann. Sci. del'Ecole Norm. Sup. 17 (1984), 157-189. , personal communication, 1983.

[11] E. Fabes, M. Jodeit, Jr., and Nf Riviere, Potential techniques for boundary value problems on C domains, Acta Math. 141, (1978), 165-186. [12] E. Fabes and C. E. Kenig, On the Hardy space H1 of a C 1 domain, Ark. Mat. 19 (1981), 1-22.

[13] E. Fabes, C. E. Kenig and G. C. Verchota, The Stokes system on a Lipschitz domain, in preparation. [14] C. Fefferman and E. Stein, HP spaces of several variables, Acta Math. 129 (1972), 137-193. [15] A. Gutierrez, Boundary value problems for linear elastostatics on c 1 domains, University of Minnesota preprint, 1980. [16] D. S. Jerison and C. E. Kenig, An identity with applications to harmonic measure, Bull. AMS Vol. 2 (1980), 447-451. [17]

, The Dirichlet problem in non-smooth domains, Annals of Math. 113(1981), 367-382.

[18]

, The Neumann problem on Lipschitz domains, Bull. AMS Vol. 4 (1981), 203-207

[19]

, Boundary value problems on Lipschitz domains, MAA Studies in Mathematics, Vol. 23, Studies in Partial Differential Equations, W. Littmann, editor (1982), 1-68.

[20] C. E. Kenig, Weighted HP spaces on Lipschitz domains, Amer. J. of Math. 102(1980), 129-163.

[21]

, Weighted Hardy spaces on Lipschitz domains, Proceedings of Symposia in Pure Mathematics, Vol. 35, Part 1, (1979), 263-274.

[22]

, Boundary value problems of linear elastostatics and hydrostatics on Lipschitz domains, Seminaire Goulaouic-MeyerSchwartz, 1983-84, Expose no. XXI, Ecole Polytechnique, Palaiseau, France.

[23]

, Recent progress on boundary value problems on Lipschitz domains, to appear, Proceedings of Symposia in Pure Mathematics, Proceedings of the Notre Dame Conference on Pseudodifferential Operators, Volume 43 (1985), 175-205.

[24] V. D. Kupradze, Three dimensional problems of the mathematical theory of elastocity and thermoelasticity, North Holland, New York, 1979. [25] 0. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, 1963.


183

[26]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. Vol. XIV (1961), 577-591.

[27]

J.

Necas, Les methodes directes en theorie des equations elliptiques, Academia, Prague,1967.

[28] L. Payne and H. Weinberger, New bounds in harmonic and biharmonic problems, J. Math. Phys. 33 (1954), 291-307. [29]

, New bounds for solutions of second order elliptic partial differential equations, Pacific J. of Math. 8 (1958), 551-573.

[30] F. Rellich, Darstellung der Eigenwerte von ~u + Au durch ein Randintegral, Math Z. 46 (1940), 635-646. [31] J. Serrin and H. Weinberger, Isolated singularities of solutions of linear elliptic equations, Amer. J. of Math. 88(1966), 258-272. [32] E. Stein and G. Weiss, On the theory of harmonic functions of several variables, I, Acta Math. 103 (1960), 25-62. [33] G. C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, Thesis, University of Minnesota, (1982), also, J. of Functional Analysis, 59(1984), 572-611.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS Steven G. Krantz* §1. Three basic methods for obtaining integral formulas We begin by discussing three ways to think about integral formulas on domains in C 1 , with a view to finding techniques which might generalize to en . These are (I)

Exploit symmetry of the domain;

(II)

Use differential forms and Stokes's theorem;

(III) Use functional analysis. Discussion of I. Let 11

=

lz E"C: lz I < 11. For n E" Z 1 define cpn(rei())

=

rlnleinO. Then direct calculation shows that

ciJn(O)

=

2~

J cpn(ei~d() 277

1

all n.

(1.1)

0

Now if f is harmonic on a neighborhood of 11

1

then f has an L 2

convergent Fourier expansion

(1.2) n=-oo

By linearity 1 (1.1) and (1.2) yield

*Work supported in part by the National Science Foundation. The splendid lecture notes prepared by Li Hui Ping, Li Xin Min and Ye KeYing greatly simplified the task of writing this paper. 185

186

STEVEN G. KRANTZ

J f(ei~dO. 21T

f(O) =

irr

(1.3)

0

Of course formula (1.3) holds in particular for holomorphic f; then we rewrite (1.3) as •

f(O) "' 1 .

2iTi

J~ 27T

elO -0

(iei 0d0) = 1 .

0

where y(O)

=

t

2m j

f((;) d' ,_0 y '

(1.4)

eiO, 0 ~ 0 < 277.

Now (1.4) is the Cauchy integral formula on the disc for the point z

=

0. In order to obtain the general Cauchy formula, we exploit more

symmetry. Recall that if z £A is fixed then the function

called a Mobius transformation, has the following properties: (a)

cp: A .... A

is biholomorphic (i.e. holomorphic, one-to-one, and

onto, with a holomorphic inverse). (b) cp(O) = z

(c)

cp,

¢- 1 are smooth on a neighborhood of ~.

If f is holomorphic on a neighborhood of

X,

define g(~) = f orf>(~).

Then (1.4) applied to g yields

f(z)

= g(O) = 2;i

f g(~) d~ ~ f f(rp~)) d~. =

y

Change variables by ~ =

y

cp - 1((;) = ('-z)/(1-z (;).

Then

187

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

and logarithmic differentiation of ¢- 1 gives

f(z) =

2 ~i ~ f('') [:~z + 1 ~ ,:] d'

(1.5)

y

Now the numerator of the second integrand in (1.6) is a holomorphic function of '

which vanishes at 0. By (1.4), the second integral

vanishes. Formula (1.6) now becomes

which is the Cauchy Integral Formula for the disc. REMARK

1. If f is only assumed to be harmonic, then we cannot argue

that the second integral in (1.6) vanishes. Instead, a little algebra applied to (1.5) gives

This is the Poisson Integral Formula. REMARK

2. Among bounded domains in C, only the disc (and domains

biholomorphic to it) has a transitive group of biholomorphic self maps. (This follows from the Uniformization Theorem; see also [32].) Thus approach I has serious limitations in C1 . In en the limitations are even more severe. Indeed in Section 7, after we develop a lot of machinery, we shall return to the concept of symmetry in en and gain some new insights.

188

STEVEN G. KRANTZ

Discussion of II. We need some notation. Recall that in real differential

analysis on R2 we use the basis Jx ,

t

for the tangent space (i.e. all

linear first order differential operators are linear combinations of these) and dx , dy for the cotangent space. We have the pairings

< axa• dx >

=

a • dy > < ay

=

1 •

< axa. dy >

=

a• dx > < ay

=

0.

In complex analysis, it is convenient to define differential operators

The motivation for this notation is twofold. First,

Secondly, if f( z) = u(z) + iv(z) is a C 1 function, with u and v real valued, then

(au _av

ar =0 ~ ax = dY az

and

auay =_- av) ax •

(1.7)

which is the Cauchy-Riemann equations. Thus af = 0 means thett f is

az

holomorphic. We also define

dz = dx + idy , dz = dx - idy . It is immediate that

=
= 1 ,

< aa. dz > = < ~. dz > z

az

=

0.

189


An arbitrary 1-form is written

(1.8)

u(z) = a(z)dz + b(z)dz and we define the exterior differentials

au = ~ dz az Clearly du = au

+au.

STOKES' THEOREM.

A

dz 1

au = az aa dz

A

(1.9)

dz ,

Recall If {l CC Rn is a bounded domain with smooth

boundary and u is a smooth form on fi then

In our new notation, if

n ~ C 1 ~ R2

and u is a 1-form as in (1.8), (1.9),

then Stokes' Theorem becomes

(1.10)

I(~

-;)

dz

A

dZ .

n Now we can prove THEOREM.

If {l ~ C is smoothly bounded and f is holomorphic on a

nei~hborhood of

fi then

f(z)

=~ 21Tl

If(() d(. all z ( ( -z

an

n.

190

STEVEN G. KRANTZ

Proof. Fix z

and

!(')

.,-z

t:

0. Let e 0' am> E I. If E is sufficiently small, say

Discussion of Ill. If define

{}E =

lz

l {}:

dist(z,

0 < E < Eo I then ne will also be smoothly bounded. Define

H 2(0)={f holomorphicon 0:

Jlf(()l 2 ds(() (}{}

is normal projection then (flan)

a

=1, uz j 1 az j 1

and all other pairings are 0.

j=1, .. ·,n,

194

STEVEN G. KRANTZ

If a= (a 1,···,ak),

fJ

= (f3 1 ,···,fJe) are tuples of non-negative

integers (multi-indices) then we write dza =dz

a1

A··· Adz

ak'

cfZ fJ

=azfJ 1 A···

Aazt:;!o ·

,...L

A differential form is written

u

=I aaf3 dza Acrzf3

(2.1)

a,fJ

with smooth coefficients a a{3" (If 0 ranges over

lal

=

p,

1{31

=

:S p ,q c Z and the sum in (2 .1)

q only, then u is called a form of type

(p,q) .) We then define

au =

~ aaafJ

B ~ -r:- dzj A dza A Oz' , ~ . U.t: ) a, f3 ,J

By a calculation (or functoriality), du = called holomorphic if d"u

=0.

=

~ iJaafJ

.J3 .

~ - - Qz j A dza A dz f3 oz.J a, ,j

au + au.

A C 1 function

(Note: this means that df ifi:.

is

=0,

J

j

=1, ·· · ,n,

so f is holomorphic in the one variable sense in each

variable separately.) Finally, we introduce two special forms: if w = (w 1,···,wn) is an n-tuple of smooth functions then we define the Leray form to be ~

. 1

71(w) = ~ (-1)1+ wj A dw 1 A ... A dwj_ 1 A dwj+l A··· A dwn. j=l

Likewise cu(w)

= dw 1 A · · · A d wn .


195

We define a constant

J

W(n) =

lU(I) " lU(() .

B(O,l)

Here B(z ,r) = I( len: ,, -z I < rl. Now we may formulate a generalization of approach II in Section 1. THEOREM (The Cauchy-Fantappie formula). Let 0 ~en be a smoothly

bounded domain. Assume that w = (wl····,wn)

l

C 00(0

X

n\~)'

wj =Wj(z,(), and n

~ wj(z,() • ((j-zj) ~ 1 on

0 X 0\ ~.

(2.2)

j=l

Iff fc 1

di>

is holomorphic on n, then lor any z f0 we have

f(z)

=

nW~n)

If(() Tf(w) "6J(() .

an Before proving this result, we make some detailed remarks. REMARK 1. In case n = 1 , then w = w1 = - 1-

(-z

The Cauchy-Fantappie formula becomes

f(z) = 1 . Jf(() d( ,

2m

(-z

an which is just Cauchy's formula.

(of necessity). So

(2.3)

196

STEVEN G. KRANTZ

REMARK

2. As soon as n

~

2, the condition (2.2) no longer uniquely

determines w. However an interesting example is given by

Let us calculate what the theorem says for this w in case n = 2. Now

which by direct calculation

(2 .4)

Thus, by the theorem, we get a form of the Bochner-Martinelli formula:

for f (

c 1( n).

holomorphic on

n.

Now we turn to the proof of the Cauchy-Fantappie formula. For simplicity, we restrict attention to n = 2. Let


If a 1 , a 2

f

1

197

then we define

B(a 1 , a 2 ) = det (a 1 , 1 the system is then "over-determined" and a compatibility condition is necessary. For n = 1 the system is not over-determined. The three basic considerations about a POE are existence, Wliqueness and regularity. It is easy to check that

a is elliptic on functions in the

interior of a given domain; hence, if u exists, it will be smooth whenever f is (we will see this in a more elementary fashion later). So interior regularity is not a problem. Also, since the kernel of

a consists of all

holomorphic functions, uniqueness is out of the question. So, for us, existence is the main issue. The following example shows that the compatibility condition does not by itself guarantee existence of u. EXAMPLE.

Let {l C C 2 be given by

n = (B(0,4)\B(0,2)) U B ((2,0), ~)

at ~ 0

202

STEVEN G. KRANTZ

v

Let U=B((l,O).o and V=B((1,0),H asshown. Let 1jfC;(u) satisfy 11

=1

on V. Finally, let

Then f is smooth and

aclosed on

holomorphic on supp (Ji,) then the function h

n n.

= u - _1__1 zl-

n\(s ((1,0), }) n lz 1 =11).

0 since

1 is well-defined and z 1 -1

If there existed a u satisfying ~ = f would be holomorphic

(dh ~ 0)

on

But u would necessarily be smooth near

(1 ,0) (since f is) hence h has a singularity at, for instance, (1 ,0). Thus we have created a function holomorphic on 8(0,4)\8(0,2) which does not continue analytically to (1 ,0). This contradicts the Haitogs extension phenomenon (an independent proof of this phenomenon will be given momentarily). o Now that we know that ~

=

f is not always solvable, let us turn to

an example where it is useful to be able to solve the

a equation.

203


EXAMPLE. Consider the following question for an open domain 0 CC en:

If w = n n lzn = Ol -10 and if g is holomorphic { on w (in an obvious sense), can we find G

(3.1)

holomorphic on 0 such that Giw = g ?

If 0 is the unit ball, then the trivial extension G(zpz 2 ,···,zn)"' g(z 1 ,···,zn_ 1,0) will suffice. However if {} = B(0,2)\B(0,1) s; e 2 then g(z 1,0) = 1/z 1 is holomorphic on o> but could not have an extension G (else the Hartogs extension phenomenon would be contradicted). o THEOREM. Suppose that w C Cn is a connected open set such that

whenever f is a smooth a-closed (0.1) form on

ali

=

n

then the equation

f has a smooth solution. Then the answer to (3.1) is "yes."

Proof. Let 77: en-> en be given by 77(Zl····,zn) = (zl····,Zn-1'0). Let B = lz ( n: 17Z I wl. Then B, w are disjoint relatively closed subsets of

n,

tive neighborhood of w and ¢ Define

n

so there is a C 00 function ¢ on

'"" F

=0

such that ¢

=1

on a rela-

on a relative neighborhood of B.

on 0 by

'"" F(z)

{¢(z) · f(11(z))

if

0

else.

z

£

supp ¢

=

Then F gives a C 00 (but certainly not holomorphic) extension of f to 0.

204

STEVEN G. KRANTZ ~

~

We now seek a v such that F + v is holomorphic and F + vi(L) =f. With this in mind, we take v of the form zn · u and we want -~

a(F+v)=O or

Now f

is holomorphic on supp ¢ and z n is holomorphic so all that

o TT

remains is

acp . (f 017)

t-

zn .

au = 0

or

(- #).

(f 017)

(3.2)

zn The critical fact is that, by construction,

0

n {zn =01

acp

=

0 in a neighborhood of

so the right-hand side of (3.2) is smooth on {}. Also it is

easily checked to be

a closed.

Thus our hypothesis is satisfied and a ~

u satisfying (3.2) exists. Therefore F

=F +v

has all the desired

properties. o Our two examples show that solving the

a equation is (i) subtle and

(ii) useful. Thus we have ample motivation to prove our next result. LEMMA.

Let ¢

l

C~(C), k ~ 1 , and define f

u(z)

satisfies u (

c k(C)

Proof. We have

and

1 =- 2m -.

au = f.

=

cp(z)dZ. Then

lJ¢1

and let 0, flnna(P,e) is unbounded. It is useful to be able to construct singular functions. Often we ·can

nearly do this in the sense that we can find a neighborhood U cl P and a holomorphic function on

un0

which is singular at p (this is "called

a local singular function). Then the problem reduces to extending local singular functions to global ones. LEMMA.

Let 0 ~ en be a domain on which the

uniform estimates. If p (

ao

a operator satisfies

and there exists a local

si~ular

at P which is bounded off any B(P,e) then there exists a

function

~lobal

one.

211


Outline of proof. Let g: U n 0 ... C be a local singular function at P. Let V be an open neighborhood of P such that satisfy ¢

=1

near P and ¢

a problem to find a bounded

=0

off V. Set f

V £ U. =

Let ¢ £ C 00(U)

¢ · g + u and solve a

u. Then f is a global singular function at

P. Fix a strongly pseudoconvex domain 0. We sha 11 prove later that {i) uniform estimates for the

a operator hold on

0 and {ii) local singular

functions satisfying the hypotheses of the lemma exist for each p £

ao.

By taking a suitable root of the local singular function and applying the lemma, we may construct for each P £ aG a singular function Fp at P which is in L 2{0).

Now we will prove that there is an L 2 holomorphic

function F on 0 that cannot be holomorphically continued past any boundary point. This shows that 0 is a domain of holomorphy and essentially solves the Levi problem (se17 [31 ]). For the construction of F, let !Pi lj: 1 be a countable dense set in 00. Let Hij be the L 2 holomorphic functions on 0 U B (Pi,T) , j = 1,2, ....

Let A 2 (0) be the L 2 holomorphic functions on 0. Con-

sider the restriction map Yij : Hij -> A 2 (0). Define Xij

=

image Yij

£ A2 (0).

Because FP. exists for each i, Xij /. A2 {0) for all i,j. We claim that 1

.u. Xij /.A2 (0). Assume the claim for now. Take F £A2 (0)\ .u. Xij' l,J

l,J

This is the F we seek. The claim now follows from: PROPOSITION.

Let X and Y be Banach spaces, and T: X-> Y a

continuous linear map. Then the following are equivalent: (1) T(X) is not of first category in Y .

(2) T is an open mapping. (3) T is onto.

Proof. This a variant of the Open Mapping Theorem for Banach spaces. (I am grateful toR. Huff for this proposition.) o

212

STEVEN G. KRANTZ

§5. Convexity and pseudoconvexity Let n ~ RN be an open set. Then n is called geometrically convex ifwhenever P,Q£n and

O~t~1

then (1-t)P+tQfn. Incalculus,

however, a C 2 function y = f(x) is called convex if fq? 0. How are these ideas related? If n has smooth boundary, then we may think of n as given by

n = !x

£

RN: p(x) < Ol

for a smooth function p with \lp i 0 on

an

(Exercise: use the

implicit function theorem). The function p is called a defining function

(an,

for n. If p

Tp(an)

let =

j 0 so small that

n=lz (en: dist (z, ll)
(z,()\

provided ( is near

TTZ ,

~~

1TZ,

1TZ ,

we conclude that

(\Re w(z,()\ + \Im(z,O\)

say \( -1TZ \ < r 0 • Then

which

224

STEVEN G. KRANTZ

The second integral is trivially bounded since when lz-'1 ~ r 0 then A 1 is bounded. The first is majorized by

c

J f

=C

+C

f

Now

c'r 0

0.

Therefore we may set

By a calculation (see [31 ]), the matrix (gij(z)) gives a non-{]egenerate Kahler metric on {} (called the

biholomorphic

mappin~s.

Ber~man

metric) which is invariant under

In particular it holds that if «: 0 1 .... f! 2 is

biholomorphic then

As a result, metric geodesics and curvature are preserved. The Bergman metric and kernel are potentially powerful tools in function theory, provided we can calculate them. To do so, we exploit the idea of Kerzman and Stein [231 to compare K with the Henkin kernel. However a complication arises: the Henkin integral (7.1) is a boundary

232

STEVEN G. KRANTZ

integral while the Bergman integra 1 is a solid integral. How can we compare functions with different domains? What we would like to do is apply Stokes' theorem to the Henkin integral and turn it into an integral over 0. However, for z

l

0 fixed, Henkin's kerne 1 has a singularity at (

=

z . So

Stokes' theorem does not apply. The remedy to this situation is to use an idea developed in [19], [30],

[33]: for each fixed z ( n' let

Now construct a smooth extension 'liz of 1/Jz to 0. The CauchyFantappie formula is still valid with 'Pz replacing takes place on the boundary where 'II z =

.Pz

(since the integral

ifJ z ). Thus Stokes' theorem can

be applied to the new Henkin formula containing 'II z. The resulting solid integral operator on L 2(0) can be compared with the Bergman integral via the program of Kerzman and Stein (details are in [33]). The result is that

K(z,()

=

'l'z(() +(terms which are less singular).

As a result, curvature, geodesics, etc. of the Bergman metric may be calculated. Also the dependence of these invariants on deformations of

an

can be determined (see [12], [13]; it should be noted that the methods

of [1] or [6] may be used for the deformation study instead of the KerzmanStein technique). The following are the three principal consequences of these calculations for a smoothly bounded strongly pseudoconvex (a) As Q

J

z

->

an,

n:

the Bergman metric curvature tensor at z converges

to the constant Bergman metric curvature tensor of the unit ball. The convergence is uniform over

an.

(/3) The kernel and the curvature vary smoothly with smooth perturbations of

an.

(y) 0, equipped with the Bergman metric, is a complete Riemannian

manifold.

233

INTEGRAL FORMULAS IN COMPLEX ANALYSIS Now we conclude this paper by coming full eire le and discussing

once again the topic of symmetry of domains. The reader should consider that, up to now, all of our effort has been directed at obtaining (a), (fJ), (y). Now we use those to derive concrete information about symmetries. If {} s;_ en is a domain, let Aut fl denote the group of biholomorphic self-

mappings. If two domains

nl

and

n2

are biholomorphic we will write

nl ~ n2. THEOREM (Bun Wong [41 ]). If {} CC en is smoothly bounded and

strongly pseudoconvex and if Aut fl acts transitively on !1, then

n ~ball. Proof (Klembeck). Let Pol n be any fixed point. Let IPjl Pj .... afl. By hypothesis, choose ¢j

(constant curvature tensor of the ball).

(*)

Thus the Bergman metric curvature tensor is constant on !1. We now use THEOREM (Lu Qi-Keng [34]). If M is a complete connected Kahler

manifold with the constant holomorphic sectional curvature of the ball then M ~ball. This theorem, together with (*), completes the proof. o THEOREM (Greene- Krantz [13] ). If fl

an

..\II 'S llx t: Rnl Na(u-un) >..\/211

~ [2CaAp.\- 1 llu-unllhP]P. Since p < oo, llu-u

II

n hP

... 0

as n ... "", and since ..\ > 0 is arbitrary, it

follows that

By taking a countable sequence ci a's which increase to infinity, we obtain a proof d Fatou's theorem. It is clear that the family of Euclidean balls plays an important role in this theorem, not only in the definition of nontangential approach regions, but also crucially in the definition of the Hardy-Littlewood maximal operator and the proof of its boundedness. We now recall how these

248

ALEXANDER NAGEL

balls are also involved in studying the fundamental solution for the Laplace operator. An important fundamental solution for /'t.. is given by the Newtonian potential:

N(x) = if n = 2

lrr log lxl where wn = 2rrn 12 if

cp

f

;r(T) . Then

C ~(Rn) cp(x) =

I

~N

=

8 as distributions. In particular,

N(x-y)/'t..cp(y)dy

(2)

R" and if

1/;(x) =

f

N(x-y)cp(y)dy

(3)

R" then /'t..f/;

=

cp .

Proofs of these facts can be found in Folland [5], Chapter 2. A great deal is known about the operator

f-+ N

* f(x) =

f

N(x-y)f(y)dy .

R" Basically, the fundamental idea is that, when measured with appropriate norms, N

*f

has two more orders of smoothness than f itself. For

example, if f satisfies a HCilder continuity condition of order a, 0 0 define It is easy to check that

We associate to the family of dilations a pseudometric p((x,t),(y,s)) =

~

lzjl 2 = lzl 2

Recall that n is the image of the unit ball B =

!

l(w 1 ,···,wn+l)l~tlwl+d(w,'lOJ.

What do the conesponding balls B(z, 8) look like? We see that

\V t B(Z,8) is essentially equivalent to the pair of inequalities: lz-wl < 8

Fix

z = (z,zn+l) tan.

The complex tangent space to

an

at z is

given by the equation (t)n+l- zn+l- 2i<w,z> If w = (w ,wn+l)

(an'

=

0

0

the distance from \V to this complex tangent plane

is essentially

=

\Re (wn+l-zn+l)+i(\w\ 2 + lz 12 -2Cw,z >)I

=

IRe (wn+l-zn+l-2i)+i\w-zl 2 1

~ lw-z[ 2 +(Re(wn+Czn+l-2i)l

< 82

""'

0

261

VECTOR FIELDS AND NONISOTROPIC METRICS

Thus the balls B(z, 8) are essentially "ellipsoids" of size 8 in the directions of the complex part of the tangent space to

an

at

z,

and of

size 8 2 in the orthogonal real direction, and hence in particular

Thus the doubling property of the balls is verified, and d{l (or Hn ) equipped with the pseudometric d is indeed a space of homogeneous type. We now want to discuss the analogue of Fatou's theorem for boundary behavior of holomorphic functions in n. This problem was first studied by Koranyi [9] for domains like {}, and was later generalized by Stein [17] to general smoothly bounded domains in en. Here we want to emphasize the role of the nonisotropic balls on the boundary, in analogy with the role of Euclidean balls on Rn in Fatou's theorem. We begin by defining appropriate nonisotropic approach regions in G. Let rr : {} ... aG be the projection

For a> 0 and

VI= (w,s+ilwl 2 ) l an let

Aa( VI) = l(z ,zn+t) l n:rr(z ,zn+l) l B( VI, ap(z,zn+ 1 ) 1 12 )1 where of course B(w,8) is the nonisotropic ball defined by the pseudometric d. It is clear that this definition is analogous to our earlier definition of nontangential approach regions ra<xo) in n~+l. Now (z,t+ilzl 2 +iy)

f

Aa-II~ A 1>--111fll,

THEOREM

oa

it

P = 1.

8. Suppose u is continuous on {} and pleurisubharmonic on

0. For a > 0 there is a constant Ca (independent of u ) so that for all

'(an

263


Theorem 7 of course follows from Theorem 5 and the Marcinkiewicz interpolation theorem (see Stein [16], Chapter 1), since we already know

an

is a space of homogeneous type. Before proving Theorem 8, we point

out some of the consequences of these results. For 1 ~ p ~ "", we let Hp(O) denote the space of holomorphic functions on

n

which satisfy

sup J\F(z,t+i\zl 2+iy)[Pdu(z,t): y>O

1\Fil~

AI=

U lNaFE>AI

eSf (see Koninyi and Vagi [10]). Finally, we consider fundamental solutions. On Hn let

a a X·=...,-+2y-"""!;", J ox, J 01:

a

J

J

where we write z · J

=X·

J

a

a T="""t:

Y-=...,--2x·~·

oy.

J

J

ot

01:

+ iy-. These vector fields form a basis for the J

left invariant vector fields on Hn. Put - = -1 (X -+iY .) - (X ·-iY.), Z. Z · = 21 J

and consider for a

t

C.

J

J

J2

J

J

267


This second order operator arises in the following way: if we identify Hn with

an,

the vector fields

zj

annihilate the boundary values of holo-

morphic functions. Thus, in analogy with the operator

a,

we consider

n

abf

=I zid(i,

on functions,

j=l

and we extend this in the usual way to (O,q) forms on

an.

In L 2 (Hn)

we can define a formal adjoint (~)*, and the Kohn Laplacian is then

On q forms, o{,q> acts diagonally, and is given by the operator ~a where a = n - 2q. The operator c::1tJ is not elliptic but Kohn's fundamental work [8a] showed that one can obtain subelliptic estimates for ~a. Folland and Stein [6] discovered a fundamental solution for ~a· Define:

THEOREM 10 (Folland and Stein). ~acf>a

tions, where ca is a constant, and ca

=

cao in the sense of distribu-

.J 0 if a t

±n, ±(n+2), ±(n-t-4),

etc. Thus except for the exceptional values of a, solution for ~a, and it is easy to verify that

where

o = d((O,O), (z,t)).

1 cf> is a fundamental ca a

One also obtains corresponding estimates for

derivatives of cf>a, so that again there is a complete analogy with the estimates (4) for the Newtonian potential.

268

ALEXANDER NAGEL

In the case of the Heisenberg group, the basic vector fields are xl,···,Xn, y

I•"' ,Y n

I

and the vector field T which is given "weight"

two. We shall later see how the general construction applied to these vector fields gives the nonisotropic pseudometric d. Part II. Metrics defined by vector fields Our object in this part of the paper is to outline the construction of metrics from certain families of vector fields. Many details of the arguments will be omitted, and complete proofs can be found in [13]. In [7J, Hormander studied differentiability along noncommuting vector fields, and used the techniques of exponential mappings and the Campbell-Hausdorff formula. The case of vector fields of type 2 was studied in [11]. Balls reflecting commutation properties c:i vector fields have also been studied by Fefferman and Phong [4a], by Folland and Hung [Sa], and by SanchezCalle [lSa]. Let

nc

RN be a connected open set, and l:_t y l•'",Yq be C 00

real vector fields defined on a neighborhood of 0. We associate to each vector field Yj an integer d j = d(Y ;)

? 1 which we call the formal

degree, and we make two fundamental assumptions about this collection of vector fields. -

N

(1) For each x dl, the vectors IY 1 (x),· .. ,Yq(x)l span R . (2) For all j, k, we can write [Y;,Y k] =

e

I

c~k(x)Ye

where

de:Sd/dk 1

-

cjk c C 00(0). Here [X,Y] =- XY- YX is the commutator of the two vector fields. There are several basic examples to keep in mind.

=i;

and let dj =1 for 1 ~ j 0, there are

(3) Given x and y, there

curves ¢,1/1: [0,1] .... 1} with ¢(0)=x, ¢(1)=y, t/J(O)=y, t/J(1)=z, q

q

cp'(t)=Ia/t)Y/¢(t)), t/J'(t)=Ibj(t)Yj(lj!(t)), with 1

la;(t)\

1

S (d(x,y)+e)

d(Y .) 1 ,

lb;(t)\

S d(y,z).+e)

d(Y .) 1

•

Define 8: [0,1] .... fi by ¢(at) O(t) = {

.;, (at-1) ! (3y) 2

JJ J 2rr

-y/2 0

We now need the following result, which is where the hypothesis

lirb

~

0

is used: There arc constants C 1 , C 2 which depend only on rn so that for 8 > 0,

LEMMA.

(i)

(ii)

o < c 1 ~ A(O,cS)/t(cS) ~ c 2 )

We defer the proof for a moment, and return to the estimate for u(O,iy). In our last integral, when r is between between

~

J.

and 3

o~

and

o 3J , t(r)

is

It follows from the lemma that in this range,

A(r) ~ y/5(y) 2 , and it follows from part (iii) of the lemma that the range of s integration in the integral is contained in 1\s \ < Cy I for some constant C. Thus: 277 C5(y)

iu(O,iy)I(z) is a real polynomial of degree at most m with 0

Sj

~ m, and if !'!c/J(z) 2' 0 for lzl ~

ai~ (0) az'

=

0,

8 0 then for 8 < 8 0 •

In fact, if we can prove this for 8 = 8 0 = 1, then given c/J, we apply the result to 1/J(z) = c/J(8z), and the result for general 8 follows, so it

.

suffices to study 8 = 8 0 = 1. But now we let

,I= ~c/J(z)

real polynomials of degree < m such that

iAb

--:- (0) = 0, 0 < j < m; 6.cf>(z) > 0 if lz\

az1

and

- -

I

-

I

(!'- +13c/J jazaaz13 co>

=

l

1\ .

:S 1 ;

296

ALEXANDER NAGEL

It is easy to see that }": is a compact set of polynomials and that the

maps

II 27T

c/J-+}

0

1

c/J(reiO)rdrdO

0

are continuous on I. Moreover, these functions are strictly positive

!l.c/J ? 0 and !l.c/J i 0, on jz I 'S 1 , and so the functions are bounded below by a constant Am > 0. The general result now follows by dividing since

a general polynomial

c/J by I

I

fP+f3c/J (0)!. ifzaazf3

Finally, in order to show

when for

c/J is a polynomial of degree :S m.

o = 1.

it again suffices to check this

But

is a norm on this space of polynomials. Hence lllc/JIII

:S CA(0,1) :S Ct(1).

As an easy corollary of the theorem we obtain an estimate for the Szego kernel S(z,() for the domain 0 on the diagonal z = (. Recall that the orthogonal projection S of L 2 (an) onto H2 (U) is called the Szego projection, and formally, this projection is given by integrating against a kernel: Sf(z) =

J

f(()S(z,()da((),

an

z(n.

297


In fact S(z,() = ~ cPj(z)c/>j(() where

lc!>jl is a complete orthonormal

basis for H2 (Q), and this series converges uniformly on compact subsets of {} x {}. (See Krantz [lOa}, Chapter l, for further details.) Now it is easy to check that for z

f {}

S(z,z) 1 12 =sup IF(z)l

..

where the supremum is taken over all F by our theorem, if F

H 2 (D) with IIFII 2 ~ 1 . But H

H 2 (D),

f

IF(z)l

f

:S CIBI- 1

J

IF(()! da(()

B

< CIBI- 1 12

I

1/2

IF(()I 2 da(()

an

where B = B(rr(z),o) is the ball centered at the projection rr(z) of z, and A(rr(z),o) = -p(z). Thus we obtain: COROLLARY.

If z

f {}

S(z,z)

:S CIB(rr(z),n)l- 1

§11. Estimates for the Szega kernel on aD As a final application, we show how one can make estimates of the Szego kernel S(z,() on the boundary, at least for certain very special domains {}. Thus, let

298

ALEXANDER NAGEL

where ¢ is a subharmonic, non-harmonic polynomial of degree m. We also make the very restrictive assumption that !:i¢(z) = !:i¢(x+iy)

is actually independent of y. Our approach is the following: if

then L is a global tangential antiholomorphic vector field, and

so we identify H2 with the kernel of the differential operator. When we identify

an

with

cX R

in the usual way' and write z

operator becomes:

We now make a change of variables on C x R ~(x,y,t) =

(x,y,t-A(x,y))

where

J lC

A(x,y) = -

~ (t,y)dt .

0

Then if we put

b'(x) =

f 0

lC

!:i¢(t)dt

~ X + iy

' this

299


and

"'L dXa .[dYa b ,(x) dta] +

=

+

1

it is easy to check that L(f ou(x.y.t)dydt

R2 so that

u(x,y,t) =

JJ

e 211 i(Yl7+tr)u(x,n,r)dndr.

R2

cr-1:-cr

LU=J

LJU

where

Let t/l(x, 11 ,r) = e-211(71x+b(x)T)

300

ALEXANDER NAGEL

and let h\pg(x,77,r)

=

r/I(X,'fl,r) g(x,77,r) .

Then

Now

and

are isometries. Thus L is similar to the operator

9x,

acting on func-

tions which satisfy

The kernel of this operator thus consists of functions g(77, r) so that

Let

Thensince b"'(x)?O, l=l(77,r)\r1

TJ,Tg-

I J ""'

g(r) e411(7Jr+Tb (r)) dr

0()

e4TT(71r+rb(r))dr

-0()

Thus

J

0()

P17 ,rg(x) =

-0()

g(y) K17 ,r(y)dy

302

ALEXANDER NAGEL

where

J

if (71, r)

E

l:

""

e47T(l1r-+f'b(r))dr

-00

if (71,r)/I.

0

Ff(x,y ,t)

=Iff

f(r,s,u)S((x,y,t); {r,s,u))drds dy

where

S((x,y,t); {r,s,u)) =

=

Jfe277i[(y-s)l1+(t-u)r]e2~[l1(x-r)+r(b(x}-b(r))]

J"" e -277r[(b(x)+b(r))+i(t-u)] 0

I I 00

e217l1{(x+r) + i(y-s))

00 ~------------d71

-oo

e47T(l1r-rb(r))dr

-00

This is the kernel we have to estimate. We begin by estimating the inner integral. For r > 0, set

dr.

303


Then replacing r by r +}, and 'T/ by 'T/ + rb'(~} it follows that F(.\+it,r) =

J

00

t :+.

-~

.

e21Ti'Tiid'T/

~•'(})- 0, (1-iA)-Y:r-V2

= ..\-'f,...-V 2 (1/.\-ir%-f; 2 ,

where we have fixed the

principal branch of z-'h---f/2 in the plane slit along the negative halfa'X!is. The pewer series expansion of (w-i)-V:r--f/2 (which holds for \w\ < 1 ), then gives the desired asymptotic expansion (1.4). Step 2. Observe next that if TJ ( C~ and f is a non-negative integer, then

I

(1.5)

""

-00

To prove this let a be a C"" function with the property that a(x) for \xi ~ 1, and a(x)

=

0 when

The first integral is dominated by

lxl

=1

~ 2, and write

cl+l.

The second integral can be

written as

with

~f

=

~ d~ ~x . A simple computation then shows that this term

is majorized by eN -

..\N

J

1X 1e-2N dX= ctN/\, -N cf-2N-l

lx\~s

if f - 2N < - 1 . Altogether then the integral in (1.5) is bounded by

314

E. M. STEIN

CNief+t +A-Nl- 2 N+lJ and we need only take

(with N >

e2t)

,

E

= .\- 1 12 ,

to get the conclusion (1.5).

A similar (but simpler) argument of integration by parts also shows that

J

ei.\x 2e(x)dx = 0(.\ -N), every N :::- 0

(1.6)

whenever

e(s' and e vanishes near the origin.

Step 3. We prove the proposition first in the case cf>(x)

=

x 2 • To do this

write

where

'J;

is a C~ function which is 1 on the support of 1/J. Now for

each N, write the taylor expansion

Substituting in the above gives three terms 00

I

(a)

-oo

-oo

:\ 2

e~x

2 .

e-x xl dx

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

315

For (a) we use (1.4); for (b) we use (1.5); and for (c) we use (1.6). It is then easy to see that their combination gives the desired asymptotic expansion for

J e j,\x 2rp(x)dx.

Let us now consider the general case when k = 2. We can then write rf>(x)

=

c(x-x 0 ) 2 + O(x-x 0) 3 with c

.J 0

and set rf>(x) = c(x-x 0) 2 [1+e(x)],

where e is a smooth function which is O(x-x 0 ), and hence Ie(x)l < 1 when x is sufficiently close to x 0 . Moreover, rf>'(x) J 0, when x J x 0 , but x lies sufficiently close to x 0 • Let us now fix such a neighborhood of x 0 , and let y = (x-x 0 ) (1 + e(x)) 1 12 • Then the mapping x .... y is a diffeomorphism of that neighborhood of x 0 to a neighborhood of y = 0, and of course cy 2 = r/>(x). Thus

with

'J;

f

C~ if the support of r/1 lies in our fixed neighborhood of x 0 .

The expansion (1.3) (for k=2 ), is then proved as a consequence of the special case treated before. REMARKS:

(1) The proof for higher k is similar and is based on the fact that

I~ eMx\-x\1 dx ~ ck,f(l - ;.\ )-(f+l) /k 0

(2) Each constant aj that appears in the asymptotic expansion (1.3) depends on only finitely many derivatives of rf> and r/1 at x 0 . Note e.g. that when k = 2, we have a 0 = Vn"(-irf>'(x 0 ))-l 12 rp(x 0). Similarly the bounds occurring in (1.3') depend on upper bounds of finitely many derivatives of rf> and r/1 in the support of r/1, the size of the support of rp, and a lower bound for rf>(k)(x 0 ).

316

E. M. STEIN

References: The reader may consult Erdelyi [8], Chapter II, where further citations of the classical literature may be found.

2. Oscillatory inteArals of the first kind, n

~

2

Only some of the above results have analogues when n ?: 2, but the extension of Proposition 1 is simple. Continuing a terminology used above we say that a phase _function

rp

defined in a neighborhood of a point x 0

in R" has x 0 as a critical point, if (\/rp)(x 0 ) ~ 0. PROPOSITION 4. Suppose "' ( C~(Rn), and ¢

is a smooth real-valued

function which has no critical points in the support of "'. Then

1(,\)

=

J

eiArp(x)"'(x)dx

= 0(,\-N),

as A-+oo

for every N

~ 0.

Rn Proof. For each x 0 in the support of "', there is a unit vector ~ and a

small ball B(x 0 ), centered at x 0 , so that (~, 'Vx)r/J(x)?: c > 0, for X (

B(xo). Decompose the integral

reiA¢(x)l{!(x)dx

as a finite sum

where each "'k is C"" and has compact support in one of these balls. It then suffices to prove the corresponding estimate for each of these integrals. Now choose a coordinate system xl'x 2 , ···,xn so that x 1 lies a long ~ . Then

But the inner integral is 0(,\ -N) by Proposition 1, and so our desired conclusion follows.


317

We can only state a weak analogue for the scaling principle, Proposition 2; it, however, will be useful in what follows. PROPOSITION

multi-index

5. Suppose 1/J c C~, ~ is real-valued, and for some

a, \al > 0,

throughout the support of 1/J. Then

(2.1)

with k =\a\, and the constant ck(~) is independent of A and r/1 and remains bounded as long as the ck+l norm of ~ remains bounded. Proof. Consider the real linear space of homogeneous polynomials of degree k in Rn. Let d(k,n) denote its dimension. Of course lxatla\=k is a basis for this space. However it is not difficult to see that there are d(k,n) unit vectors e- has a non-generate

PROPOSITION

critical point at

X O.

small neighborhood of

(2.2)

J

If t/J

x0 ,

f

C~ and the support of t/J is a sufficiently

then 00

eiAcf>(x)l/l(x)dx"-'.\-n;z

~aj.\-i,

as ,\ ... ..,

Rn where the asymptotics hold in the same sense as (1.3), (1.3' ).

Note. Again each of the constants aj appearing in the asymptotic expansion depends on only finite many values of derivatives of cf> and 1/J at x 0 . Thus e.g. a 0 = frrn/ 2 . \' llt, ll2' ... , lln

-~

J-1

(-illj)- 1 12) .

are the e1genvalues of the matnx

•

r/J(x 0), where

h azcf>(x o) l 12 axjaxk r

Similarly

each of the bounds occurring in the error terms depend only on upper bounds for finitely many derivatives of cf> and r/1 in the suppcrt of r/1, the size of the support of "' , and a lower bound for det

~ a;~ 1 J

v2n

D (f) j'

l af =~ ~... Ili\X j

OJ
1 , since 71 is in 3° and Vrf>(x) = O(x) as x .... 0; thus. IVxcl>(x, 71)j ~ c'> 0, if the support of ';;J is a a sufficiently small neighborhood of the origin. Hence for the 71 in region 3° we may use Proposition 4 to conclude that the left-side of (3.2) is actually O(A.-N) for every N. The proof of Theorem 1 is therefore concluded. REMARK. We have used only a special consequence of the asymptotic formula (2.2), namely the "remainder estimate" analogous to (1.3') when N = r = 0. Had we used the full formula we can get an asymptotic expansion for d~({); its main term is explicitly expressible in terms of the Gaussian curvature at those points x direction

t

or

t

S , for which the normal is in the

-t.

(2) We shall now consider the problem in a wider setting. Here S will be a smooth m-dimensional sub-manifold, with 1

~

m S_ n-1, and our

assumptions on the non-vanishing curvature will be replaced by the more

324

E. M. STEIN

general assumption that at each point S has at most a finite order contact with any hyperplane. We shall call such sub-manifolds of finite type. (These have some analogy with the finite-type domains in several complex variables, which are also discussed in Nagel's lectures [21 ].) The precise definitions required for our considerations are as follows. We shall assume that we are considering S in a sufficiently small neighborhood of a given point, and th"en write S as the image of mapping

cp : Rm

... Rn,

defined in a neighborhood U of the origin in Rn. (To get a smoothly embedded

s

we should also suppose that· the vectors

aacp , Xl

c:!2 ,... ,~

UAm

UA

are linearly independent for each x, but we shall not need that assumption.) Now fix any point x 0 ( U C Rm, and any unit vector 71 in Rn. We shall assume that the function (cp(x)- cp(x 0 )) • 71 does not vanish of infinite order as x .... x 0 • Put another way, for each x 0 unit vector 11, there is a multi-index a, with 1 'S

(Jxt

f

U and each

\al , so that

(cp(x)· 1/)lx=xo f. 0. Notice that if (x',.,') are sufficiently close to

(x 0 ,71), thenalso

c~r¢(x')·71'1x=x'JO.

eachunitvector TJ then 3a, be called the type of the type of

cp

cp

lal~k,

with

Thesmallest k sothatfor

~(cp(x)·11)l 0 =10 axa x=x

will

at x 0 • Also if U1 is a compact set in U,

in ul will be the least upper bound of the types for x 0

in ul. THEOREM

2. Suppose

finite type.

Let dlL

(3.3)

S is a smooth m-dimensional manifold in

= c/fdo, \d~(e:)\

and in fact we can take e =

with c/f ( C;;'(Rm).

'S A\e:l-e,

lor some

Rn of

Then

e> 0,

1/k, where k is the type of S inside the

support of c/f. Proof. By a suitable partition of unity we can reduce the problem to

showing that


I

ei¢(x)·e-~(x)dx

=

325

O(ie-1-1 /k)

Rm with ¢ as described above, and the support of Now we can write there is an

e- = ATJ,

a with 1

Ia! :S

with

!TJI

=

k 1 so that

'J;

sufficiently small.

1 , and >.. > 0. Then we know that

(Jx)a ¢(x) ·

Tf

I 0 whenever x is 1

in the support of ~ (once the size of the support has been chosen small enough). Thus the conclusion (3.3) follows from (2.1) of PropositionS. References.

Theorem 1 in its more precise form alluded to in the remark

goes back to Hlawka [14]. See also Herz [13], Littman [18]. Randol [25]. and Hormander [16]. When S is a real-analytic sub-manifold not contained in any affine hyper-plane, then it is of finite type as defined above. For such real-analytic S estimates of the type (3.3) were proved by Bjork [2].

4. Restriction theorems for the Fourier transform The Fourier transform of a function in LP(Rn), 1 < p :S 2 is most naturally thought of as an Lp' function (via the Hausdorff-Young Theorem) and so at first sight it is viewed as defined only almost-everywhere. This impression is further supported by the case p ""' 2 , when clearly the Fourier transform can be completely arbitrary on any given set of zero Lebesgue measure. It is therefore a noteworthy fact that whenever n 2: 2 and S is a sub-manifold of Rn (with some appropriate "curvature") then there exists a p 0 = p(S), p 0 > 1, so that every function in LP, 1 -::; p-::; p 0 has a Fourier transform restricting to S (i.e. with respect to

the induced measure on S ). Let us make this precise. Suppose that S is a given smooth sub-manifold in Rn, with da its induced· Lebesgue measure. We shall say that the LP restriction property holds for S, if there exists a q

=

q(p), so that the inequality

326

E. M. STEIN

(4.1)

holds for each f

S,

f

whenever S 0 is an open subset of S with compact

closure in S. THEOREM

3. Suppose S is a smooth hypersurface in Rn with non-zero

Gaussian curvature. Then the restriction property (4.1) holds for

1 < p < 2n + 2 , (with q - - n+3

= 2 ).

.

Proof. Suppose 1/1 ~ 0 and 1/1 ( C~. It will suffice to prove the

inequality (4.2)

for p 0 = 2 n + 32, and f n+

l

S;

the case 1 < p < p0 will then follow by -

interpolation.* By covering the support of

-

1/J

by sufficiently many small

open sets, it will be enough to prove (4.2) when (after a suitable rotation and translation of coordinates) the surface S can be represented (in the support of

1/J) as

a graph: ~n = 4>(~l'····~n- 1 ). Now with d~t = 1/fda we

have that

where (Tf )(x)

= (h K)(x),

with

*1n fact the interpolation argument shows that we can take q so that {4.1) holds with q = (n- 1) p', which is the optimal relation between p and q.

n+1

327


Thus (4.2) follows from Holder's inequality if we can show that (4.3) where

p~

is the dual exponent to p 0 .

To prove (4.3) we consider the function Ks (initially defined for Re(s) > 0) by

es2_ (4.4) K x _ _ s< ) - f'(s/2)

f Rn

Here we have abbreviated (~ 1 , .. ·,~n- 1 ) by ~'; we have set ';/;(~') =

rp(~')(1 + I'V¢(~')! 2 ) 1 12 ,

so that

VJ(~')d~' = d~;

also 71 is a

C~(R)

function which equals 1 near the origin. Now the change of variables ~n ... ~n + ¢(~') in the above integral shows that it equals

with

-00

Now it is well known that ~s has an analytic continuation in s which is an entire function; also ~0

= 1; and i(s<xn)l :S. clxni-Re(s),

where lxnl ? 1 , and the real part of s remains bounded. From these facts it follows that Ks has an analytic continuation to an entire function s (whose values are smooth functions of x 1' ... , xn of at most polynomial growth). One can conclude as well that

328

E. M. STEIN

(a) K 0 (x)

=

K(x) ,

(b) IK_n/2+it(x)l ~A, all x

l R0 , all real t , n (c) IKl+it(t")l ~A, all ~ l R , all real t . ~

In fact (c) is immediate from our initial definition (4.4), and (b) follows from Theorem 1. Now consider the analytic family Ts of operators defined by Ts(f) =

h Ks. From (b) one has (4.5)

I!T_n/ 2 +it(f)ll

L

00

~ Al!fll 1 , all real t , L

and from (c) and Plancherel's theorem one gets (4.6)

liT l+it 0, and the metric could be defined in terms of the usual distance. The second are the dilations (z,t) ... (pz,p 2t), and the appropriate quasi-distance (from the origin) is then (lz14 + t2) 1 14 . The latter dilations and metric are closely tied with the realization of the Heisenberg group as the boundary of the generalized upper half-space holomorphically equivalent with the unit ball in en+l. This point of view, as well as related generalizations, is elaborated in Nagel's lectures [21].

336

E. M. STEIN

In the present context the first type of dilations and corresponding metric would be appropriate if one considered expressions related to ordinary potential theory in Hm viewed as R2 m+ 1 • However the two conflicting types of dilations (and related metrics) occur in e.g. the solutions of du.

=

f. (One sees this for example in Krantz's lectures [17],

where in the formula of Henkin we have a kernel made of prQducts of functions each belo~ging to one of the two above homogeneities.)* Other expressions of this type occur in the explicit formulae for the solutions of the a-Neumann problem (see [1], Chapter 7). Let us now consider the simplest operator on the Heisenberg group displaying simultaneously these two homogeneities. The prime example is given by Tf =hK

(6.1)

where convolution is with respect to the Heisenberg group, and the kernel K is a distribution of the form (6.2)

K(z,t)

=

L(z)B(t).

L(z) is a standard Calderon-Zygmund kernel in em

=

R 2 m, i.e. L(pz) =

p- 2 m L(z), L is smooth away from the origin, and L has vanishing

mean-value on the unit sphere. Here B(t) is the Dirac delta function in the t-variable, and in an obvious sense is homogeneous B(pt) = p - 1 B(t). Thus K is homogeneous at degree -2m- 1 with respect to the standard dilations, and at the same time homogeneous of degree -2m - 2 with respect to the other dilations; in both instances the degrees are the critical ones. We turn next to the question of proving that the operator (6.1) is bounded on L 2 (Hm). The most efficient way is to proceed via the Fourier transform in the t-variable. This leads to the problem of showing that the family of operators T,\ defined by

*In particular the terms A 1 and A 2 that appear in §6 of [17].

337


(TA,)(F)(z) =

(6.3)

J

L(z-w)eiAF(w)dw ,

em (with < z ,w > the anti-symmetric form which occurs in the multiplication law for the Heisenberg group) is bounded on L 2 (Cm) to itself, uniformly in A, -oo 0. Con-

We fix a curve t .... y(t) = (t, y 2 (t)), 0 y(t)

£

c2 ,

sider the transformation T, which maps function on the interval [0,1] to functions on R 2 , given by

Je~·y(t)f(t)dt. 1

(7 .1)

(T£)(0 =

0

THEOREM 9. Under the assumption above T is bounded from LP[0,1] to L q(R 2 ), whenever 3/q + 1/p ~ 1 and 1 :S p

< 4.

(Note that when p .... 4, then q .... 4 in the above relation between p and q .)

345


Proof. Write

F(~) = ((Tf)(~)) 2 =

(7.2)

f Je~·(y(s)+y(t))f(s)f(t)ds 1

0

1

dt ,

0

and we shall try to apply Plancherel 's theorem (more precisely, the Hausdorff-Young inequality) to F. To do this break the above integral into two essentially equal parts according to t

>s

or t 'S_ s , which

divides [0,1} x [0,1] into the union of two regions R 1 and R2 • We then consider the mapping of R 1 .... R2 given by x = y(s) + y(t), i.e. x 1 = s + t, x 2 -= Y/s) + y 2(t). It is easy to verify on the basis of our assumptions that this rna pping is one-one, and its Jacobian J satisfies IJ I -= \y;(s)-y;(t)\ ~ c\s-t\. Therefore

(7.3)

I

e ~·(y(s)+y(t))f(s)f(t)dsdt =

f e~·xf(x

x )dx dx

1' 2

1

2

R2

R1

with f(x 1 ,x 2 ) = f(s)f(t)\J\- 1 • So if we denote by F1 (~) the quantity appearing in (7 .3) then whenever 1 'S_ r (7.4)

:S

2, and 1/r' + 1/r = 1 , we know that

IIFlll Lr' (R2) :S cll£11 Lr (R2) ·

However

=

Jlf(s)\r lf(t)ifiJ\ 1 -r dsdt

346

E. M. STEIN

To estimate the last integral we need to invoke the theorem of fractional integration in one dimension in the form

llf\1~ whe~ p

So we take g(t)"' \f(tW, then \\gllu

=

ur. Then if we fix

a so that -1 +a= 1-r, then 32r. =

~, and p = 3 ~r. The limitation

=

0 < a becomes r < 2, and with q == 2r' we obtain from (7.4) that

IIF1 11 r' 2 L

(R )

:S c'

(J1

)2/p

\f(t)\P dt

,

0

with a similar estimate for F2 (0 which is the analogue of (7.4), but taken over R 2 . Since F = F1 + F2 and F = (Tf ) 2 we obtain IIT(f >II q

2

:s A1\fll L p[0,1] .

,

so

L (R )

Note that ~ = -1. = 3r- 3 = 1 - !. q 2r' 2r p 1

J. ~ !.p = 1 ,

q

and the limitation

:S r < 2 is equivalent with 1 :S p = 32~ r < 4. Theorem 9 is therefore

proved. It is clear that inequalities for the Fourier transform play a key role

in the above argument. If we want to generalize Theorem 9 it is natural to look for a corresponding extension of the L 2 boundedness of the Fourier transform and the Hausdorff-Young theorem. One result along these lines is as follows. Suppose we consider the family of operators T, depending on the parameter .\ , .\ > 0, defined by

T>,(f)(()

=

J Rn

ei.\tf)(x,O!/I(x,()f(x)dx,


347

where 1/J is a fixed C~(Rn x Rn) cut-off function; ell is a real-valued C"" phase function which we assume satisfies the assumption that its Hessian is non-vanishing, i.e.

(7.6)

PROPOSITION:

(7.7) COROLLARY: (7.8) where 1 ~ p ~ 2, and 1/p + 1/p' = 1.

REMARK. The boundedness of TA for any fixed A is trivial, but what is of interest is the decrease in the norm as A _. "". This decrease is consistent with the special case when Cl>(x, () is bilinear (and nondegenerate); when we take A ->

oc

in that case we recover the usual

(LP,LP') inequalities for the Fourier transform. Notice also that the corollary follows from the proposition by the use of theM. Riesz convexity theorem. To prove (7. 7) we argue as in the proof of Theorem 7; as in the treatment of the operator T"" it suffices to show that the operator norm of T~TA is bounded by AA-n. Now this operator has as its kernel the

function KA ((,n) given by

(7.9)

KA((,71)

=

J Rn

eiA(CI>{x,71}-CI>(x.())¢(x,J7)1/1(x,()dx .

348

E. M. STEIN

Now since

we can find a= (a 1 , .. ·,an), so that the aj depend smoothly on x and

l~(x,g,fl) ~ cle-111 .oo the support of K~(g,fl)· Set Dx = i~ (a,Vx>· Then since (Dx)~ei.\(«(x,fl)-«(xl)) = eiA(«(x,fl)-«(xl)), we can integrate by parts N times in (7.9) and obtain (7.10) It follows from (7.10) with N = n + 1, that the operator T~T,\ which has kernel K,\ has a norm bounded by A,\-n and the proposition is proved. We shall now formulate some theorems for oscillatory integrals of the form (7.11)

(T,\f)(0=J

eiA«(tl)l/l(t,g)f(t)dt, g(Rn,

Rn-1

which will generalize the restriction theorems (Theorem 9 above, as well as Theorem 3 in §4) and also give results for Bochner-Riesz summability. Notice that (7.11) are mappings from functions on Rn- 1 to functions on Rn. The basic assumptions on the real phase function «!> are as follows: We consider for each fixed (t 0 ,g ~ the associated bilinear form B(u,v) definedby B(u,v)=(v,Vt)(u,Vg)(«)(t 0 l~. with U£Rn- 1 , v

f

Rn. Our first assumption is that B is of rank n - 1 .

Thus there exists (an essentially unique), ii

f

Rn,

lui

= 1, so that the

scalar function t .... (U,Vg«(t,g}) has a critical point at (t 0 ,g 0 ). We shall also assume that this critical point is non-degenerate, i.e. we suppose the non-vanishing of the (n-1)x(n-1) determinant:


349

These assumptions will be supposed to hold at all (t 0 ,g~ in the support of r/J(t, 0, where r/1 is a fixed function in c;(Rn- 1 x Rn). THEOREM

10. Under the assumptions above the operator (7.11) satisfies

(7 .13) with

q=(~:Dp', ~+f=1,

(a) when n = 2, if 1

(b) when n

~

) (0) = 0, then the conditions (7.12) are near the origin equivalent with the non-vanishing Gaussian curvature of the graph tn = ¢(t 1 , .. ·,tn_ 1 ). If we apply the result (7 .13), letting A ...

oo,

it is not difficult to recover

Theorem 9 from part (a), and Theorem 3 from part (b). (2) The proof of part (a) follows the same lines as the proof given for Theorem 9, once we use (7.8) as the substitute for the Hausdorff-Young theorem; further details as well as relations with Bochner-Riesz summability may be found in the papers of Carleson and Sjolin [3] and Ht>rmander [15]. Since part (b) has not appeared before, we will outline its proof. This will also serve as a good review of many c:i. the notions we have discussed here. Proof of part (b). It suffices to prove the case p = 2, since the case

p = 1 is trivial and the rest follows by interpolation. Now the case p = 2 is equivalent by duality to the statement

(7.14)

IIT~(F)II

2 Rn-1

L (

)

~ AA-n/r'IIFII r Rn L (

)

350

E. M. STEIN

with r

o:

2 0 (0 so that (u,9~)cl> 0(~) i 0 to increase the rank of 9x 9~~ to n. Now, as in the proof of Theorem 3 in §4, we form K~ defined by 2

Kf(~,q) = ~~:/2)

f eiA(a>(x,77}-a>(x,~)).p(t,q)rf;(t,01xnrl+SJJ(xn)dx, R"

with dx

~

dt dxn, and where v is a C'(;' function which equals 1 near

the origin. We easily verify (7.16) since a>(x, 0 = cl>(t, 0, when x = (t,O) . Next (7.17)

K~+it is the kPrnel L 2 (Rn)

of a bounded operator from

to itself with norm ~ M-n/2.

This follows by applying the estimate (7.7) of the proposition above and using the non-degeneracy of the Hessian of iP(x, ~). Finally we claim that (7 .18)

00

f

-oo

352

E. M. STEIN

Then since lil_n/ 2 + 1 / 2 +it(u)\

'5. cluln/2-l/2 ,

as u

"" we see that to

prove (7.18) it suffices to show that (7.19)

1n proving this estimate for the integral KA given by (7.15) we may suppose that the integrand is supported in a sufficiently small neighborhood of a given point t = t 0 , (for otherwise we can write it as the sum of finitely many such terms). When we write ll>(t,71)

=

ll>(t,O

= (\7~ll>)(t,71)

·

(71-~) + 0(71-~) 2 we see that these are two cases to consider as in the

proof of Theorem 1 in §3: 1° when the directions 71 - ~ or ~ - Tf are close to the critical direction u arising in condition (7.12b); or 2° in the opposite case. In the first case we use stationary phase (i.e. Proposition 6 in §2) to obtain (7.19). 1n the second case, we actually get O(A-177-~ 1)-N, for every N ~ 0 as an estimate, by Proposition 4. This

completes the proof of (7.18), and shows that K~n/2+l 12+it is the kernel of a bounded operator from L 1 (Rn) to L 00(Rn), with bounds uniform in A. The proof of the theorem is then concluded by applying the interpolation theorem, as in the proof of Theorem 3. 8. Appendix Here we shall prove Lemma 2 and Theorem 6 which were stated in §S. First let Pd denote the linear space of polynomials in Rn of degree 0, !log u! (8.3) implies (5.4) whenever mp

=

=

log+u + log+fr ~ u + }u-r. Therefore

1 , and so that result also holds in

general. E. M. STEIN DEPARTMENT OF MATHEMATICS PRINCETON UNIVERSITY PRINCETON, NEW JERSEY 08544

REFERENCES (1]

M. Beals, C. Fefferman, and R. Grossman, "Strictly pseudo-convex domains," Bull. A.M.S. 8(1983), 125-322.

(2]

J. E. Bjorck, "On Fourier transforms of smooth measures carried by real-analytic submanifolds of Rn," preprint 1973.

354

E. M. STEIN

[3]

L. Carleson and P. Sjolin, "Oscillatory integrals and a multiplier problem for the disc," Studia Math. 44(1972), 287-299.

[4]

M. Christ, "On the restriction of the Fourier transform to curves," Trans. Amer. Math. Soc., 287(1985), 223-238.

[5]

R. Coifman andY. Meyer, in these proceedings.

[6]

R. Coifman and C. Fefferman, "Weighted norm inequalities for maximal functions and singular integrals," Studia Math. 51 (1979), 241-250.

[7]

S. Drury, "Restrictions of Fourier transforms to curves," preprint.

[8]

A. Erdelyi, "Asymptotics Expansions," 1956, Dover Publication.

[9]

C. Fefferman, "Inequalities for strongly singular convolution operators," Acta Math. 124 (1970), 9-36.

[10] C. Fefferman and E. M. Stein, "HP spaces of several variables," Acta Math. 129 (1972), 137-193.

[11] D. Geller and E. M. Stein, "Estimates for sin gular convolution operators on the Heisenberg group," Math. Ann. 267 (1984), 1-15. [12] A. Greenleaf, "Principal curvature in harmonic analysis," Ind. Univer. Math. J. 30 (1981 ), 519-537. [13] C. S. Herz, "Fourier transforms related to convex sets," Ann. of Math. 75 (1962), 81-92. [14] E. Hlawka, "Uber Integrate auf konvexen Korper. 1," Monatsh. Math. 54 (1950), 1-36. [15] L. Hormander, "Oscillatory integrals and multipliers on FLP," Ark. Mat. 11 (1973), 1-11. [16]

, "The analysis of linear partial differential operators. I," 1983, Springer Verlag.

[17] S. Krantz, "Integral formulas in complex analysis," in these proceedings. [18] W. Littman, "Fourier transforms of surface-carried measures and differentiability of surface averages," Bull. A.M.S. 69(1963), 766-770. [19] G. Mauceri, M.A. Picardello, and F. Ricci, "Twisted convolutions, Hardy spaces, and Hormander multipliers," Supp. Rend. Cir. MatPalermo 1 (1981), 191-202. [20]

J. Milnor, "Morse Theory," Annals of Math. Study

tt51, 1963,

Princeton University Press. [21] A. Nagel, "Vector fields and nonisotropic metrics," in these proceedings.

OSCILLATORY INTI!:GRALS IN FOURIER ANALYSIS

355

[22] D. H. Phong and E. M. Stein, "Singular integrals related to the Radon transform and boundary value problems," Proc. Nat. Acad. Sci. USA 80(1983), 7697-7701. [23]

, "Hilbert integrals, singular integrals, and Radon transforms,'' preprint.

[24] E. Prestini, "Restriction theorems for the Fourier transform to some manifolds in Rn in Harmonic analysis in Euclidean spaces," Proc. Symp. in Pure Math. 35, part 1(1979), 101-109. [25] B. Randol, "On the asymptotic behaviour of the Fourier transform of the indicator function ci. a convex set," Trans. Amer. Math. Soc. 139 (1969), 279-285. [26] E. M. Stein, "Singular integrals and differentiability properties of functions," 1970, Princeton University Press. (27] E. M. Stein and S. Wainger, "The estimation of an integral arising in multiplier transformations," Studia Math. 35 (1970), 101-104. [28] E. M. Stein and G. Weiss, "Introduction to Fourier analysis on Euclidean spaces," 1971, Princeton University Press. [291 R. S. Strichartz, "Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations/' Duke Math. ] .

44(1977), 705-713. [30] P. A. Tomas, A restriction theorem for the Fourier transform," Bull. A.M.S. 81 (1975), 477-478. [31]

, "Restriction theorems for the Fourier transform in Harmonic Analysis in Euclidean spaces," Proc. Symp. in Pure Math. 35, part 1 (1979), 111-114.

[32] S. Wainger, "Averages and singular integrals over lower dimensional sets," in these proceedings.

[331 A. Zygmund, "On Fourier coefficients and transforms of functions of two variables," Studia Math. 50(1974), 189-201.

AVERAGES AND SINGULAR INTEGRALS OVER LOWER DIMENSIONAL SETS Stephen Wainger< 1 ) I.

Introduction These lectures deal with work primarily due to Alex Nagel, Nestor

Riviere, Eli Stein, and myself dealing with certain averages of and singular integral operators on functions, f, of n variables, n

~

2. These

averages and singular integrals differ in character from the classical theory in that the integration is over a manifold of dimension less than n.

Let us begin with an example of the type of problem we have in mind. The classical differentiation theorem of Lebesgue asserts for any locally integrable function f

f(x) = lim - 1r-oO IQrl

J

f(x-y)dy

a.e.

Q,

(where Qr is the square, Qr = !xt:Rn!sup!xil ~rl, and !Qrl denotes the Lebesgue measure of Qr ), and

f(x) = lim - 1r-oO IBrl

J

f(x-y)dt

Br

(where Br is the ball, Br = lxllxl <Sri).

* Supported in part by a grant from the National Science Foundation.

357

a.e.,

358

STEPHEN WAINGER

Our first problems are the following: Problem lA: Does lim

1)

r ... o

J

f(x-y)dor(y) == f(x)

a.e.?

aQr Here aQr denotes the boundary of Qr

~nd

dor is n-1 dimensional

Lebesque measure on aQr normalized so that dor(aQr) == 1 . Problem IB: Does 2)

lim r-+0

J

f(x-y)dttr = f(x)

a.e.?

asr

Here dllr is the unit rotationally invariant mass on aBr. 1) and 2) trivially hold if f is continuous, and the questions only become interesting when we consider functions in a class like L"", L 2 or

,

L1 .

In questions lA and IB, we are considering certain averages

3)

MQ/(x)

=I

f(x-y)dor(Y)

aQr

and

4)

Ma/(x)

=I

f(x-y)dllr(Y).

aar We are asking if 5)

MQ f(x) ... f(x) r

a.e.

AVERAGES AND SINGULAR INTEGRALS

359

and M8 f(x)

,6)

->

f(x)

a.e.

r

The standard approach to this type of problem involves considemtion of appropriate maximal functions. We define the maximal functions 7)

and

m8 f(x) = sup M8

8)

r>O

mB

(Jfj){x) . r

is called the spherical maximal function. Since 1) and 2) hold for f which are continuous 1) would follow for

every f in LP, 1

'S p < oo, if we could show

for every f in Lp, and 2) would follow if we could show

The argument showing that 9) and 10) imply 1) and 2) is the same as the argument showing that Lebesgue's differentiation theorem follows from the weak type inequality for the Hardy-Littlewood maximal function given in chapter 1 of [S]. While it is not quite as well known, there are appropriate estimates on maximal functions that guarantee 1) and 2) hold for all L 00 functions. In our case this means the following: Let E be a measurable set and XE its characteristic function. Then if (11)

360

STEPHEN WAINGER

where C(A) may depend on A but not on E, then 1) holds for every f in L"". H

12) then 2) holds for every f in L ""·. A discussion of this can be found in [BF]. Let us try to see if 9) or 10) could be true in some simple cases. We consider for example the one-dimensional case. Here Br = Qr = lxl-r<xQf(x) =

:Jil 8 f(x) = oo for every x. We can

also see that 11) and 12) are false in one dimension. We just take Ee = lxl~x::;el. Then !Eel .... 0 but

for all x. We could still ask if 1) and 2) hold in some interesting class even though 9), 19), 11 ), and 12) fail. However an important idea of Stein shows that the failure [Sl] of 9), 1 0), 11 ), and 12) implies that 1) and 2) fail even in the class of locally bounded functions. The statement of the ·main theorem of [SI] requires that the underlying space be compact. But if 1) or 2) were true for an LP class on Rn, it would also hold for the

corresponding LP class on the torus. Furthermore, the theorem of Stein requires the hypothesis that 1

:S p :S 2. However due to the positive

nature of the averages under consideration, his ideas can be modified to show that 1) fails for at least some L 00 functions. See [SW]. Thus we obtain negative results in one dimension. Similar reasoning gives the same negative conclusion for question lA in any number of dimensions.


361

One need only consider F(x 1 ,. .. ,xn) = f(x 1 )h(x 2 ,··· ,xn) where f is as above and h is a nice function. So there are no interesting positive results in problem lA. However as we shall see later there are positive results for problem 18 in 3 or more dimensions. One might ask if there is a simple geometric reason why . there should be positive answers for the sphere and only negative answers \

for the boundaries of squares. It turns out that the underlying basic reason that we have positive results for the boundary of balls and negative results for the boundaiy of squares is that spheres are round and boundaries of squares are flat. In other words an important word for us will be CURVATURE. We will come back to the role d curvature in our problem in a little while, but first we shall discuss the other problems that we will consider. Problem II: Let y(t) be a curve passing through the origin in Rn. Is it true that lim

kJ

0h

f(x-y(t))dt = f(x) a.e., for f in L.,., or L2 or L 1 ?

Problem III: Let v(x) be a smooth vector field in Rn. Does

J h

lim _hl h->0

f(x-tv(x))dt = f(x)

a.e.

0

for f in L""' or L2 or L1 ? Corresponding to problems II and III there are interesting singular integrals. We let y(t) be a curve and v(x) be a smooth vector field as in problems II and III. We set

13)

Hi(x) =

I

a

f(x-y(t)) dtt '

-a

(where sometimes we wish to think of a as finite and sometimes as ""' ), and

STEPHEN WAINGER

362

J 1

Hvf(x) =

14)

f(x-tv(x)) d( .

-1

We call Hy the Hilbert transform along the curve y and Hv the Hilbert transform along the vector field v(x). We then have the following two problems: Problem II': Can we have an estimate 15)

for some p's ? Problem III': Can we have an estimate 16)

for some values of p ? The classical development of singular integrals and maximal functions suggests that problems II' and III' should be related to problems II and III. In fact the progress on problems I, II, III, II', and III' is all interrelated. We have presented our problems as variants of Lebesgue's Theorem on the differentiation of the integral. These particular variants arose from other considerations. Riviere was led to problem II' from the consideration of a problem of singular integrals, namely from trying to generalize the method of Rotations of Calderon and Zygmund. Calderon and Zygmund developed the method of rotations to reduce the study of operators

where K is a kernel having "standard homogeneity" that is 17)

A>O


363

(K(x) is a function on Rn ) to the one-dimensional Hilbert transform

Hf(x) =Jf(x-t) dtt

( f a function on R 1 ). We will explain how the method of rotations can lead to problem ll'. Let K(x,y) be a function of two variables x and y which is odd, K(-x,-y) = -K(x,y) and which has a "parabolic homogeneity," that is 18)

We wish to consider the LP boundedness of the transformation

JJ 00

Tf(u,v) =

19)

00

-DO

f(u-x,v-y) K(x,y)dx dy .

-00

We now introduce parabolic polar coordinates into 19) x = rcos ()

and find

JJ ""

20)

Tf(u,v)

=

0

277 f(u-rcos (), v-r 2sin ())

0

where r 2 N(O) is the jacobian factor in the change of variables. N(O) is smooth and N(0+77) = N(O). By 18 we see that

364

STEPHEN WAINGER

J

211

21) Tf(u,v) =

I

DO

N(O)K(cosO,sinO)dO

0

}f(u-rcosO,v-rsinO)dr

0

-I

=

I} ~

217

N(O)K(cos(0+TT),sin(O+rr))d0

0

f(u-rcos O,v-rsin(O))dr

0

since K is odd. Thus

J

217

Tf(u,v) =-

J ~

N(O)K(cosO,sinO)dO

0

}f(u+rcosO,v+rsinO)dr

0

since N(O+rr) = N(O) . Finally

J

211

21A) Tf(u,v) =

J} DO

N(O)K(cost,sin O)dO

0

f(u-rcos O;v-r 2 sin O)dr .

-oa

Now adding 21) and 21A) we find that 211

Tf(u,v)

=}I

J

DO

N(O)K(cosO,sinO)dO

0

}f(u-rcos0,v-r2 sin0)dr.

-~

If K(cos O,sin 0) is in L 1 of [0,2rr], we can apply Minkowski's inequality 1T

IITfiiLP

:scI 0

where

dO

IIHofiiLp


Hef =

I

365

00

f(x-yo(r)) d: ,

-00

with

Now we prove

by showing

This is a problem of the type II'. Stein was led to consider problem II by his study of Poisson integrals on symmetric spaces. We are not going to launch into a discussion of symmetric spaces, but instead we consider an example. Let

If K(x,y) were dominated by a decreasing, radial, L 1 function, the

classical theory would imply lim Me f(x,y) = f(x,y) e~o

see [SWE]. However

a.e.

366

STEPHEN WAINGER

so the smallest ~adial majorant of K is

1 which is not integrable. l+X2+y2

In effect K has too much of its mass along the coordinate axis. The extreme case of this phenomena would be to have a kernel with all of its mass on the coordinate axis. In other examples, kernels have too much of their mass along curves, and the extreme case of difficulties arising in problems of Poisson Integrals on symmetric spaces lead to Problem II. Appropriate positive results to problem III would have implications for the boundary behavior of functions holomorphic in pseudoconvex domains in en. The natural balls in these problems are long, thin and twisting. The idealized situation is that of a vector field. In the case of a strictly pseudo convex domain, the balls satisfy the standard properties that ensure that the usual covering arguments apply. See [SBC). For progress in the case of pseudoconvex domains see [NSW) and [NSWB). Now that we have seen some of the roots of our problems, let us consider why these problems don't fit into the framework of the standard theory of Maximal functions and singular integrals as presented for example in [S]. In the standard treatment of averages over Euclidean balls an important geometric property of the Euclidean balls is used. If two balls B 1 and B 2 of the same radius, r, intersect, then B;, the ball having the same center as B 2 but having radius 3r contains B 1 . To see how badly this property fails for our problems let us suppose we were considering averages 1

he

Jh Jt2 +E

0

f(x-r,y-s)drds

t2

over slightly thickened parabolas or balls


367

We now consider the intersection of two of these balls of the same size

Clearly one of these "balls" is not contained in a fixed multiple of the other (uniform in

E ).

We can also see the difficulty of using the Calderon-Zygmund theory to study Problem II'. Suppose y(t) is the parabola (t,t 2 ) in R 2 and

Tf(x,y) = Jf(x-t,y-t 2 )d: .

Then we may formally write

Tf(x,y)

=

=

If

Jf

f(x-t,y-st 2 ) 8(1;s) ds dt

f(x-t,y-v)

~ 8 (1- t~) dv dt .

Or 22) where 22A)

K(x,y) =

1.. 8 {1-x3

\

.!) , x2

368

STEPHEN WAINGER

The Calderon-Zygmund theory deals with convolution operators with kernels

K(x~y)

less than

but in their theory K(x+h 1y+k)- K(x 1 y) should be much

I

K(x~y)

if h and k are much smaller than x or y. However

for our K if (X Y) is a point on the curve y : x 2 and (x+h y+k) is not 1

1

on the curve, no cancellation in the difference K(r+h,y+k)- K{x,y) can occur no matter how small h and k are. I

The Calderon-Zygmund Theory is based on 4) tools a) The Fourier transform (The Fourier transform is even used in the L 1 theory) b) Interpolation c) Covering lemmas d) Calderon-Zygmund decomposition. Perhaps the natural attack on our problems would be to find appropriate covering lemmas and suitable variants of the Calderon-Zygmund decompcsition. Some progress in finding covering lemmas for related problems was made by Stromberg [Str] and [STRO] and Cordoba [CORl], [COR2], Cordoba and Fefferman [CFl], [CF2] 1 [CF3] 1 and Fefferman [FEf]. Our approach will however be different. We shall try to use the Fourier transform or other orthogonality methods and interpolation to reduce our problems on averages and singular integrals to the more standard averages and singular integrals. In retrospect we see that some of these ideas occurred in [SPL] 1 {CS] 1 and in [KS]. We have said earlier that curvature and Fourier Transform would be important for us. Actually they go together. If one has a nice measure on a curved surface, the Fourier transform of that measure decays at infinity even though the measure is singular. Let us consider some examples. Define, for a test function ¢,

IJ.(¢):

I 0

1

¢(tiO)dt.


ll

369

is supported on a straight line, namely the x-i:! xis, and

~(~,71) = ll(ei~xei71Y)"'

Jei~tdt 1

0

which is independent of 71 and hence cannot decay at infinity along the 71-axis. Now let us consider a measure supported on a parabola, 00

23)

v(c!J) =

I

-oo

Then

=

I

00

e-t2e~tei71t2 dt

.

-oo

This integral may be computed exactly by completing the square, and it is easy to see that

Thus V(~,71) tends to zero at infinity. Another example is afforded by rotationally invariant Lebesgue measure on the n-1 dimensional sphere lxl this measure by diL, we have for

gl

dll(g) = CnJn- 2 2

= 1 in Rn. If we denote

Rn

Cl~l) 1~1- {n~2)

•

370

STEPHEN WAINGER

See [SWE]. Thus

Of course we want to have a tool to estimate the Fourier transform of measures in general, not in just a few specific cases. This tool is

a lemma of Van Der Corput.

VANDER CORPUT'S LEMMA. Let h(t) be a real function. For some j, assume lh(j)(t)l ? .\. in an interval a ~ t ~ b. If j

=

1, assume also that

h'(t) is monotone, then

II I

b

exp (ih(t))dt

For the proof of Van Der Corput's lemma for j = 1 and 2 see [Z]. The proof for higher j is similar. Let us consider the measure

d~-t(c;b) =

24)

f

2

c;b(t.t 2 )dt.

1

Then

This integral cannot be evaluated explicitly, but we wish to see that Van Der Corput's lemma may be applied. We take h(t) = e-t-+ 71t 2 • First we use the fact that h .. (t) = 71· Thus by VanDer Corput's lemma with j = 2, we see


A ldtt((.T/)1 ~

25)

c

1

I I 112

(1 + T/ )

Now if 1111 < 1.{!, 8

So by Van Der Corput 's lemma with j = 1 , 26)

if

Putting 25) and 26) together we have 27)

for some 8 > 0. Stein pointed out in retrospect that we can already see from an estimate like 27) that dtt has interesting properties from the point of harmonic analysis -namely even though dtt is singular, Tf = dtt

*f

maps LP into L 2 continuously for some p < 2. For

371

372

STEPHEN WAINGER

The second integral is bounded if q' is sufficiently large which means for some q > 1. But then the first integral is bounded for f c LP where

!.p

+~

2q

=

1.

II. The Hilbert transform

alon~ curves

The first progress in our series of problems was made on the Hilbert transform along curves. The Hilbert transform along a curve can be thought d as a multiplier transformation 28) where

29)

my(e-) =

I

00

e~·y(t) dtt .

-oo

To see that 29) is true we may either substitute the formula

373


f(x)

=fe

-ifx f(~ )d~

into 13) or recognize the fact that Hf = D

*f

where D is a distribution

D¢ =f¢(y(t))dt .

So A

AA

Hf =Of,

and D may be computed by evaluating D on an exponential. Thus to prove that Hy is bounded on L2 one needs to show that my is bounded. The first result of this type was obtained by. Fabes [F]. Fabes showed Hy is bounded on L 2 in 2-dimensions for the curve y(t)

=

(t, ltlasgnt) ,

a> 0.

So Fabes' proof consisted in showing that the integral

J exp(it~+ilt\a(sgnt)Tt) d: 00

m(~,Tf) =

-oo

is uniformly bounded in ~ and Tf. To this end Fabes employed the method of steepest descents. The method of steepest descents is a method of obtaining very precise asymptotic information for large A about integrals of the form

J

exp(iAh(t))dt

374

STEPHEN WAINGER

by contour integration. However to employ the method one has to have very precise information on where the real part of h(z) is positive and negative in the complex plane. Thus already to employ the method of steepest descents for the curve (t,t 2,t 3 ), one would have to understand the zero set of

uniformly in ~ 1 , ~2 and ~ 3 • So it is hard to imagine using the method of steepest descents, and for the curve t, t 2 , t 3 , t 4 , t 5 it would seem close to impossible. Fabes' result was very important in that it gave the first clue that problems such as II and II' could have positive answers. However a better method would have to be found -a method that needed less precise information about h(t). The next step was to . a1 a2 an-1 show that 1f y(t),. (t,t ,t ,···,t ) , 1 < a 1 < a 2 < ··· < an_ 1 a. a. a. (here t J can mean either It! J or ltl J sgnt ) then Hy was bounded on L 2(Rn) [SWA]. Here we had to prove the boundedness of the integral

I

00

exp[i(e-1t+···+entan-1)] dtt.

-oo

The proof was by way of the Van Der Corput lemma but was unnecessarily complicated because at that time we only knew the lemma for j = 1,2. Let us see how Van Der Corputs "lemma works in the case y(t) = (t,t2, ... , tn). We then have to show that

30)

J exp(~1 t+···+i~ntn)~t ~

C(n).

e ~Jt!~R

where C(n) does not depend on E,Rl 1,···ln. We shall prove 30) by induction on n. By changing variables, replacing t by ___!__ 1 1 we l~nl n

375


may assume f n = ± 1 in 30). Then by using Van Der Corput's lemma with j = n , we find

J t

31)

exp(if1s+···+ifn_ 1sn-l±isn) :::c(n).

1

An integration by parts together with 31) shows that

J

32)

exp (if 1t + ··· + ifn_ 1tn- 1 ± itn)

~ < C(n) .

1:::1 t!:=:R

'I

Now

f

exp (if1 t + ... +if n-1tn-1 ± itn) dtt

e 0. Hence by Stein's interpolation theorem we would know that T 0 = T was bounded in all Lp, p S 2.


381

Then by a duality argument T would be bounded in all LP , 1 0.

Let us try to understand why the kernel for Tz, Re z > 0, might be a little better than the kernel for T0 . T is essentially Hy y = (t ,t 2 ), and so the kernel K 0 of T0 is essentially

from 22) and 22A). If we introduce "parabolic polar coordinates" y

=

r 2 sin 0 x

=

rcos 0 , we see

where sin 0 0 - - =1. cos 2 0 0 We might expect if Re z

=

e > 0 Kz to be e better than K 0 . So we

might expect

Now we would like to explain why a

1

IO-Ooll-e

singularity is better than

a 8(0-0 0 ) singularity. To see the situation more clearly, let us examine the analogous situation for the standard polar coordinates with 0 0 = 0. Suppose 53)

where x = rcos 0 and y = rsin 0 , and

382

STEPHEN WAINGER

54) (The factor _!. for ordinary polar coordinates plays the same role as 1 r2

r3

for parabolic polar coordinates.) We are trying to see whether

I

55)

)K(x,y)-K(x-h,y-k)l < C .

For either K = K0 or K = K£ . Let us take h = 0 and k = 1 and consider first K = K0 • Let us look at the contribution from y's which are very close to 0. We have

JJ

r>C nearo

If y is very close to 0 [0'1 "'

~ : y

r

0

1

~I ~y-1

So the left-hand side of 55) is at least f~

} dr

= "".

So 55) can't

hold. Let us put the matter a little differently. If we consider the 8's with ()

~

0 where the difference

offers no cancellation, we find there is only one bad 8, ()

=

0. But still

383


J•l•

jK 0 (,, 6)- K 0 ( 0 an operator of CalderonZygmund type. This proved that HY was bounded in LP 1 0.

h~mobeneity

is a powerful friend not

"-'

to be tossed away lightly. However Dp does not have this homogeneity. This situation can be remedied by defining

59) where 00

J

60)

-00

Note that for A> 0 61) Let us see how formula 61) can help us. We would like to show

62) if Re z > - ~. By formula 61) we may assume Then by Van Der Carput's lemma with j

=

Tf =

± 1 , let us say

2 , we see that

Tf =

1.


=I

t

ei(seis2 dsj :S C .

1

So an integration by parts shows

Joo (l+t4)-z/4ei(tei71t 2 d t 1

:S C(z)

1

(·1J=l). Now

-1

J 1

:S C(z)

t2

ftti 'S C(z) .

-1

But we already know that

I -1

1

e~tei11t2 ~t

0. Then

f J ""

00

dt

T

-00

-00

=

l

e-iqy (1 + 17 2t4)-E/4 ei7Jt 2

-00

J~

e i"(x 2-y )(I + •'x•) _,/4 d"

-00

where Pe/ 2 is a modified Poisson kernel. Pe/ 2 decays exponentially fast at oo and Pe; 2(u)"' C as u ... 0. See [SWE]. lull-E/2 Thus Ke<x ,y) has a singularity near the curve (t,t 2) of the form 1 which is just the improvement over the a-function that we llH:J 0 11-e/2 seek. A modification of these ideas worked for curves


y(t) = (t a1

387

at a2 an •t •...• t )

< a 2 ,- ··, < an . See [NRW]. However, there is a natural generalization of these curves. All of

these curves satisfy an equation of the form

63)

y'(t) =

~ y(t)

where A is a real nxn matrix such that the real parts of the eigenvalues of A are positive. For example if y(t)

=

(t 1 t 2 )

A=(~ ~). A curve satisfying 63) 1 where all the eigenvalues of A have positive real part is called a homogeneous curve. A will generate a group of transformations TA = exp(A log>.). Then

64) where

65) Moreover there is a distance pA (x) defined on R 0 such that

In the case of the cure (t 1 t 2 ) we may 1 as we said before take

388

STEPHEN WAINGER

Then

and

It turns out that in the case of a general homogeneous cure, we can

obtain a satisfactory analytic family of operators by defining

66) where

67)

p A

*

is the distance function corresponding to A*, the adjoint of A. For a detailed description of the argument see [SW]. Here we shall

just make a comment. If some of the eigenvalues of A have non-zero imaginary part, y(t) can be an infinite spiral. For example the curve y(t) = (tacos (/3logt), flsin ({3logt)) is an example. So one could believe it might be rather messy to prove integrals involving exp (ig·y(t)) to be bounded. It might be difficult to show that at each t some derivative of

~ · y(t)

However, if one makes a change of variables t

would be non-zero. =

eu, we would be led to

consideration of integrals involving ~ · 71(u) where 71(u) -= y(eu). If y'(t) = ~y(t), 71(u) satisfies

68)

71'(u) = A71(u) .


389

We shall show that if 77(u) is a curve in Rn satisfying 68) where the eigenvalues of A have positive real part, then either 17(u) lies in a proper subspace of Rn or for every

1

~ j

Sn

e-1 0

and u there is a j ,

such that dj -. e. 77(u) 1- 0 . du1

69)

From 68) we see that

By the Cayley-Hamilton theorem, we can find numbers aj , 0 O

IM 8

f(x)l ,. r

394

STEPHEN WAINGER

Stein used g-functions to prove THEOREM

3:

77) if p > ~ and n > 3.

n-1

Simple exampl,es of the form f(x)

_...;1/J~(_x)_ _ n-1 Jx\ -n loga _!_

= _

ix\

where 1/J = 1 near the origin and has compact support show that p > n~ 1 is necessary in order that 77) hold. The situation for n

=

2 , p > 2 is

unknown at this time. I would like to present here Stein's original argument which proved

77) for p = 2 and n = 4 . We define

78)

Assume that we could prove Jlg(f )1\ 2 ~ C(n) \If\\ 2 •

79)

L

L

and let us see how 77) would follow. Now r

rnMrf{x)

=I fs

snMsf(x)ds

0

J r

=

n

0

J r

sn- 1 Msf(x)ds

+

sn

0

d~ Msf{x)ds .


395

Thus

Mrf{x)

~~

J r

sn-t M8 f(x)ds +

:n

0 =

J r

sn

Js Msf(x)ds

0

I{r) + II(r) .

Now I(r) is dominated by the Hardy-Littlewood Maximal function and

ll{r)<S

r~J

r

sn-l/ 2 s 1 12 M8 f(x)ds

0

~1/2

r

~ .~ ~ (

~

s 2 "- 1

d~

g(f )(x)

Cg{f)(x).

So if we assume 29), we have llsup Ml(x)ll 2 ~ Cilfl\ 2 • L L We turn now to the proof of 79).

396

STEPHEN WAINGER

m(r)

1

=

n-2 1n-2(r) .

-T

r 2

Here

J n-2

is the usual Bessel function. We shall need to know

I..

0

J~

j lm(t\e!) \f(g)j

J J lfO

r

ldiLh *f(x,y)-ifih *f(x,y)l 2 dh

I~

J e

diLh *f(x,y)dh\

0

~ g(f )(x,y) +sup 11/!h *f(x,y)l . h>O

402

STEPHEN WAINGER

A classica 1 argument (see [R]) shows

I sup r/lh * £11 h>O

L

p

:S Cpll £11 p · L

Thus by 95), we see

suP'I\

e>o

Je

dllh*f(x,y)dhl

~ ~ Cl!£11 L2

0

If

L

2 ·

f~O.

}J e

dllh * f(x,y)dh

0

kI

2h

f(x-t,y-t 2 )dtdh

h

>

!.Je

- e

0

~ !-

J'

J

e

f(x-t,y-t 2 )

1 dh

11

E/2

f(x-t,y-t')dt ,

0

So from 96) we infer

!sup~ fe e>o

0

It remains to prove 96).

~

f(x-y,y-t 2 )dt

All£11 L

L2

2 •

403

AVERAGES AND SINGULAR INTEGRALS 00

Jj(gf)(x,y)j 2 dxdy =

f

~ Jjdp.h ... f-1/lh *fj2dxdy

0

=I""'~ I ~h((•.,>-¢h 0 we may write the expression in 98) as

f

00

~ l~(,\h(,,\2h2q}-~(,\h(,,\2h2q)j2

.

0

By choosing ,\ so that ..\2( 2 +..\4 71 2 == 1, we see that it suffices to estimate 99) when ( 2 + q 2 = 1 . In this case we see

J~ 1

99)

dp.(h(.h 2.,)-¢(h(.h 271)' 2

0

A

~c

f~ 1

0 ~

since dp.(O) = 1/1(0) = 1 .

dh

~c

404

STEPHEN WAINGER

Then from 27)

for some 8 > 0. So

r

'\\! if.;(he,h'•ll ~ { h'~~g ~ c .

100)

1

~

Also r/J((,TJ) S.

( (1 +

1

c

2 N 2 N for any N, so

+., )

f "" d: l~(h(,h2 .,)1

101)

<S

c.

1

Now we obtain 98) and hence 96) by combining 100), 101) and 102). In this section we have emphasized L 2 methods. LP results for p > 1, can be obtained by combining the L 2 estimates presented here with the techniques of section 2. Altogether one can prove the following theorems: THEOREM 4. If y(t) satisfies y' = ~ y(t) where all the eigenvalues of A have positive real part,

kJ jf(x-y(t))\dtiiLp <S ClifiiLP' h

II

sup o

Beijing Lectures in Harmonic Analysis

Beijing lectures in harmonic analysis

Beijing lectures in harmonic analysis

Lectures on Harmonic Analysis

Lectures on harmonic analysis

Lectures on Harmonic Analysis (University Lecture Series)

Trends in Harmonic Analysis

Harmonic Analysis

Harmonic Analysis

Harmonic analysis

Harmonic Analysis

Harmonic Analysis

Harmonic Analysis

Harmonic Analysis

Harmonic Analysis

Harmonic Analysis

Conference on Harmonic Analysis

Harmonic Analysis, Iraklion 1978

Non-Commutative Harmonic Analysis

Commutative harmonic analysis 04

Harmonic analysis on semigroups

Noncommutative harmonic analysis

Non-Commutative Harmonic Analysis

Non-Commutative Harmonic Analysis

Principles of harmonic analysis

Harmonic Analysis on Semigroups

Noncommutative harmonic analysis

Noncommutative Harmonic Analysis

Euclidean Harmonic Analysis

Non-Commutative Harmonic Analysis

Harmonic Analysis and Applications

Beijing Lectures in Harmonic Analysis