Beijing Lectures in Harmonic Analysis

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<1 a(x), realvalued.

x + A(x), A '(x) =a . We then have

where

Of course, in this case, if we use the Fourier transform we can define

ct>(t :x)f =

J

""

-00

eixg cp(g)f(g)dg.

6

R. R. COIFMAN AND YVES MEYER

This gives, for example

J ~dt 00

sgn

{t tx)f =Jeix( sgn ( f(()d( =! p.v.

=

H(f).

-oo

Thus we can define

I

00

f(t)(1+a(t) dt x-t+A(x)-A(t)

-oo

(where we used the observation that cp(ULU- 1) = Ucp(L) u- 1 ). We view F(a) = sgn La as an operator on L 2 (R) and wish to know whether it is analytic on L 00 or if we can replace a by complex a and still have a bounded operator. lla lloo < 1 , we find

lf we do this, writing a =a + i{J

F(a)f

=f

f(t) (l+ia+i8> dt x-t+A(x) + iB(x)- A(t)- iB(t)

=I

f~ t)[(1 +a)/(1 +a))(l +a) dt x+Ax)-t-A(t) +i(B(x) -B(t))

where

h = x+A(x),Cf =

f

00

f(t) (l+Bl.(t)) x-t+iB1(x)-iB1(t)dt

-00

f 1(t)

'=

f(t) 11+a - 1 +a Bl.(t)

81

=

Boh- 1 .

7

NON-LINEAR HARMONIC ANALYSIS

Since Uh is bounded on L 2 it would suffice to prove that C is bounded on L 2 forall B suchthat B' issmall. We could also try to prove this by expanding

-i1T sgn(L )f

a

=f

J

00

f{t) ~+a(t)) dt (x-t) + (x)-A(t)

=

~ (-l)k

~

(A(x)-A{t)\kf(l-Hi) dt. x-t } x-t

-oo

Observe that the operators are of the form

T(f) =ftp(A(x)-A(t)).llildt x-t x-t

We will prove Theorem 1: Let 'I'

t

=

fk(x,t)f(t)dt.

C""(C) and A(x) such that

Then the operator T is bounded on L 2 (R) (and LP 1 < p < oo

).

This

result will then be extended to Rn and other settings. We now return to the interpretation of C as the Cauchy integral for the curve z(t) = t + iA(t) where A is Lipschitz

t

t

. as we can see its boundedness in L 2 is equivalent to the analytic dependence of C(a)f on the curve a. This now is related to the lectures by C. Kenig (to which we shall return later).

8

R. R. COIFMAN AND YVES MEYER

Let us consider a more general version of the Cauchy integral. Let

r

be a rectifiable CUfVe through

01

S

be the arc length

parameter

The Cauchy integral on r

is given as:

J

""

Cr(f) = p.v.

f(t)z'(t) dt z(s) --z(t)

-00

00

I J

1

z {s) - z(t)

f(t)z'(t)dt s-t

s-t

-00

00

w(z(s)-z(t)\ fl(t) dt s-t s-t

J

-oo

if we assume

0 < o < lz(s)-z(t)l < 1

*

s-t

we can take

r/J €C 0(C)rp(z)

=}

on

-

o < lzl :51

and obtain the bounded-

ness on L 2 of Cr (from Theorem I). Condition

* is

equivalent to a

f

the so-called chord arc condition and *for 5 small is BMO with

llaii 8 MO

as an operator valued functional of a

small (see [3]). If we think of C 1

we will see that B.M.O. is the

space of analyticity or holomorphy of Ca.

§2. All the operators which we encountered previously had the form

9

NON-LINEAR HARMONIC ANALYSIS

T(f)

=

p.v.

where !K(x,y)i < -1 C I laxKI + x-y antisymmetric i.e.,

Ia

J

y

K(x,y)f(y)dy

Kl::;

_c_ , moreover )x-yj2

they were also

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

(For example, K(x,y) =

G.

David and

J.-L.

(A(x~-:(y))x~y

¢

l

C 4 , A' ( L"" .) Recently

journe found a necessary and sufficient condition for

such operators to be bounded on L 2 (or LP ). This condition is simply that T(l) must be of bounded mean oscillation. We now would like to state certain facts concerning B.M.O. and prove their theorem. Recall that b ( BMO(R) if

112 2 Jlb-m/b)l dx~ < III

llbll* =(sup..!. I

oo,

where mlb) =

Iii

I

Jb(x)dx I

and I is an interval (or a cube in Rn ), and that this norm is equivalent to the following "Carleson" norm

sup(_!_

II I

I

If I

where !{It =

t l{l(i)

1{1 (

III

1/2

ll{l t *bl2 dxdt) t

0

c~, ft/ldx

=

0 (1{1 i 0) (see [5]).

A basic reason for the frequent occurrence of functions in B.M.O. is

the following simple fact. PROPOSITION.

If T is as above atrl T is bounded on L 2 then T

maps L"" into B.M.O.

10

R. R. COIFMAN AND YVES MEYER

Proof. Let b

l

L"" and let I be given. Consider

~

b = Tb = T(bxl) + T(b(l-xl)) =

~

T = 2I and write

~

o1 + b 2 •

Clearly

1/2

+

(l:l ~ fb,-mj1> ll') 2

The first term is dominated by

For the second we observe that

<

J

_III_ dyllbll I x-y 12

< Cl\bll .

""-

00

lx-yl>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.

NON-LINEAR HARMONIC ANALYSIS

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 + I. ¢'[>I

s T.

13

NON-LINEAR HARMONIC ANALYSIS

(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

R. R. COIFMAN AND YVES MEYER

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
= 2 ~(t~)~1(t(}g(~) = (Q1,8tg)~ where

t/1 1(x) =

xcp(x) and Q 1 tg

the first term as

•

=

t/1 1 t * g . •

This permits us to rewrite

where

and

Also observe that Lt(l) = Jf1(x,y)dy = = tPt *T(l) = tPt*b (here all integrals converge absolutely since let(x,y)i ~ Pt(x-y) ). Before proceeding we remark that since let(x,y)i ~ Pt(x-y) we can think of Lt as an averaging operator on the scale t, and of P 1f as being essentially constant on that scale i.e., Lt(Ptf) would look like Lt(l) Pt(f). (This would be exact if Pt is a conditional expectation and

15

NON-LINEAR HARMONIC ANALYSIS

Lt had the correct measurability.) We are thus led to write

!

Q1 , 11L,(l)P1(f)i

d,'

0

+

I~ Q,}L,P,f-L,(llP,(f)ldi 0

I

00

Q1 ,til/It* b(x). cPt * fl dtt + E(f) .

0

We consider E(f) as an error term and prove that

(this is valid using only •). In fact, let g

£

L2

2

- p t(f )(x)) dy

~1/2 dxtdJ

A simple application of Plancherel's theorem permits us to estimate the first term. For the second we use dominate it by

*

and Minkowsky's inequality to

16

R. R. COIFMAN AND YVES MEYER

As for the first term, we proceed as above and are led to estimate

This is dominated by \\fll 2 if and only if (r/Jt *b(x)) 2 dxtdt is a Carleson measure i.e., b is in B.M.O. In fact, recall that dv(x,t) on R! is a Carleson measure if for each interval I

~

2

I= l(x,t) c R+: (x-t,x+t) :S II.

I

NON-LINEAR HARMONIC ANALYSIS

17

Carleson's lemma states that

J

FP(x,t)dv(x,t)-:::; c

J

F*(x)Pdx,

where F*(x) = sup \F(x-y,t)J.

Jy\
This is proved by observing that if 0..\

6..\

=

lx:F*(x)>..\1 then outside

F(x,t) <..\, thus

and the LP result follows by integration. We now would like to make a few comments concerning the principal term:

I~ Q,((,,. b)(,. f)) d: . ·(b;,) . 0

This is-essentially equivalent to a simpler looking version

I

00

<"'t *b)(¢t *f) dtt

0

which is the basic bilinear operation in (b, f) commuting with translations and dilations i.e.: If br(x) = b(x-r) and b..\(x) "'b(..\x) then

Since all the problems we will be considering are invariant under such transformations, it is not surprising that the bilinear operations which arise look like

TT.

18

R. R. COIFMAN AND YVES MEYER

One cannot conclude this section without mentioning that

11

is also

the principal term in the Bony linearization formula and is called paraproduct by him i.e., given F £Coo, f £ Aa, 0
The proof of this linearization formula is the "same" as for the T(l) theorem and is achieved as follows. 00

F{f) =

~~"/, F(f • ¢ 1) =-

J

[

t

i F(h ¢ 1) d1t

00

F '(f * ¢Jt) f

* .Pt dtt

(here .Pt = t

~ ¢Jt)

0 00

[

F'(f) •

¢ 1 £< Y,1 dtt + e

=

•(£; F '(f)) + e

A2a if f

£

Aa. (This method

and it is quite easy to verify the e

£

should be explored further in connection with non-linear estimates [15).) §3. Multilinear Fourier Analysis We would like to understand various explicit representations for "homogeneous polynomials" i.e. multilinear functionals. For simplicity and to avoid technicalities we consider Ak a multilinear operator defined on trigonometric polynomials on T' with values in periodic continuous functions i.e.

19

NON-LINEAR HARMONIC ANALYSIS

We will assume that Ak commutes with simultaneous translation of ti i.e.

* where

Clearly if we take t-(0) =e 1

ik-0 h 1 ,t. ((/) 1

=

e

-ihk·

1

t.(O) we find using the 1

linearity in each argument that

taking h

= ()

we get

k

iO(l: j i)

A(tl 12···tk)(O) =I···IA(jl, ···,jk)tl(jl)t2(j2)tk(jk)e jk

1

jl

where

Befcre continuing let us observe that for k = 1 , eikO form a basis diagonalizing all linear operators commuting with translation, in fact, all such operators are normal. For k?. 2 it isn't as simple. Consider for a moment a finite dimensional vector space V and a bilinear transformation A:V x V ~ V. We can ask when is it true that

20

R. R. COIFMAN AND YVES MEYER

"' defined on there exists a basis in V e 1 ···eN a linear operator A V®V diagonalized by e-®e. 1 J

A(e.®e·) =A· . e.®e. 1 J 1 ,J 1 J such that

(1 , ... , N) 2 ... (1 , · · ·, N)

i.e. we want the diagram to commute

A

V®V ----V®V

I

v

A: VxV

I·

V

where

This is clearly the situation for periodic functions. It would be interesting to understand the bilinear operators admitting such a diagonizable lift to the tensor product. This observation indicates that the hypothesis concerning commutation with translations, imposes severe restrictions on the nature of a multilinear operation. We now return to the line on which we realize multilinear operations as

and

21

NON-LINEAR HARMONIC ANALYSIS

(Note that this realization is only valid for multilinear operations verifying some mild continuity conditions.) This realization permits us for ~ dt . example to show that the study of J0 r/lt * b cpt * f T can easily be converted to the study of

I "': * ""

(r/Jt•b·¢t•f)dtt

0

which we considered previously in the proof of Theorem [II). In fact

if we take r/1 with support in 1 <

J(\

< 2 and

¢

supported in

\(J

< 110

we have

where ¢ 1(() isanyfunctionequalto 1 on 1.9<\(1<2.1. Inthis case we see that the two expressions are equal. This sort of analysis comparing frequencies and interaction of various functions can be carried out permitting one to understand the structure of some multilinear operations. We are thus led to consider the general question of studying multilinear multipliers. We state two known results.

22

R. R. COIFMAN AND YVES MEYER

PROPOSITION

1. Let !A(~ 1 ••• ~k)l <

1 k 1 then (maxl~il)

-T

maps L 2 x···xL 2 -.L 2 continuously. This is an elementary result which we leave as an exercise. It would be desirable to obtain more subtle results for L 2 • THEOREM

[5]. Let A(~( .. ~k) be such that

then

k

1-

q

=

I

-1

P·

1

1

oo

> - P·1 > 1

oo

> p1

•

'

More generally one can take operators of the form

with o(x, ~) such that

_R

l~a;- o(x,~)l ~

s

Catf3

II

1 + 1~1 a

for all a,{3 sufficiently large

and the same conclusion is valid. We now prove the result by induction on k; for simplicity we consider only k = 2.

23

NON·LINEAR HARMONIC ANALYSIS

We take f

t

LP, f 2 =a

=

[I

t

L 00 and write

k(x-t,

x-y)a(t~ f(y)dy

where k(x 1 ,x 2 ) =A(x 1 x 2 ) satisfies \k(x 1 ,x 2 )1 ~

C

X~+ X~

, \Vkl ~...f._ \xl 3

thus k{x,y) = Jk(x-t,x-y)a(t)dt verifies conditions 1°,2°, and 3° of the T(l) theorem: thus to verify the boundedness in L 2 (for a fixed) it suffices to calculate T(l) and T*(l). These can easily be calculated, in fact

which is a linear Calderon Zygmund operator applied to the bounded function a. Thus it is in BMO. Since T*, (in f, for a fixed) is given by the symbol A*(~ 1 , ~ 2 ) = A(~ 1' -~ 1 -~ 2), verifying the same hypothesis the result follows for T*(l).

Property 3° can be verified once we observe that the same induction shows that

and that this condition implies

J

IT(¢t)\dx

III

~c

on any interval of length t.

24

R. R. COIFMAN AND YVES MEYER

In fact

now assume if>t(Y) is supported in I.

if \xI

> 2lt I

thus

JIT if>t(x)l dx

~c

if dist (1,1) > t .

I

Since T(if>t) is in B.M.O. it follows that

T~ l~l JIT(¢t)(x)-mfT<¢t))l I

=9 JIT¢ 1\dx

:S c,

I

thus we obtain

provided 1/1 t cf>t are supported in I. Again we conclude with a few general observations. As you all know the Fourier transform in a basic tool in linear P.D.E. permitting the reduction for example, of initial value problems to simple O.D.E. We claim that this can be achieved also for non-linear partial differential equations. Before illustrating on an example recall also that a common method for

25

NON-LINEAR HARMONIC ANALYSIS

solving differential equations (linear and non-linear) is to plug into the equation a formal power series whose coefficients are determined recursively using the differential equation. We now sketch how these two ideas can be combined to "formally" solve the Korteweg-de Vries equation (KdV)

du + a3 u :lo.

ax3

Ol

=

-6 u du ~ux

X

(

R,

t '> -

0

-

The set of solutions is clearly invariant under x translations and can be parametrized by functions of x (for example by specifying u(x,O)

~

f(x) ).

Let us write u as a power series

where

(here v is some unspecified functional parameter) plugging into the equation we obtain

du

a3u

___k + _ _ k

at

ax3

n-1

= -3 ~- ~ u.uk . ux ~

J

-}

j=1

k

we find first that

ak(~, t) = e

it ~e 1

J

ak(O, and that

(Strictly speaking ak is determined only if we assume that it is invariant

26

R. R. COIFMAN AND YVES MEYER

under permutations of ~k• so that the equality should be true only after symmetrization.) It can be shown (not so simply) that

solves the recurrence above.

Thus

If we let Cv(f) (0 = p.v.

f/

r,+7]

vx ' t(7])f(7])d7],

and

where r.fr satisfies the integral equation

we get

NON-LINEAR HARMONIC ANALYSIS

27

This set of remarkable formulas constitute the so-called method d "inverse scattering" for solving K deV see (9], and was shown to us by B. Dahlberg. §4. Functional calculus, resolvent expansions We start by considering a small perturbation of _I). L =

_!).

I

+

b· .(x) lJ

ij

a2

~

ox,ox, 1 J

The bij(x) are complex valued functions. We would like to prove L 2 estimates for such functions as

These provide us with estimates for solutions of the initial value or Dirichlet problem for 2 ata u = -Lu, T1 ata u = Lu, ata2 u = -Lu

or more generally for F(L) where F is a bounded holomorphic function in a sector iarg

zl < 8

containing the spectrum of L (this can be shown

to be true if e0 is sufficiently small). We define formally

F(L)=2~i

I (~-L)- 1 F(t;")d~ r

where

r

is the boundary of the sector.

28

R. R. COIFMAN AND YVES MEYER

We let i,j

(~"

A

R( =+~+u)

-1

a2 ax.ax. 1

J

We thus find

To better understand such expressions we consider the first two terms

Since F is assumed to be bounded holomorphic in a wedge containing the real axis we obtain (using Cauchy's theorem) that l(m(()l < ckk and

-1(1

thus A 0 is a Calderon-Zygmund kernel operator mapping L 2 -+ L 2 , L oo -+ BMO. We now consider A 1(B) and to simplify the exposition we will assume that

r

=

iR (i.e., F is bounded holomorphic in the half

plane Re z > 0 ). This is not going to affect the argument.

29

NON-LINEAR HARMONIC ANALYSIS

J

00

A (B) 1

=

_t_ B it+~

__i!__ l

axiaxi it+l\

F(it) dt t

-oo

and consider separately t > 0, t < 0. Change variables t

1 ± 2, to s

obtain 2 terms of the form

J

=

oo_l_ B s2o.o. _1_ IJ.(S) ds •

2A

1+S u

1

J l+S • 2A u

S

0

Now write

This gives

f.0

00

-

o.o. 1- B 1 J tJ.(S) ds which we recombine with the other l+S2~ --,_;S

term corresponding to t < 0 to get

The second term is of the form

The operator in curly brackets has a kernel satisfying*· Thus, to check boundedness in L 2 we need to calculate T(l), T*(l). This again reduces to the linear case when the terms are recombined. The higher order case is much more complicated since terms appearing lack regularity and need to be replaced by more regular terms complicating the induction. The main idea is the same, see [7], leading to L2 estimates for F(L) for e0 sufficiently small.

30

R. R. COIFMAN AND YVES MEYER

In general a functional calculus is obtained by considering functions F(L) where F is holomorphic in a neighborhood of the spectrum of L, and using the Cauchy formula. This becomes impossible for an operator

-£,

like

z ( C 1 , z = xl +ix2 since

and the spectrum is the whole complex plane. We are thus led to consider for ¢

£

C~(C 1 ) expressions of the form

¢(L) = -1.

fa¢

1 --

a"X A.-L

2111

- = - -1. dAAdA 2171

c

fc

cp(A.)

~ (_!__) dAAd'X.

iJA A.-L

These formal expressions need to be given sense and one should prove that ¢(L)¢(L)

=

(¢1/J)(L). In the case L 0

=

f!

such an expression is

easily justified using the Fourier transform. Since

1 1 s;, (here we used the fact that -:2 a_ z- = u(z), where 5 is the Dirac mass 171

z

If we now consider L = 1 ~

-£

at 0 ). We see that

like to understand the nature of

where a

f

L 00 ,

II a I ~5 0 we would

31

NON-LINEAR HARMONIC ANALYSIS

cp(L) = 2 1 . 171

fcp(A) .l_ - 1-

a>.. L-A

d..\11d'X

for functions ¢ homog_:neous of d 0 0. For example, we would like (this will give us

L=

E·L

which will turn out to be very important).

We can, in this case, calculate explicitly

(L->..)- 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

R. R. COIFMAN AND YVES MEYER

= 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.

NON-LINEAR HARMONIC ANALYSIS

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

~<:~ dy

:S C(ilt'alloo+1)Niifll 2 2

for some N > 0, C > 0. In fact, this will imply that for cp(x) such that Jl ~(01(1 + l.t'l N)d; < oo I'

we have

To show that the nam of . A(x)-A(y)

J

1

TA(f) =

grows like

IIA 'II~

e

-=-x-=y:--

~<:~

dy

we start with a number of easy observations. The

kernel of T satisfies the estimates (*)and is antisymmetric, thus it suffices to estimate the B.M.O. norm of T(1).

35

NON-LINEAR HARMONIC ANALYSIS

The operator norm of T is unchanged if we add a constant to a, or replace a by -a (this in fact will imply that the result is true for

a

€

BMO ).

We also need a lemma indicating that the space B.M.O. can be characterized by a weak local estimate. LEMMA (Stromberg). Let b be measurable and assume that there exist

a

> 0,

N

> 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
36 R. R. COIFAfAN AND YVES MEYER

In that case conside, the smallest {uncu.,.,

A1 ? A with A'(y)?

~M,

2M

3

1(y) "A(y) eXcept{"' disjoint inten>als lk on Which then h.ve 2M ABut A1(y) = J.

=2M/I

I-t M/Uikl

1M IU!kl ~OM- mr
Since

-¥J S Ai(y) :S M2,

we have

We

37

NON-LINEAR HARMONIC ANALYSIS We consider the function IAi I~ M2.

2M -A'(x) = Ai Then we have m1(A])

> 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 <- M + - M + 2M 3 31·3 3

2 2 , 2 2 3M - 3M ~ AI ~ 3M + 3M . The main result is the following. THEOREM (G. David). Assumin8 that there exist 8

> 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

R. R. COIFMAN AND YVES MEYER

Then T maps L 00 to BMO with IITII

If we let u(M)

=

sup

\\A:\Ioo-<;1

L

II fe

00

,BMO

~ c.,.,co.

iMA(x)-A(y) x-y

x-y

dyllsMo a direct application

of this theorem choosing for each interval

shows that u(M)

~ cu(~~

i.e., u(M):s_c 1 (1+M)N forsome N or . A(x)-A(y)

I\Tf\IL 2 = Je

Proof. We consider f

1

x~;Y

t

L oo, f

f(y)dy L 2

S C(l+I\A'\\~IIf\IL

2•

= fx 1 + f(l-x 1) = f 1 +f 2 , 1\flloo < 1. One

checks that

J

IT(f 2 )(x)- T(f 2 )(xo>l S c\11

I

from which it follows \x d: \Tf 2 (x) -c 1i > c \ < 7'/o\1\ if c is large enough. We now claim T(f 1 ) is well approximated by T 1(f 1), which we control, so that we want now to estimate

J E'

IT(f 1 )-T 1(f 1)\dx

NON-LINEAR HARMONIC ANALYSIS

where E =I- Uii

:S

I

I

E' =I- uTi where

f

Ti

=

39

(1-tO)Ii

\K(x,y)-K 1(x,y)\dy

xE"E' yd

Jf

=I

\K(x,y)-K(x,yi)+K1(x,yi)-K1(x,y)\dy

xE"E' Ii

where we have used the fact that K(x,y) = K1(x,y) outside Ii and yi are endpoints of Ii, (K(x,yi) = K1(x,yi)). This integrand is dominated by the

Marcenki~witz

function of Uii. Consequently the integral is

bounded by c III enabling us to apply Lemma 1. All of these results can be extended to Rn by various methods. The easiest is the so-called method of rotation based on a general transference principle, valid for multilinear operators commuting with translations, and some nonlinear operatas. We state the result in general although for the case of Rn this is an easy application of Fubini's theorem. TRANSFERENCE THEOREM.

Let Ut be !-parameter group of measure

preserving transformations on a measure space (X,dx) and A(a 1·"ak,f) a k+l multilinear operation on R given by

A(a; f)=

Jf Rn

satisfying

k(x-t, x-t 2 ••• x-tk, x-y)

II ai(ti) f(y)dtdy

40

R. R. COIFMAN AND YVES MEYER

Then the multilinear operator

A

on L 00(X)k x LP(X)-> LP(X) defined by

satisfies 1\A(A,F)ll

L

p

(X)

:S

II IIAi II

00

L

)

(X

IIF II

L

p

(X)

(note that the constants are the sam e). The examples we have in mind are the following. X

=

Rn, dx Lebesque

measure, e a unit vector

If we take Hf -=

J.

R

f(x-t) dt t

'

Hf = Jf(x-te) dt. t

Or consider

Hf ~ cf£(0-t) -

1-dt. ctg .!.. 2

If we take k

A (a f) =J(A(x)-A(y)\ f(y) dy k ' ~ x y x-y

J

41

NON-LINEAR HARMONIC ANALYSIS

then

A

k,e

(A f)= rtA(x)-A(x-te)\k f(x-te) dt ' t J t ·

J\

Multiplying by U(e) and integrating on lei = 1 in Rn we get

=

Je(x~=:(y)y K(x-y)f(y)dy

where k(y) = U(yjyj) IYin

is odd for K even even for K odd. §6. We now are in a position to recall the various ingredients which we discussed previously and recast them in a general setting. We considered operators L which are "small" perturbations of an operator L 0 for example, we took L

a = -!\ and L

=

-~+~ bij

a ax.ax. 2

I

llbijlloo < e0 and

J

then proceeded to expand functions of L, F(L) as a power series in terms of the coefficients of L-L 0 i.e., we wrote

F(L) = F(L 0)

+I"" Ak(b) k=l

where the Ak(b) was an operator valued homogeneous polynomial of d 0 k in b. Of course such perturbations could be shown to converge only in a little ball around 0 (in the space of coefficients). It is then appropriate to ask how far can one extend the results and what is the natural domain of analyticity or holomaphy of the function F(L). This question is

42

R. R. COIFMAN AND YVES MEYER

meaningless if the Banach space in which we prove analyticity (in a neighborhood of 0) is not the largest possible space for which such estimates can be proved. A first task is to identify this largest space, which we will call the space of holomorphy. Once this space has been found it is natural to ask for the domain of holomorphy of the corresponding functional. Let us return dow to our first example where F(a) =

sgn6~

d:)

where a is a function on R, and F(a) is an operator on L 2 (R). The series expansion was

Let us consider for simplicity the commutator series

F ( )£ oa

= ~f(A(x)-A(y)\k f(y) dy ~ 0

x y

We wish to find the smallest norm

} x-y

Ill Ill

=

(i.e., the largest Banach space)

for which

* in particular for k

=

1 we must have

IIA 1(a)il L 2 ,L 2 ~ Clilalll · H we decide to define lllal\l = 11A 1(a)il

2

2

it certainly defines a

(L ,L )

seminorm, if it actually is a norm for which the estimate* can be proved, we would have identified the space of holomorphy. But

43

NON-LINEAR HARMONIC ANALYSIS

A ( ) f =JA(x) -A(y) f(y) d 1 a x y x-y Y ·

We have already shown that IIA 1(a)IIL 2 L 2 :S cllalloo and it can easily be shownthat llalloo isdominatedby ciiA 1 (a)ll 2 2 • (In fact it suffices L L to consider the L 2 norm of

on the interval I whose center is x 0 . By letting I shrink to x 0 and using the L 2 estimate on A 1(a) one obtains an estimate for a, for simplicity one can assume that a is smooth since the estimate does not

Ill

depend on this condition.) Thus our norm norm. The estimates

Ill is equivalent to the L 00

* have already been proved

in this norm.

We may also find the domain of holomorphy of F 0(a), in fact

Fo(a)f

=

I

f

1 x-y- (A(x)-A(y)) f(y)dy =

1 f(y)dy 1 _ A(x)-A(v) """"X"=Y · x-y

We have seen by the method of boosting for the Lipschitz constant that this will be bounded on L 2 as long as

1

A..(A(x)-A(y)\ f x-y ) or some

-1-_-A..,. . ,. .(x-7-)-_.. ,. A. .,. (y"""'") = ~

A..LCoo

~ ~

X y

. IA{x)-A(y) . x-y - 1 I> 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

R. R. COIFMAN AND YVES MEYER

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 <5 0

then 1 -8 0 <

lz(s~=~(t)l ~ 1 , and this is precisely the condition permitting us to write C(a) =

f cp lz(s) -z(t)) !ill \

s-t

s-t

dt giving rise to a bounded operator on L 2 •

Thus BMO is the natural space of holomorphy. As you recall from S. Krantz's lecture one can express the Szego projection S(a), projecting L 2 (r,ds) ~ L 2(R,ds) onto H;(f,ds) (

H;

is the space of functions in

L 2 admitting a holomorphic H 2 extension to the "left" of f) in terms of the Cauchy operator. This representation yields the result that the S(a) has BMO as its natural space of holomorphy and is an entire function on the manifold of chord-arc curves. (A similar result is true for the Riemann mapping function.) Another remarkable example leading to an "exotic" space of holomorphy and to interesting geometry involves the functional calculus in L

= 1 !-a

t.

As already seen it was necessary to assume that a ( L 00

and a * ~ ( L"" with small norm. (This was obtained by considering the z2

-

example T(a) = the norm Ilia ill

E.)

It turns out that T(a) is analytic in a, relative to

= llalloo + ~a* -\lloo, and the condition really means that z

there exists a bilipschitz map h(z):C .... c such that

t

h = a(z) + 1.

NON-LINEAR HARMONIC ANALYSIS

45

It is clear from these and other examples that the identification of

the space of holomorphy, or a detailed study of the first bilinear operation

A 1 (a)f is basic to the understanding of these functionals. R. R. COIFMAN DEPARTMENT OF MATHEMATICS YALE UNIVERSITY NEW HAVEN, CONN.

YVES MEYER CENTRE de MATHEMATIQUES ECOLE POLYTECHNIQUE PALAISEAU, FRANCE

REFERENCES (1]

A. P. Calderon, Cauchy Integrals on Lipschitz curves and related operators. Proc. Nat. Acad. Sci., U.S.A. 75 (1977, 1324-1327.

[2]

R.R. Coifman, D. G. Deng andY. Meyer. Domaine de la racine caree de certains operateurs differentiels accretifs. Ann. Inst. Fourier 33, 2 (1983), 123-134.

(3]

R.R. Coifman, Y. Meyer, Lavrentiev's curves and conformal mappings, Rep 5. 1983, Mittag Leffler lnst., Sweden.

(4]

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

(5]

R.R. Coifman andY. Meyer, Au dela des operateurs pseudodifferentiels. Asterisque 57. Societe Mathematique de France (1978).

[6]

G. David, Operateurs integraux singuliers sur certaines courbes du plan complexe. Ann. Scient. Ec. Norm. Sup. 4° serie, 17 (1984), 157-189.

[7]

G. David, J. L. journe, "A boundedness Criterion for Calderon Zygmund operators," Annals of Math. 120(1984), 371-397.

[8]

E. Fabes, D. Jerison and C. Kenig, Multilinear Littlewood-Paley estimates with applications to partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 79(1982), 5746-5750.

[9]

R. Rosales, Exact Solutions of Some Nonlinear Evolution Equations, Studies in Applied Math. 59, 117-151.

[1 0] E. M. Stein, Singular Integra Is and Differentiability Properties of Function, Princeton University Press (1970).

MULTIPARAMETER FOURIER ANALYSIS Robert Fefferman

Introduction The article which follows is an attempt to give an exposition of some of the recent progress in that part of Fourier Analysis which deals with classes of operators commuting with multiparameter families of dilations. In some sense, this field is not that new, since already in the early 1930's the properties of the strong maximal function were being investigated by Saks, Zygmund, and others. However, for many of the problems in this area which seem quite classical, answers have either not been found at all, or only quite a short time ago, so that our knowledge of the area is still fragmentary at this time. The article is divided into six sections. The first treats some basic issues in the classical one-parameter theory whose multiparameter theory is then discussed in the remaining sections. Since the reader is no doubt quite familiar with the main elements of the classics 1 theory, we have omitted references to the materials in section one. The book "Singular Integrals and Differentiability Properties of Functions" by E. M. Stein is an excellent reference for virtually all of the material there. Finally, it is a pleasure to thank Professors M. T. Cheng and E. M. Stein for all of their hard work in organizing the Summer Symposium in Analysis in China, as well as many others whose generous hospitality made the visit to China such a very enjoyable one.

47

ROBERT FEFFERMAN

48

The maximal function, Calderon-Zy~mund decomposition, and Littlewood-Paley-Stein theory

1.

We hope here to review briefly some aspects of the classical !-parameter theory of these topics. The three are inseparable and we hope to stress this. We begin with the fundamental CALDERON-ZYGMUND LEMMA. Let f(x) ~ 0, f

€

L 1 (Rn), and a> 0.

Then there exist disjoint cubes Qk such that

1 (1) a
J

f(x) dx :': 2na

Qk

(2) f(x)

~

a a.e. for

and (3) IU Qkl

~~

J

X

j U Qk

lf(x) Idx .

Rn Proof. Let Rn be subdivided into a grid of congruent cubes so large that - 1 f. f < a for all of them. Subdivide each cube in this collection 1Ql Q into 2n congruent subcubes. Select from these the cubes Q' such that

..l. f. , f >a. 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 <;; IM£ >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)

MULTIPARAMETER FOURIER ANALYSIS

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
a> 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<~

lx\.

h x/lxln+l are especially important since

they are related to HP spaces and analytic functions. For a Caldeton-Zygmund singular integral T, it is easily seen to be bounded on L 2(Rn), since K(~) f L 00 • Also, since T* is also a Calderon-Zygmund singular integral, T is bounded on the full range of LP(Rn), 1 < p < oo,

2. Littlewood-Paley-Stein Functions. The most basic, simplest of these

!/! ( C;'(Rn),

are the g-function and S function defined as follows: Let

fRn !/!

=

0. Let !/! t(x)

=

en !/!(I-)

I

for t > 0. Then 00

lhl/lt(x)\2

~t

0

s2(f)(x) =

fJ

J..

If *ifJtI2 dtdy tn+l

l(x)

where f'(x)

=

l(y ,t)\\y-x\ < tl. Then it is a basic fact that "IIS(f)il P ~ L

Cpllf!ILP and i!g(f)IILP ~ Cpllf!ILP' when 1 < p < ""· If, say, 1/1 is suitably non-trivial, (radial, non-zero is good enough) then the reverse inequalities hold: I!S(f)li P ~ cpllfll P and i!g(f)li L

L

LP

~ cpllfll

LP

.

54

ROBERT FEFFERMAN

Now take S(f). We want to point out here that S is a singular integral. In fact define K:Rn Then

S{f)(x)

t->

L 2 {r(O); dydt) by K(x)(y,t)

lh K(x)l 2

=

L
n+l

=

l{lt(x- y).

)

and K satisfies

IK{:~t+h)-K(x)l ~

C _l_h_l lxln+l

lhl < !. 2 lxl.

Also by a Fourier transform argument IIS£11 2 ~ cllfl\ 2 so the L (Rn) L (Rn) Calderon-Zygmund theorem applies. In fact, the adjoint operator also maps LP(L 2(f')) ~ LP(Rn), 1 < p < 2, because it is also C-Z, so again this explains why we get boundedness of S on the full range 1 < p < oo. 3. The Hardy-Littlewood Maximal Operata as a Singular Integral. Let cp(x)

f

C (R") and suppose for Ixi < 1, ¢(x) = 1 and for lx I > 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
(since if t > 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

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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

<

J

lf(x)\dx ·(log a)k- 1da

Mf(x) [log+Mf(x)}k- 1dx.

Qk

2. Multi-parameter differentiation theory During the first lecture we discussed some fundamentally important operators. of classical (and sometimes, not so classical) harmonic analysis: the maximal operator, singular integrals, and Littlewood-Paley-

58

ROBERT FEFFERMAN

Stein operator. These operators all had one thing in common. They all commute in some sense with the one-parameter family of dilations on Rn, x ... ox,

o > 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<8/2)x (jx 2 j<8/2) •

MULTIPARAMETER FOURIER ANALYSIS

59

If we have a weak type inequality, we must have

1

so that llf0 11 ~ c log~ and the smallest Orlicz norm for which this holds is the L(log L) nam. A similar computation in Rn reveals that for M(n) to map Lei> boundedly to Weak L 1 we must have L <; L(log L)n-t. The next theorem shows that indeed M(n) does indeed map L(log L)n- 1 boundedly into Weak L 1 . THEOREM OF }ESSEN-MARCINKIEWICZ-ZYGMUND

(1935) [1}. For

functions f(x) in the unit cube of Rn we have

The proof is strikingly simple. Define Mx. to be the 1-dirnensional 1

maximal function in the ith coordinate direction. Consider the case n ""2, which is already entirely typical. Let R be a rectangle containing the point (x 1 , x 2), say R =I xJ. Then

(2.1)

I~ I

I

jf(xl'x2)jdx2

~ Mx/(xt,x2)

J

so (2.1) is

:S.

I~!

I I

M2f(xl,x2)dx 1 :S Mxl (Mx/)(xl,x2) .

60

ROBERT FEFFERMAN

We have seen that Mx 2

that

maps L(log L)(Q~ boundedly into L 1 (Q 0 ) so

and finally

Now, this method of iteration in the proof above gives sharp estimates for M(n), and it may be suspected that the whole story r:J. the harmonic analysis of several parameters can be told by applying this iteration technique. That this is not the case should become clear as this lecture proceeds. We want to describe some of the multi-parameter theory and to do this, let us begin with maximal functions. For many years after the Jessen-Marcinkiewicz-Zygmund theorem, there was no machinery around to treat problems here, and then, only fairly recently, two such machines were created. The one we describe here proceeds by means of covering lemmas while the other, due to Nagel, Stein, and Wainger, which Wainger has described in detail, uses the Fourier transform [2]. Though the two methods seem totally different on the surface, they are really quite closely related and have in common the main theme of reducing higher parameter, complicated operators to lower parameter simpler ones, which are already well un:lerstood. The model for our method is the following, where the operator· M(n) is controlled by M(n-l). COVERING LEMMA FOR RECTANGLES OF THE STONG MAXIMAL OPERATOR [3]. Let IRkl be a given sequence of rectangles in',,

Rn ~ B(O,l) whose sides are parallel to the axes. Then there is a subse-

quence IRk} of IRk! so that

MULTIPARAMETER FOURIER ANALYSIS

61

1

(2)

!!exp(~ ~

n-1

)(o;.., ) Kk

II L 1 (B) ~

C •

Before we prove this theorem, let us show that it implies the JessenMarcinkiewicz-Zygmund result. Let a"> 0, and for each point x liM(n)f(x) > al there is a rectangle Rx containing x with

(2.2)

Witho~t

loss of generality we assume URx == URk where Rk are certain

if the Rx 's. Apply the covering lemma to get Rk with properties (1) and (2) above. Then by virtue of (1) we need only show that

By (2.2), IRk!

~ ~ f'Fl !£! and summing we have k

luRk

I~~

f

1£1 I, X~k ~ k II£11L( 1og L)n-111 I, XRJexp(L 1/(n-1)) •

Qo

Now, let us prove the covering theorem. We shall proceed by induction on n. Assume the case n-1 . Let R 1 ,R 2 , .. •, Rk, .. • be ordered such that the xn side length decreases. For a rectangle R, let Rd denote the rectangle whose center and xi side lengths, i

< n, are the same as

those of R , but whose x0 side length is multiplied by S. Then we describe the procedure for selecting the Rk from the Rk: Let ~ 1 == R 1 • Suppose Rl'R 2 ,· .. ,Rk havealreadybeenchosen. Wecontinuealong the list, and each time we consider the rectangle R we ask whether or not

62

ROBERT FEFFERMAN

where the above union is taken over all the ~j, j ~ k for which Rj n R

'* 0

If the answer is no, we move on to consider the next rectangle on the list. If the answer is yes, we make the rectangle R = Rk+l, and start the

process over again. Now, we

~UOVe

(1) as follows: If R is an unselected rectangle, then

Let us slice all rectangles with a hyperplane perpendicular to the xn ax is. Then if slices ~re indica ted by using S 's instead of R 's ,

so that M(n-l )(X

U
)

>!. 2

on UR1., where M(n- 1 ) is acting in the

xl'x 2 ,···,xn_ 1 coordinates. Bytheboundednessof M(n- 1 ) on,say, L2 (by induction) we have

To obtain (2), notice that the Rj's satisfy

If we again slice with a hyperplane perpendkular to the xn axis,

iS'k n

U ~jl <} fSkl j
63

MULTIPARAMETER FOURIER ANALYSIS

so that, if Ek = 'S'k-

cp

j~k 'S'j

we have !E'ki > ~ iS'ki and if

L(log L)n- 1(S 0) we shall show that

l

(2.3)

which will give

I r(~ xs-)

1 /(n-1)]

exp Lc

dx 1 .•• dxn_ 1 .:;

c.

so Integrating this estimate in xn finishes things. To obtain (2.3) we write

II s0

xil'.dx,···dxn-1

~ ~ L"' ~ c ~ Jil:ki l~kJ"' ~ ::;k


I

M(n- 1 >(cp)dx 1dx 2 ···dxn_ 1 < C'lle/>11 -

L(log L)n

so

Notice that in our argument above the slicing was the most important idea. If you try the proof without it, you will not wind up with the estimate you want, on the exp( ) 1 /(n- 1 >norm, but rather on the exp( ) 1 /n norm instead. Also the slicing is the mechanism by which we control M(n) by the lower parameter operator M(n- 1 ) and here this enables us to proceed by induction. Of course, in the end the theory of the boundedness of M(n) had been known for some 40 years before the covering lemma. But the lowering of the number of parameters, and the induction procedure will be used in what follows as the key ingredient to prove new theorems.

-1 ·

64

ROBERT FEFFERMAN

To illustrate the method of the machine, we consider a maximal operator whose relation to multiplier theory will be studied below. Suppose in R 2 we consider the class 2! of all rectangles of arbitrary side lengths which are oriented in one of the directions ek

k

=

=

2-k,

1 ,2, ·· ·, measured from some fixed direction, say the positive

x-direction. Define the maximal operator sup -11

mf(x) =

x€R€0

R\

m by

I

\f(t)\ dt .

R

This operator was considered following the ideas of the covering approach outlined above by Stromberg [4] and Cordoba-Fefferman [5]. Somewhat later Nagel, Stein, and Wainger [6] used Fourier transform methods to extend the result we shall discuss below. What we prove is that

\lmf(x)>all S ~ llfll 2 2 a2

2 ·

L (R )

The proof consists of showing that, given a sequence of rectangles {Rkl belonging to W, there exists a subfamily IRk! such that

and (2)

II~ k

Xu';\<

I

2

2

t
'S IURkl 112

. '"'-'

To prove this we give a rule for selecting Rk, given that we have already selected

'"'-' Rj

for j < k. Assume that the

Rk

all have their

longest side in a direction in the 1st quadrant, and are ordered so tha~ their longer side lengths are decreasing. Then consider the rectangle' R following Rk-l. Consider in particular

_!_I IRUR-1 IRI j
l

=1-

IRI

J I X~ R j
dx. j

MULTIPARAMETER FOURIER ANALYSIS

If this is less than l/2 , select R as

Rk.

65

If not go to the next

rectangle on the list, and apply the same test to it. In this way we obtain 'V

the desired Rk. Notice that

This is (2). To show (1), we let R be some rectangle which was unselected. This implies that

Now draw the following picture. Let S be the envelope rectangle to R whose sides are parallel to the coordinate axes.

FigUTe 1

66

ROBERT FEFFERMAN

Then by dilating S if necessary, we can assume that

'"" Rj

is centered at

the same center as S. Now the point is that

IRjns1 lSI -

IRjnRI IRI

-->c--.

(2.5) In fact,

and

and our inequality (2.5), taking into account that £'"" L, is

-1 > -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

MULTIPARAMETER FOURIER ANALYSIS

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>
Not long ago, using the methods we have just discussed, Cordoba was able to prove this [7]. Let us give the proof. Suppose IRk I is a sequence of rectangles with side lengths s,t, and cp(s,t) in R 3 • We must show there exists a subcollection {Rkl such that (1) lURk! > cjURkl

(2)

II~ XR:

llexp(L) k

~C

·

69

MUL TIPARAMETER FOURIER ANALYSIS

To prove this, order the Rk so that the z side lengths are decreasing. With no loss of generality, we may assume that IRkn

[i~k Rjll < ~ IRkl,

that there are finitely many Rk and that the Rk are dyadic. (In fact, we may assume this because if _l_ f. lfl >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~<)} ve} +·,[exp(Ix~<)} ve}

71

MULTIPARAMETER FOURIER ANALYSIS

so

So far what has been done suggests the following general conjecture of Zygmund which says: Let (¢i(tl't 2 ,···,tk)l = 4ll, i = 1,2,···,n be functions which are increasing in each of the variables ti > 0 separately. Define a k-parameter family of rectangles Rt t ... t 1 2'

' k

by

and a maximal operator on Rn by

Then

llxllxl < l,~(f)(x) >all 'S~ llfll L(log L)k- 1

•

Quite recently a beautifully simple counterexample to the general conjecture was given by Fernando Soria of the University of Chicago [8]. Soria's counterexample was: in R 3 , consider all rectangles S of the form s x t¢ 1 (s) x t¢ 2 (s) where ¢i are increasing on [0,1]. In fact we claim that given any small rectangle R with side length x,y and z such that x < y < z 'S 1 in R 3 , there exists a rectangle S f S such that R C S and IS I S C IR 1. Clearly, then

Ms cannot be any better than the

3-parameter operator M( 3 ). To do this let ¢ 1 and ¢ 2 be increasing, ~ 0, and continuous on [0 ,1] so that on each interva 1 2-(k+l) < s < 2-k,

¢

(s)

¢:(s) takes every value between 1 and C·2(k+ 1 )

Then given three

side lengths p 1 , p2 , p 3 'S 1 of R , simply choose s according to the

72

ROBERT FEFFERMAN

following: say Tk-l < p 1 s;2-k. Choose s ( (rk-l,Tk) satisfying cp2(s)

1/) 1 (s)

P3

=P2

k 1

P

(since 1 < 2 < - P2 -

< 2 + we can do this). Then choose

P1

so that tcp 1 (s) = p 2 • This guarantees tcp 2(s)

=

t

p 3 , and we are finished.

cp2(s) k We can make cp 1(s) assume every value between 1 and C2 on [2-k- 1 ,rk] bt letting cp 1(s) = e- 1 /s and cp 2(s) = cp 1(s) on

[~

· rk- 1, 2 -k] and

cp 2(s)~cp 1 (~·2-k- 1 ) 3. Multiparametet

wei~ht-notm

forall

s([2-k-1.~.2-k].

inequalities and applications to multipliers

In this lecture we want to describe further applications of the ideas centering around the covering lemma for rectangles previously described. We shall begin with more about maximal operators, and then move on to multiparameter multiplier operators, and the connection they have with our maximal functions. The first topic we take up is that of classical weight norm inequalities, which have proven of enormous importance throughout Fourier analysis. Here, we want to know which locally integrable non-negative weight functions w(x) on R 0 have the property that some operator T is bounded on LPw(x)dx. The most basic examples are the Hardy-Littlewood maximal operator, and Calderon-Zygmund singular integrals Tf = f * K. The theory was developed in R 1 by Muckenhoupt [9] and Hunt, Muckenhoupt and Wheeden [1 0], and in Rn by Coif man and C. Fefferman [11 ]. We will present only a small segment of that theory now and list some relevant facts for which the interested reader should see the Studia article of Coifman-C. Fefferman [11]. It is no coincidence that the class of weights w for which the HardyLittlewood maxima 1 operator is bounded on LP(w) is exactly the same as the class ri. w for which all Calderon-Zygmund operators are bounded on LP(w). This is the so-called

#

class of Muckenhoupt.

73

MULTlPARAMETER FOURIER ANALYSlS

A nonnegative locally integrable function w(x) on Rn is said to belong to AP if and only if for each cube Q ~ Rn

_l_ ( IQI

J dx~ (...!_ Jw- 1 /(p-1 )dx~pw

1

IQI

Q


-

Q

The smallest such C is called the AP norm. We say that wE A00 if and only if, whenever Q is a cube and E C Q, if IEI/IQ I > 1/2 then w(E)/w(Q) > 'Y1 for some

'Y1

> 0.

Let us list some properties of AP classes: (a) If p > 0 and w

E

AP then pw

E

AP with the same norm as w.

({3) H w E AP and 8 > 0 then w(8x) If w

(y)

E AP

then w- 1 /(p-1)

(8) If w ( AP then w

E

A

00

enough to show that if IQI w(E)

•

=

E AP'

E AP

where

with the same norm as w .

!. ~ ..!,= 1. p

p

In fact, if w E _AP by (a) and ({l) it is

1, and

JQ w = 1

then lEI > 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~


74

ROBERT FEFFERMAN

So far all the properties of AP weights listed are obvious and follow straight from the definitions. There are some deeper properties which though not difficult to prove are not immediate.: (e)

If w lAP then w satisfies a Reverse Holder Inequality:

(

_1

- IQ I

~ 1/(1~) - 8 _!_Jw IQI (

Jw 1 ~ Q

close to 1 as

Q

o > 0.

for all cubes Q with

)


The constant C5 may be taken arbitrarily

o > 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


~ C(Mf)a(x) (for w fA 1

implies w f AP for all p > 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

MULTIPARAMETER FOURIER ANALYSIS

(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


w-1/(p-1))

-

Q

Conversely, assume w £ AP. Calderon-Zygmund decompose f heights ck where k

£

£

LP(w) at

z, and c is large (to be described later) and

get Calderon-Zygmund cubes 1Qf1j: 1 so that

1

IQ~I J

I

f "-' ck and {Mf > yekl

Q~

c u(J'~ -

1

J

( y is a large constant dependent only on n ). Then

Now w £ AP ~ w

£

A00 therefore w({J'~) < CMw(Q~) and so the above J

-

J

expression is

~ CM'~ w(Q~) k J.

o(Q~)) (-+

(~ IQ. I o(Q. >

•

J

J

I

fo- 1 odx)p

~by the AP condition on

k

Q.

J

So far we have used only arithmetic. Now we come to the main point.

If

E~ = Q~- U Q~ then choosing C large enough insures that J

J

f>k

J

IE~I > !. 2 IQ~I , and since o £ AP' J J and so

~o £A

00

we have o(E~) > 71o(Q~) J J

w)

77

MULTIPARAMETER FOURIER ANALYSIS

where MJf)(x) =sup

a~) (Q

lfJda. Now the same proof that works to

xfQ

show that the Hardy-Litt lewood maximal operator is bounded on LP(Rn) proves that if a satisfies a doubling condition, then for p > 1 , Ma is bounded on LP(da) . It follows that the second inequality is

--- -- -~~--.

Note that the operator MILf(x) =

=~ IL(~)

(Q lfidiL and its boundedness

on LP(diL) enter in a crucial way the proof of the weight norm inequalities for the Hardy-Littlewood operator. This operator MIL is interesting in its own right, since it is natural to ask what happens if we replace the Godgiven Lebesgue measure by another measure diL. In fact, if IL is any measure finite on compact sets and if, in the definition of MIL we insist that the balls be centered at x then MIL is bounded on LP(IL), p > 1. The proof of this remarkable theorem relies on a refinement of the usual Vitali covering lemma due to Besocovitch. We should also remark that the original proof of the weight norm inequalities for M on Rn made use of MIL as well. In fact, if w hard to see that

f

AP it is not

78

ROBERT FEFFERMAN

(Indeed,

..!... IQI

J

f dx = _!_ Jfw 11Pw-11Pdx

IQI

Q

=

~ IQI

<..!... -IQI

JfPwdx)

Q

1 /p

~w(Q) 1

~ 1 /p,

1 /p ( (

Jw-p'/pdx

Q

1 /p

-

ffPw)

~_!_ IQI

Q

)(p-1)/p Jw- 1 /(p-l)dx < CM (fP) 11P(x) .)

-

Q

w

Q

The proof would be complete if Mw(fP) 11P were bounded on LP(w). Unfortunately, ~(fP) 1 /p is bounded on L q(w) only for q > p. But if we notice that w £ Ap

:> w £Ap-e then the proof is complete.

Anyway, it is clear that the operator Mil arises naturally. Next we shall study its n-parameter variant M~n) just the strong maximal operator, only taken relative to the measure ll :

M~n)(f)(x) =sup

(k)

x£R ll

I

lfl

d~t .

R

R a rectangle with sides parallel to the axes. Unlike the case where n = 1, restrictions must be placed on ll whether or not R is centered at x. The following result gives conditions on ll which are rather unrestrictive, and which guarantee the boundedness on LP(I£) on M~n) [13]: THEOREM.

Suppose ll is absolutely continuous and non-ne~ative on Rn,

and that the Radon-Nikodym derivative of ll, w(x) satisfies an A00 condition in each variable separately, uniformly in the other variables. Then M~n) is bounded on LP(dl£) for all p > 1 . We require the following lemma.

MULTIPARAMETER FOURIER ANALYSIS

LEMMA.

79

If w(xl'···,xn) is uniformly in ADO in each of the variables

separately, then w satisfies the following: If R is any rectangle with its sides parallel to the axes and E C R is such that ~ > -1 2 , then IRI

rE w

-

> lJ• for

JR W

some lJ

> 0.

Proof. The proof is by induction on n. Assume this for n -1 . Consider a rectangle R as above, R "'IxJ where I is n-1 dimensional and J one-dimensional, and a subset E of R such that \EI 1 IRI > 2. For each x £I let Jx = l(x,y)\y €] I be a vertical segment. Since

~

> -12 it is easy to see that for x in a set I' of measure ? 1010 III we IRI .. 1 ' have IE nJxl > 100 IJxl· Now. since w is uniformly ADO in the xn

variable, (3.1)

But also if we fix any xn £ J then

(3.2)

by induction. It follows by integrating (3.1) in x' £I' that

80

ROBERT FEFFERMAN

and integrating (3.2) in xn

l

J gives,

J

J

w dx 2: r(

I'xj

which shows that

w dx ,

Ixj

J, w > 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

j
so since w fA"" in x 1,·",xn_ 1 we have w(S'k- U S'1.) > 77W(Sk) (if j
w(E) =

JE

w(x 1 ,···,xn-l'xn)dx 1 , .. ·,dxn-l) and we test

~C

J M~- 1 >(cp)wdx. ~s

k

(~k = Sk- U S'1.)·. By induction M~- 1 ) is bounded on LP(w) so this j
last integral is

This shows that

Ill X"N II ,

o:>k 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:

MULTIPARAMETER FOURIER ANALYSIS

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 < p < oo, just because of the boundedness of the Hilbert transform on the LP spaces in R 1 • The case that remains is the one we consider here. Let

01 > 02 > 03 ... >on> on-1 _, 0 be a given sequence of angles 0 and let

85

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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
below, since JEkl2'

and let fk =

x~ .

Then looking at the picture

J:!.k

~IRk! on at least a set

Figure 3

of measure>

l~O i'tki the

88

ROBERT FEFFERMAN

segments pictured contain at least 1 /100 of their measure in Ek. If we duplicate the rectangle Rk as shown, on these segments Tkfk > 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 <2k+ 1 1. Then if Sk alsostandsforthe multiplier operator corresponding to Sk, we see that

But in R 1 we have the classical weight norm inequality for the Hilbert transform:

It follows that (3.3) is

(3.4)

Me

is bounded on L(q 12 )'/( 1 +E) if e is sufficiently small. It follows

that (3.4) is

89

MULTIPARAMETER FOURIER ANALYSIS

proving that

4.

To

is bounded on Lq .

HP spaces - one and several parameters In this lecture we wish to discuss another chapter of harmonic

analysis relating to differentiation theory and singular integrals, namely Hardy Space theory. In this lecture, we shall discuss the one-parameter theory, and, in the next, the theory in several parameters. In the beginning when HP spaces were first considered, they were spaces of complex analytic functions in R;

=

lz =X+ iy lx fR 1, y fR 1 , y >01 which satisfies

the si.21e restriction

J

)1/p

+oo

(

IF(x +iy)iPdx

:S: C for all y

f

R~ .

-00

One of the main reasons for introducing these spaces was the connection with the Hilbert transform. If F(z)

=

u(z) + iv(z) is analytic with u and

v real, and if F is sufficiently nice then F will have boundary value u(x) + iv(x) where v(x) is the Hilbert transform of u(x). It turns out, since

increases as t .... 0, we have +oo

IIF II

H

1

d~f sup

t>O

I

IF(x +it)ldx""' llull 1

1 + llvll 1 1 · L (R } L (R }

-oo

So we may view the space H 1 through its boundary values as the space of a 11 real valued functions f as well.

f

L 1 (R 1) whose Hilbert transforms are L 1

90

ROBERT FEFFERMAN

If we want a theory of HP(R 0 ) then, following Stein and Weiss we may consider the functions F(x,t) in R~+l = {(x,t)jxcR 0 , t >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

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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}
From this our claim follows. This is because for Ag(a) = lllgl >all we have

~ J~ ..-

Hu. n

1 Au .(a)da

p -

f

0

j .•- f.~ j t 1

0

<

J

J

00

f3As(u)(f3)

0

L.

,<•>

0

00

\

fli>.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 <

=

u

r(x)

xils(u)>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

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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
So what we must show is that

To show this let us define some notation. In R 1 , if f( x) has Poisson integral u, we let Qt be the operator (or kernel) which takes f to t\lu{x,t), so that

104

ROBERT FEFFERMAN

s2(f)(x) =

ff

lhQt(y)l2

d~~t.

r(x)

Then going back to our present situation where f is given on R 2 , we define

J

00

-Qlf(xl'x 2) =

f(x 1 -y,x 2 )Qt(y)dy

-00

and

Q[

similarly. Then let us define a Hilbert space valued function

by

Now we know that in the one-parameter case

IIS 1fllp ~ cp\!fllp

and a

glance at the proof of this fact reveals that it remains valid for Hilbert space valued functions. Fix x 2 • Then

-00

-00

and integrating this in x 2 ,

(5.1)

If

S 1 (F)(xl'x2)Pdxldx2

~ cp JJIF(xl'x2)1~2(l) dxldx2 .

R2

R2

But fixing x 1 , since jF(x 1 ,x 2 )1 2 ~""'

L (J )

integral of f(x 1 , ·) at x 2 , we have

is the value of the one-parameter

MULTIPARAMETER FOURIER ANALYSIS

105

and so (5.1) is greater than or equal to

On the other hand when the S 1 operator acts on the first variable we have

the two-parameter area integral c:f f. Next, let us prove that IIS(u)il

LP

~ cpllu*ll P [21 ]. The proof is a L

simple iteration of the one-parameter case given previously. To begin with, we recall Merryfield's lemma: Let ¢ [-1, +1] and have

f¢

=

l

C;'(R 1) be supported in

1. Then there exists 1/J

l

C;'(R 1 ) whose support

is also contained in [-1, +1] with ft/1 = 0 and such that if u = P[f],

Introduce the notation t\lu(x,t)

= Q~(x)

, u(x,t)

=

P tf(x), g * ¢t(x)

=

Pt(g)(x) and g*I/Jt(x) ~ Qt(g)(x), Q~, i ~ 1,2, will denote the operator acting in the ith variable. Then we estimate

106

ROBERT FEFFERMAN

If

(5.2)

Fix x 2 , t 2 • Then (5.2) is

Now

+

lJ f

1 2"' 2 dtl (Pt f(x)) ('lt g(x)) dx 2 dx 1 . 1

1

t1

Then

J

ff
xl

So the inequality we seek in the product case is

JJ

£2 (x)g 2 (x)dx 1 dx 2

•

107

MULTIPARAMETER FOURIER ANALYSIS

It is now easy to see that IIS(u) II L P :S CP llu* I LP, P > 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 )
108

ROBERT FEFFERMAN

J

S2 (u)(x)dx when u

~- P(f]

M(Xu*>a)
where R*

=

l(y,t)jjR(y,t)nlu*>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 * <:f>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).

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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

MULTIPARAMETER FOURIER ANALYSIS

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.

MULTIPARAMETER FOURIER ANALYSIS

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
(u J..ry RdR

k

2ICR)

fJr(y,t)¢(y,t)dy

J...

td~ 1 2

k ~(R)

l£(y,t)l2dy

~~

J

12

/2(

u

RdR

k

ry J.,

WCR)

)1/2 l¢
119

MUL TIPARAMETER FOURIER ANALYSIS

I

(6.2)

To show this

Suppose that (y, t)

l

~(R), R

enough, then (y,t)

l

r(x) for all x

dRk. Then if the aperture of l

R. Since

we see that

for (y,t) c

U

~(R),

and this proves (6.1).

RcRk

But then

I

S¢(f)2(x)dx

flk;nk+l and combining this with (6.1) yields

~ (2k+l)2jflkl

r

is large

120

ROBERT FEFFERMAN

As for

u

Rf\\

iJ

l¢(y,t)l 2 dy

~\

•

1 2

~(R)

if we observe that for any R

f

fflk, ~(R) is contained in S(flk), we get

IJak· ¢dxl SC

by (4). This shows that

and completes the proof that (4)

implies (1 ). We shall show next that (2) implies (4). Let g

f

L ""'(R 2). We claim

that if U C R 2 , then

If

lg(y,t)l 2 dy

t~:2 Cllgii~IUI •

s(U) where g(y,t) = g *r/lt t (y). To show this, observe that since 1 2

supp(rp); [-1,1], supp(r/lt t (·-y))~R(y,t). Hence, if (y,t) fS(U), 1' 2

g *r/Jt(y) = (gxu) * rpt(y) and so

An easy application of Plancherel's formula says that this is, in turn,

J 'f'"' d;.., 00

II gxn 11 22

1·/.(1:)12

0

121

MULTIPARAMETER FOURIER ANALYSIS

Since,

f+ 1 if!= 0, -1

(/J(O)

=

0 and (/J

£

C 00

{ (/J(~) = 0(1~1-N) as 1~1

->

oo, for each N

> 0.

so

Joo ~~(~)1 2 ~~ <

00 •

0

It follows that

JJ

lg(y,t)l 2 dy

t~! 2 s cllgll~lnl

s as claimed. If we wish to prove that the same Carleson condition holds for g

replaced by Hx Hx (g). then we proceed as follows. Observe that 1

where IJI

=

2

Hx Hx (t/1). The function IJI splits into a product of 1

2

'l'(l)(x 1)·'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
11( \) 2

and for 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

MULTIPARAMETER FOURIER ANALYSIS

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<2 ><x

w

)>-

1- • Then 1010

Rn S I

0.

MULTIPARAMETER FOURIER ANALYSIS

A* cf>t t (x) 12

=

I,

. 2 3;js1 l
A~1

* cf>t t (x)

I,

t-

12

. 2 1 ;js 2 l
-

~

A~J

f

12

1 2

IR11 iSJ < P,

where 18 C 9lk consists ci rectangles R so that Thus R ~~ for all R

* cf>t t (x)

fR * cf>t t (x)

m

REv

R'ns .f. 0.

125

IR21

IS 2 I < P

and

18, and the reason this subtracted

term occurs is that we have double counted these fR whose R sides are both very small. In order to estimate A~* cf>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

MULTIPARAMETER FOURIER ANALYSIS

(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.

MULTIPARAMETER FOURIER ANALYSIS

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
-12, as the smoothness of the domain

decreases. We will then show the flexibility of our methods by considering extensions of our results to systems of elliptic equations in Lipschitz domains. The specific systems of equations that we will study are the Navier systems which arise in the linear and infinitesimal theory of elasticity, when the displacements or the surface forces are given on the boundary of a homogeneous and isotropic elastic body D. These systems are the prototype of the second order elliptic systems of equations. We will also

134

CARLOS E. KENIG

study the so-ca lied Stokes problem; this is the linearized stationary problem of the mathematical theory of viscous incompressible flow. Before going on to study the general situation, we will formulate appropriate theorems, by examining a model case, namely the Laplacian in the unit ball 8. In this case we have a lot of symmetry at our disposal and everything can be done explicitly. Let do denote surface measure of

as.

THEOREM. Suppose that 1 < p ~

oo

and f

l

LP(aB,do). Then, there

exists a unique harmonic function u in 8 such that lim u(rQ) lor almost every Q (

I

{*)

as , and

I

u*(Q)P do{Q) :::: CP

dB where u*{Q)

~

r->1

f(Q)

\f(Q)\P da(Q) ,

dB

sup \u(rQ)\.

=

o-:;r
=

u(rQ) converges to f(Q) not only in

LP norm, but also in the sense of Lebesgue's domina ted convergence. In the analogous estimates to (*)in the Neumann problem, u is replaced by the gradient ci u. In that case the estimate fails fa p even if

~

= oo,

is continuous.

In both the case of the Dirichlet problem and the Neumann problem, the radial limit can be replaced by a non-tangential limit: if X tends to Q with \X-Q\ < (1+a) dist(X, aB) 1 for some fixed

a> 0 then u(X) 1

has the limit f(Q) for almost every Q. The theorem is most easily proved by writing down a formula for the solution, u{X) =

faa

1 1-\Xi 2 P(X,Q) f(Q) da(Q), where P(X,Q) = wn IX-Q I"

·

The estimate now follows as an easy consequence of the Hardy-Littlewood maximal theorem. An analogous formula holds for the Neumann problem.

ELLIPTIC BOUNDARY VALUE PROBLEMS

135

This time, it is more difficult to obtain the estimates. One needs to use the Calderon-Zygmund theory of singular integrals, and the HardyLittlewood maximal theorem. The case of the Laplacian in the ball is relatively easy because of the existence of explicit formulas for the solution. What should we do in the case of a general domain, where explicit formulas are not available? What should we do to study systems of equations? What happens to our solutions as the domains become less smooth? We hope to give a systematic answer to these questions in the rest of this paper. §1. Historical comments and pceliminaries (a) The method of layer·potentials for Laplace's equation on smooth domains.

DEFINITION. A bounded domain D is called a Lipschitz domain with Lipschitz constant less than or equal to M if for any Q laD there is a ball B with center at Q, a coordinate system (isometric to the usual coordinate system) x'=(x 1 , .. ·,xn_ 1 ), xn, withoriginat Q anda function cf>: Rn- 1 --. R such that c/>(0)=0, lc/>(x')-cf>(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

ELLIPTIC BOUNDARY VALUE PROBLEMS

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) =

<3J>

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(
t

I+ K is invertible modulo

a finite dimensional subspace of C(aD). H D and co are connected, T is actually invertible on C(
139

ELLIPTIC BOUNDARY VALUE PROBLEMS

1 u(X)=wn

I

g(Q)

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

ao where g = T- 1(f). The operator K is compact on C(OD) even in C 1 ,a

,N: >I <- IP-Qin-1-HZ} C \

domains {because in this case I< Q-P \ IP-QI

and so this

procedure solves the classical Dirichlet problem in that case too. If D is a coo domain, K is compact from ck,a(OD) to ck,a(OD), k=l,2,···,0), ucC 00(D). Thisapproach can also be used to obtain results for f c LP(OD) in C 00 domains, and even on C t,a domains. We will now sketch the extension of the theorem for the ball stated in the introduction to C 1 •a domains. We first define non-tangential approach regions as follows: r{3(Q) =!X tD: IX-Q I < (1 +/3)dist (X, aD) I . The non-tangential maximal function, with opening {3, of a function w defined in D is

Because of the estimate

I

I np
domains, it is easy to see that K is compact as a mapping on LP(OD). Also, standard arguments show that

where M is the Hardy-Littlewood maximal operator on aD , and w is the double layer potential of f. Finally, from LP bounds for M, K and T- 1

=(~

I+K}-

1,

one obtains:

140

CARLOS E. KENIG

THEOREM.

and u(x)

=

Let D be a C l,a domain, 1 < p < "". If f f LP(aD, da), <X-Q,NQ> w1n f.:.v 0 IX-Q In T- 1f(Q)da(Q), then IINaull ~ t-' LP(da)

Cpllfll P

L (da)

and the harmonic function u tends to f non-tangentialiy.

The difficulty in the case of C 1 and Lipschitz domains, is that, in this case the size-estimate on the kernel of K is

-P,Np >I c I
fan

must have

~u = 0 in

D,

~ian =f. By Green's formula, we

fda= 0. When D and co are connected this is the only

compatibility condition needed. We will only consider that case for simplicity. A good first guess at the solution u is the so-called single layer potential of f given by v(x) = en f.:.

f(Q)

vD

~ (Q)

=

=

1 da(Q). Once again v is harmonic in D IX-Qin-2

I

and

~ f(Q)-K*f(Q), where K* is the adjoint of K above, i.e.

Q

*

fao f(Q)F(X,Q)da(Q)

1 K f(P) = -

f.:~ UJn vD

< P- Q,N P > IP-Q

ck,a(ao) to ck,a(.D)

In

*

f(Q)da(Q). Thus, K

is compact from

and Fredholm's theory says that

I

T' = ~ 1-K*

is

invertible on the subspace of ck,a (aD) of functions of mean value 0. Therefore the solution to the Neumann problem can be written

u(X)

=

J

(T- 1 f)(Q)F(X,Q)da(Q),

dD and if f

£

C""(aD), u

f

C""(5). If D is C 1 ,a, T is also invertible on

ELLIPTIC BOUNDARY VALUE PROBLEMS

141

the subspace of LP(aD) of functions with mean value, 1 < p < oo, Hence:

Let D be a C 1 •a domaih, ( D and

THEOREM.

1
fa 0

co

connected). Let

fda=O. Then,

u(X) = fao (T- 1f)(Q)F(X,Q)da(Q) is harmonic in D, V'u(X) has nontan~entiallimits

\IN,iV'u)ll P L

fJ

(da)

\lu(Q) for a.e. QtOO, f(Q)=, 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])

ELLIPTIC BOUNDARY VALUE PROBLEMS

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-<:alled Neumann function for D , by using the De Giorgi-Nash regularity theory for elliptic equations with bounded measurable coefficients. This, combined with the use of the socalled 'atoms' yields the desired results. These results will be explained in the second part of Section 2. Also in 1984, B. Dahlberg, C. Kenig and G. Verchota ([8)) and E. Fabes, C. Kenig and G. Verchota ([13]) were able to extend the ideas of Verchota to be able to obtain results for L 2 boundary value problems for some systems of equations on Lipschitz domains. The systems treated are those that arise in linear elastostatics and in linear hydro-

ELLIPTIC BOUNDARY VALUE PROBLEMS

145

statics. The results obtained had not been previously available for general Lipschitz domains, although a lost of work had been devoted to the case of piecewise linear domains. (See [24], [25] and their bibliographies.) For the case of

c1

domains, these results for the systems

of elastostatics had been previously obtained by A. Gutierrez ([15] ), using compactness and the Fredholm theory. This is of course, not available for the case of Lipschitz domains. The authors use once more the method of layer potentials. lnvertibility is shown again by means ri. Rellich type formulas. This works very well in the Dirichlet problem for the Stokes system (see part (b) of Section 3), but serious difficulties occur for the systems of elastostatics (see part (a) of Section 3). These difficulties are overcome by proving a Korn type inequality at the

boundary. The proof of this inequality proceeds in three steps. One first establishes it for the case of small Lipschitz constant. One then proves an analogous inequality for non-tangential maximal functions on any Lipschitz domain, by using the ideas of G. David ([10]), on increasing the Lipschitz constant. Finally, one can remove the non-tangential maximal function, using the results on the Dirichlet problem for the Stokes system, which are established in part (b) of Section 3. As a final comment, I would like to point out that even though throughout this paper we have emphasized non-tangential maximal function esti-

mates, also optimal Sobolev space estimates hold. All the Sobolev estimates can be proved in a unified fashion, using square functions and a variant of some of the real variable arguments used in part (b) of Section 3. The details will appear in a forthcoming paper of B. Dahlberg and C. Kenig, [7].

§2. Laplace's equation on Lipschitz domains (a) The L 2 theory A bounded Lipschitz domain D C Rn is one which is locally given by the domain above the graph of a Lipschitz function. Such domains satisfy

146

CARLOS E. KENIG

both the interior and exterior cone condition. For such a domain D, the

fJ

non-tangential region of opening

at a point Q

f

ao

is rfJ{Q) =

IXfD: \X-Q\ <(l+fJ)dist(X,aD)I. All the results in this paper are valid, when suitably interpreted for all bounded Lipschitz domains in Rn, n :?: 2, with the non-tangential approach regions defined above. For simplicity, in this exposition we will restrict ourselves to the case n:?: 3 (and sometimes e~en to the case n = 3 ), and to domains DC Rn, D

=

l(x,y): y >cf>{x)}, where cf>: Rn-l

->

R is

a Lipschitz function with

Lipschitz constant M, i.e. \c/>(x)-c/>(x')\ ~ M\x-x'\. o- = l(x,y):y{x)l. Forfixed M'<M, re(x)=l(z,y):(y-c/>(x))<-M'\z-x\ICD- and ri(x)= l(z,y): (y-cf>(x)) > M'\z-x\1 CD. Points in D will usually be denoted by X, while points on rJD by Q = (x, cf>{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

ELLIPTIC BOUNDARY VALUE PROBLEMS

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


I \x-z\>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

< p
(z-x,~{z)-¢{x))J1 + IV~(x)\ 2 g(X )d X

\z-x l>e

[\z-x 12 + [~(z) _ ~{x)]2]n/2

LP(
=

exists a.e. and in

LP(dD,da)

, 1 < p
±} g{Q) + Kg{Q)

I

)±; )=+!. ( )N z + 1... 1'tm ("S v g \Z - 2 g Z
(z-x,~(z)-~{x))'l/(1 +IV~(x)\ g(X )d X [\z-x \2 + [~(z) _ ~(x)]2]n/2

lz-x\>e COROLLARY

2.1.4. {Nz\]Sg)±{z)

L 2 (il>, 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

ELLIPTIC BOUNDARY VALUE PROBLEMS

\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

ELLIPTIC BOUNDARY VALUE PROBLEMS

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 ( )


argument used in the proof of 2.1.5 now proves 2.1.13. THEOREM 2.1.14. Given f with IIN(V'u)li 2

L (A)

f

:S C IIV'tfll

L~(A), there exists a harmonic function u,

, 2 L (A)

and such that V'tu

= V'tf

(a.e.) non-

tangentially on A. u is unique (modulo constants), and we can chose u

=

S~), where g

t

L 2 (A).

153

ELLIPTIC BOUNDARY VALUE PROBLEMS

The existence part of 2.1.14 follows directly from 2.1.13. (b) The LP theory

We will start out our treatment of the LP theory by discussing some counterexamples. Let z = x +iy

€

C, and for 0 < {3 < 2TT, let

Df3 = {z €C: [arg zl < {3/21. We will consider the holomorphic function f(z)

=

z 171 f3, which maps 0{3

conformally onto OTT, the right-hand plane. We will also consider a bounded domain 0{3 C Df3, with the property that d0{3 \O is smooth, and such that d0{3

n {[z[ < 11 = dD{3 n {jz[ < 11.

Let u(x,y) = Ref(z), and

u(x,y)[ 0 · v is harmonic in 0{3, and v is identically zero near '{3 the corner of d0{3, and is smooth everywhere else in 0{3. Let s be v(x,y)

=

the arc length parametrization of artf3, starting at 0. Then, it is clear that

~ € L (d0{3). Let w(x,y) 00

equations,

=

Im f(z)[ 0 . By the Cauchy-Riemann '{3

However, N('ilv)(s) = N('ilw)(s) ~ s-l+TT/{3. This function belongs to LP(ds) if and only if p TT/ {3- p > -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<-;.

~ +~

=

1 . Choose (3 so that

Let B = {lzi
Let ill = ill * be harmonic measure evaluated at X*, and k = ~. We first claim that k {}(3 with pole at

I LP(ds). In fact, let ~*' Then, for s near

G(X) be the Green's function of 0, k(s) =

~ {s)

lim G(s+eN)-G(s) = lim G(s+eN) > C lim v(s+eN~ = C e->0

e

e->0

e

-

e--+0

e

av

aN

=

{s) ~ 8 -1+TT/(3,

where the first inequality follows from the fact that both G and v are positive, and harmonic on B, and 0 on an(3 n B (this is Lemma 5.10 in [19]). Assume now that ~I+ K were invertible on Lq(ds). Let g 2: 0

l

C(a1l(3), and h(X) be the solution of the Dirichlet problem with

data g. Then, h(X*) =

J J gdill

a~ also, by the L 2 -theory, h(X*) =

gkds ,

=

anf3

K[{~ l+~- 1 (g~ (X*)'

where

K

is

the double layer potential. Let U be a ball centered at X*, contained in {}(3. By the mean value property of harmonic functions and Harnack's principle, we have

h(X.J

~ 1~1

£~ £ h

c(

~1/q

N(h)•d/

t'

~ c~ •••;

x/q

,

because of the second formula for h(X*), and the assumed Lq boundedness of (~ I+~ - 1 . But this implies that k l LP(ds), a contradiction. We now turn to the positive results. They are:

155

ELLIPTIC BOUNDARY VALUE PROBLEMS

2.2.1. There exists e = e(M) > 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 < p ::; 2 + e, ( ±

LP(aD, da), 2-e < p < -

oo,

~ I+ K)

is invertible in

and S: LP(CD, da) ... LP(aD, da) is invertible 1

l
=

Sf, f nice, then, for 1 < p ~ 2 +e, 11\?tu II P ~ L

(oD,do)

~

. This will be done by proving the following two theorems: l ~rll u1, LP(()o ,da) THEOREM 2.2.5. Let ~u = 0 in D. Then \\N('i]u)\\ P

C ~~~rll

un LP(iJD,da)

'

1 < p <; 2+E.

THEOREM 2.2.6. Let ~u = 0 in D. Then IIN(\?u)\1 C Ntull

LP(iJD ,do)

L

P L

S (iJD,do)

(iJD,do)

:S

, 1 < p ::; 2 +e.

We first turn our attention to the case 1 < p < 2 of Theorem 2.2.5. In order to do so, we introduce some definitions. A surface ball B in A isasetoftheform (x,cp(x)), where x belongstoaballin Rn-l. DEFINITION 2.2. 7. An atom a on A is a function supported in a surface ball B, with l\ai\L""<;1/a(B), and with fA ada=O. Notice that atoms are in particular L 2 functions. The following interpolation theorem will be of importance to us.

157

ELLIPTIC BOUNDARY VALUE PROBLEMS

THEOREM

2.2.8. Let T be a linear operator such that IITfll 2 A < L ( ) -

C llfl\ 2

,

L (A)

and such that for all atoms a, UTa II 1

L (A)

:S C. Then, for

1
(

)

L (

)

For a proof of this theorem, see [3]. Thus, in order to establish the case 1 < p < 2 of 2.2.5, it suffices to show that if a

=

~ is a atom, then 1\N(Vu)ll Ol~

L

1

translation invariance we can assume that c/>(0)


=

0, supp a C B 1 =

((x, cf>(x)): lxl < 11. Let B* be a large ball centered at (0,0) in Rn, which contains (x, cf>(x)), lxl < 2. The diameter of on M. Since lla II 2

:S

L (A)

faons*

1 1 a(B 1)1 2

s*

depends only

= C, by the L 2 .Neumann theory,

N(Vu) :S C
f.

estimate

c8.

na

o

N(Vu)da). We will do so by appealing to the regu-

larity theory for divergence form elliptic equations. Consider the biLipschitzian mapping : D

-->

o-

given by cll(x,y) = (x, c/>(x)- [y---cf>(x)]).

Define u* on D- by the formula u* = uo~- 1 , u* verifies (in the weak sense) the equation div(A(x,y)Vu*) = 0, where A(x,y) = J
B(x,y)

=

Because

e> 2' Cl~\ 2 •

~

~

=

L 00(0-), and

Notice also that supp ~~ C 8 1 < B* n

for (x,y)

~A(x,y)

£

f

u(x,y)

D

aD.

for (x,y)

Define now

£

D

, and U'(x,y) ={ for (x,y)

0 in

f

ao \B*

D-

I

u*(x,y) for (x,y) £ D-

it is very easy to see that u is a (weak)

solution in Rn\B* of the divergence form elliptic equation with bounded measurable coefficients, Lti' (and hence Vu) at H. Weinberger ([31 ]).

oo,

~

div B(x,y) V'U

=

0. In order to estimate u 1

we use the following theorem of

J.

Serrin and

158

CARLOS E. KENIG

u

THEOREM 2.2.9. Let 1\ull DO L

(R

n\B*)
solve Lu = 0 in Rn\s*, and suppose that

Let g(X) solve Lg = 0 in lXI > 1, with

g(X)~JXI 2 -n. Then, u(X)=uDO+ag(X)+v(X), where Lv=O in Rn\B*, and Jv(X)J ~ C !lull

L

00

\

•

(Rn B*)

IXI 2-n--v, where v > 0, C > 0 depend

only on the ellipticity constants of L. Moreover, a = C JB(X) \lu(X) · 'Vt/J(X), where

t/1 c C""(Rn),- t/1 =0 for IX I in 2B*, and t/1

Let us assume for the time being that that if B(X)

=

a is

u

,,1

for lar~e X.

is bounded and let us show

a = 0. Pick a t/J as in 2.2.9. In D, I, and so J, B\lu\lt/1 = J, \lu· W = lim f_ e \lu· \lt/1, where as

in 2.2.9, then D

e ... o

D

D

p

D~ =l(x,y):l(x,y)Jc/>(x)+d, and pis large. The right-hand

f

sideequals lim e-->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
1 < C cL 2(A), andso u cLDO(D 1 ). IX-Q\n-2 -1 + IQin-2

Let now B be any ball in Rn so that 28 C Rn\B*,

B is of unit size,

and such that a fixed fraction of B is contained in D 1 . Since N(\lu) c L 2(A), with norm less than C, f2ano l\7ul 2 ~ C, and moreover on sno 1 , lu(X)J~C. Therefore,bythePoincareinequality

ELLIPTIC BOUNDARY VALUE PROBLEMS J2B \ul 2 S C. But, since lr solves L\1 = 0,

m~x lui'S C
llu\1 L ""(ftn\ B*)-
([26]). Therefore, \1 c L""(Rn\B*),

159

Hence, since

a=O, 'i]u;,\lV,and \v(x,y)!SC/(Ixi+IY/)n- 2w, v>O. For R?:Ro= diamB*, set b(R)= f N('Vu) 2 , where, AR=I(x,¢(x)):R
or= l(x,y); ¢(x) < y < ¢(x) + CR, rR < lx I< L 2 -Neumann theory in fi 7 ,

f.

AR in r from 1/4 to 1/2 gives

I

N1('Vu)2da 'S

~

r- 1RI,

N 1('i]u) 2 daS C

I

IVul2dX

fY\

c(~,

H.

By the

\Vu\ 2 da. Integrating

OUT

J

'S RC3

0 114 \fi 112

AR

T

"-2 u ,

C 1R
since Llr = 0 (see [26] for example). The right-hand side is bounded by 1 J;;. Rn = CR 1-n- 2v, and hence b(R) < CR 1-n- 2v. Then, R3 R2(n-2)-2v n-1

f

AR

N('i]u) < C( f AR

N('i]u) 2) 112

R2

< CR-v. Choosing now R = 2j , -

and adding in j , we obtain the desired estimate. We now turn to the case 1 < p < 2 of 2.2.6. We need a further definition. 1

.

....

DEFINITION 2.2 .1 0. A function a is an H 1 atom 1f A= 'Vta satisfies

....

(a) supp A C B, a surface ball, (b) 1/A/1 < 1/a(B), (c) L DO We will use the following interpolation result:

{Ada=

0.

160

CARLOS E. KENIG

THEOREM 2.2.11. Let T be a linear operator such that \ITf\1

Cil£11 L 2 (!\) :\

and IITall 1

< C for all

L (t\)-

1

1 < p < 2, IITf\1

< c llfll

LP(A)-

Li(A)

<

2

L (A)-

1

H 1 atoms a. Then, for

.

Hence, all we need to show is that if 11u = 0, 'iltu = 'ilta, and a is 1

•

a unit size H 1 atom, N('Vu)

1

l

L (A). But note that if we let u(x,y)

(x,y) £ D

u(x,y) = {

-u*(x,y) (x,y) ( othen

u

is a weak solution of Llr = 0 in Rn\a*, since ulao \a• = 0.

Then, \r = lr 00 + ag + v, but a

=

0 since lr -lr 00 must change sign at

oo.

The argument is then identical to the one given before. Before we pass to the case 2 < p < 2 + e, we would like to point out that using the techniques described above, one can develop the SteinWeiss [32] Hardy space theory on an arbitrary Lipschitz domain in Rn. This generalizes the results for n results for

c1

=

2 obtained in [20] and [21], and the

domains in [121

Some of the results one can obtain are the following: Let H~t(aD) II •\ai: II\ I< oo, ai is an atom!, Hi_at(aD)

= II\ai: II\ I<

=

+oo,

ai is an H~ atom I. THEOREM 2.2.12. a) Given f

£

H!t
function u, which tends to 0 at oo, such that N('Vu) such that NQ'Vu(X) .... f( Q)

non·tan~entially

£

L 1 (aD), and

a.e. Moreover, u(X) =

S(g)(X), g£H!t· Also, u\a 0 lHtat(aD). b) Givenf£HLat• there exists a unique (mcxlulo constants) harmonic function u, such that N('Vu) g

£

£

L 1(aD), and such that 'iltulao = 'iltf a.e. Moreover, u

H!t, and

=

S(g),

~ £ H!t(aD). c) If u is harmonic, and N('iJu) £ L 1 (aD),

161

ELLIPTIC BOUNDARY VALUE PROBLEMS

then

~ f H~t(aD),

u !00

H} ,at(aD). d) f c H!t
f

(~

N(\7Sf) c L 1 (aD), if and only if

1- K*) f c H!t
We turn now to the LP theory, 2

< p < 2 +e.

In this case, the results

are obtained as automatic real variable consequences of the fact that the L 2 results hold for all Lipschitz domains. We will now show that I!N(\7u)ll P A

L ( )

~ CII~NII P A Ul'l L ( )

for 2

< p < 2 +e.

The geometry will be clearer if we do it in R~, and then we transfer it to D by the bi-Lipschitzian mapping ci>: R~

->

D , «<>(x,y) -= (x ,y + ¢(x)).

We will systematically ignore the distinction between sets in R~ and their images under ci>. Let y = l(x,y) c R~: lxl

< yl, y*

= l(x, y) c

R~: alxl < yl, where a

is a small constant to be chosen. Let m(x) =

sup !Vu(z,y)\, and £x+y

(z,y)

m*(x) =

sup !Vu(z,y)j. Our aim is to show that there is a small lx+y*

(z,y)

e 0 > 0 such that

au = dN.

f

Let h

f m2 +Edx

~c

= M(f 2) 112 ,

f

\f! 2 +edx, for all 0

< e ~ e0 ,

where

where M denotes the Hardy- Littlewood

maximal operator. Let E..\= lx c Rn 1 : m* (x) >..\I. We claim that

~m*>..\,h 0 !

m2

~ C..\2 \E,.\1

+ Ca

~m*>..\l m 2 •

Let us assume the claim, and

prove the desired estimate. First, note that

by the claim. Choose now and fix a so that C·a

f.

E..\

e

m2 :SC..\2 IE,.\I t CJ.I

f0

00

..\.e- 1

h~..\.1

f.E..\ m2 d..\ <- C

< 1/2.

Then,

m 2 . For e>O, Jm 2 +e=ef. 00 ..\e- 1 0

f ( ' " ..\ 1 +E

0

\lm* >..\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 <1/2, for e<e 0 , fm 2 +e~ C

f m2 he.

If we now use Holder's inequality with exponents 2 ~e and 2

2 ~ e , we see tha!

f

m2H

~

2+e

e

C(J m2 H) 2 +e(JM(f 2) -2-) 2+e, and the

desired inequality follows from the Hardy-Littlewood maximal theorem. It remains to establish the claim. Let IQkl be a Whitney decomposition of the set EA = lm*>AI, such that 3Qk C EA, and I3Qkl has bounded overlap. Fix k , we can assume that there exists x that h(x) ~ A, and hence, f2Qk f2 ~ CA2 1Qk I. For 1 ~ r rQk, and Qk,r cll(Qk ,7)

)

=

l(x,y): x

£

£

:s 2,

Qk such let Qk,r =

rQk, 0 < y < r length (Qk)l. Qk,r (and

is a Lipschitz domain, uniformly in k, r. Also, by construction

of Qk, there exists xk with dist(xk,Qk)::::: length (Qk), and such that m*(xk) ~A. Let

'"" ,T Ak ,T = aQk

so that dQk,r

=

n xk + y*

Qk, r U Ak,r U Bk,r. Note that the height of Bk,r is

dominated by Ca length (Qk), and that IV'ul :SA on Ak,r. Let m1 be the maximal function of \7u, corresponding to the domain Qk ,7 (i.e.

where the cones are truncated at height ::::: f(Qk) ). Then, for x m(x)

~

m1 (x) +A. Also,

Jm~ ~ J m~ ~(using Qk

al)k

. r

the

£

Qk,

ELLIPTIC BOUNDARY VALUE PROBLEMS

163

Integrating in r between 1 and 2 , we see that

Thus,

f.

f.

Qk

m2 ~ Ca

f2

Qk

m2 + C..\2 1Qkl· Adding in k, we see that

< C..\2 IE..\l + Ca f. m2 which is the claim. Note also lm•>..\1 '

lm•>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

ELLIPTIC BOUNDARY VALUE PROBLEMS

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'

1{:' +

(J~

)-(P)

.!:

=

2"1

--+

--+

g(P) + Kg(P)

(~~ (Sg)j) ± = ±{A2C ni(P)gj(P)- ni(P) nj(P) < N(P), g(P)>} + •

where Kg(P)

=

p.v.

~.v.

fao

[

:.,

l'(P-Q)ol(Q)do(Q~j

,

IT(Q) f'(P-Q)It g(Q)da(Q), and A, C are the

constants in the definition of the fundamental solution. Thus, just as in Section 2, part (a) is reduced to proving the invertibility on L 2(00, da) of

± ~ I+ K,

±

~ I+ K*, and the invertibility from

L 2(ao, da) onto L~(aD, da) of S. Just as before, using the jump relations, it suffices to show that if \I(X) = Sg(X), then [ITii'll 2

::::;

L
[I'Vtii'll

•

2

L (aD,da)

Before explaining the difficulties in doing so, it is very

useful to explain the stress operator T (and thus the boundary value problem 3.1.2), from the point of view of the theory of constant coefficient second order elliptic systems. We go back to working on Rn, and use the summation convention. Let a~~, 1 lJ

<:::

r, s

ellipticity condition

<:::

m, 1

:S

i, j

<:::

n be constants satisfying the

166

CARLOS E. KENIG

and the symmetry condition a~~ = a~.r. Consider vector valued functions 1J J1 .; = (u 1 , •.. , urn) on nn satisfying the divergence from system

~ a~J a~.

us

=

0 in D. From variational considerations, the most

J

1

natural boundary conditions are to Dirichlet condition (~I an= f) or the Neumann type condjtions,

aus t.... = ni a~j (jx, = fr.

The interpretation of J problem (2) in this context is that we can find constants a~~, 1 :S i, j ~ 3, 1J 1 :S r, s :S 3, which satisfy the ellipticity condition and the symmetry condition, and such that ~II+ (A+p.)V div

II= 0

axa

a ..

i

rs aus a .. ax = 0 in D 1J j

in D if and only if

.... and with Tu = dv u. In order to obtain the II

equivalence between the tangential derivatives and the stress operator we need an identity of the Rellich type. Such identities are available for general constant coefficient systems (see [29], £27]). LEMMA 3.1.5 (The Rellich, Payne-Weinberger, Neeas identities). Suppose rs ax. us = 0 rn · D ' a rs t ha t ax. aij' .. = a sr .. ' h.... is a constant vector in Rn , 1 J 1J J1 .... and u and its derivatives are suitably small at oo. Then,

a

a

Proof. Apply the divergence theorem to the formula

REMARK 1. Note that if we are dealing with the case m = 1 , aij = I,

....

and we choose h =en, we recover the identity we used before for Laplace's equation.

ELLIPTIC BOUNDARY VALUE PROBLEMS REMARK 2.

167

Note that if we had the stronger ellipticity assumption that

a~~ e.re_s > C I I~JI 2 , we would have, if ~ = l(x, cp(x)): cp: Rn- 1 -• R, lJ 1 J f, t L IIVc/JIIoo

~ Ml,

... 2 ~ that IIVtull L (uu,do)

I

~ ~--, .:~...1 2 . uv L ( 0 D,do)

In fact, if we take

->

h =en, then we would have

~

J

1Vurl 2 do

iJD

~C

J

hfnf

a~j tri '*:do=

iJD

For the opposite inequality, observe that, for each r,s,j fixed, the vector hinfafj - hf.nfa~j is perpendicular to N. Because of Lemma 3.1.5,

Hence,

f00

2 do) 112 (f I -- 2 112 , and so !Vu-- 12 do~ C(fao I\7tu-- 1 00 \7u 1 do)

REMARK 3. In the case in which we are interested, i.e. the case of the

systems of elastostatics,

168

CARLOS E. KENIG

which clearly does not satisfy a~~~{~~?: C I 1~l1 2 , since the 1J J tt quadratic form involves only the symmetric part of the matrix (q). In this case, of

cour~e

:

= T~ = A (div

REMARK 4. The inequality IIV'\i II 2 L

~ N + ~tiV'it +viti IN . (do ,do)

~ CIIV't~ll 2 ~

L (uu ,do}

holds

in the general case, directly from Lemma 3.1.5, by a more complicated

fan

algebraic argument. In fact, as in Remark 2,

rs aur aus hfnfaij ax. ax. do= J

1

r rs rs auf aus 2 Jao (hfnf aij - hi nf afj) ~ · dXj do, and for fixed r, s, j,

(hfnfarr- hinfafJ) is a tangential vector. Thus,

c Uao

... 2

IV't ul do)

1 12

(

fao IV'ul ... 2 do) 112 .

fao hfnfa~j ~~do"::

.

.

Cons1der now the matnx drs =

(a~~ nin1.)- 1 . This is a strictly positive matrix, since a~~ ~i ~1-rl"rf > 1J 1) -

c l"'cl21 ., 12 .

M d (au\ {au\ ore over' rs Cfj;}r \'av}s

rs aur aus

- aij axi

sm aum dXj = drsni aijrt aut dXj . nffk axk

rs iJuf aus rt aut St auT tT aut auT d rt aut a .. ax. ax.= drsnkakfc1X:. nmamv ax -a fd}[ ax = rsnkakvd}[ 1J 1 J v v v v sr aur tr aut aur I rt st tr I aut aur nmamf'dXf- avf~ drsnkakvnmamf- avf I dXV Now, note

e

e

dXf.

axe=

that for t, r, f fixed, ldrsnka~~ma~tf - a~fl is perpendicular to N, by our definition of drs, and the symmetry of a~j: drsnka~ma~~nv- a~r.nv = rt d st tr akvnlfv rsamfnm- 8 mfnm a!enm

-a~t'm = 0.

=

tr d sr tr _ B st tr _ av!fvnk rsamfnm -- amt'm- ts 8 mfnm-amenm-

Therefore,

fao hfnfdrs{~(~~ do~

169

ELLIPTIC BOUNDARY VALUE PROBLEMS

.... We now choose h =en, so that hyny? C, and recall that (drs) and

(a~~nknj) are strictly positive definite matrices. We then see that

Now, as I'Virl 2 = I'Vtlrl 2 +

1:1

2 , the remark follows.

REMARK 5. In order to show that suffices to show that

fao

I'Vtul 2 da

fao I'V~I 2 da~ C fao

~

C

fao \T~\ 2 da,

\A(div U')I + lll'Vll'+VU'tl\ 2 da.

In fact, if this inequality holds, we would clearly have that C

fao I\lu. . + \lu....tl2 da

it

fao

IVU'\ 2 da ~

(Korn type inequality at the boundary). The Rellich-

Payne-Weinberger-Necas identity is, in this case (with h =en ),

fao nnl~ lvll'+'V~tl 2 But then,

ii)2 tda=2 fao ~ ·IA(div ~)N+Ili'Vir+'VirtiNida. .... . . ....t 1 2 1 12 1\7u+\7u fao I\lu-- 12 da~ C
The rest of part (a) is devoted to sketching the proof of the above inequality. THEOREM3.1.6. Let U' solve ~U'+(A+/l)'VdivU'=O in D, II=S(g),

where

g is nice.

Then, there exists a constant C, which depends only

on the Lipschitz constant of ¢

so

that

170

CARLOS E. KENIG

I

I\JII\ 2 do

<_S

C

ao

I

lA. (div II)I + p.l \JII + \JIItl\ 2 do.

ao

The proof of the above theorem proceeds in two steps. They are: LEMMA

3.1. 7. Let II be as' in Theorem 3.1.6. Then,

. J ao

N(\]u) 2 do~ C

LEMMA

f

3.1.8. Let

II

J ..

N(A.(div u )I+ p. I\]u. . + \]u....t 1) 2 do .

ao be as in Theorem 3.1.6. Then,

N(A(div u--> 'I+ p.I \]u.... + \]u->t I) 2 do

<_S

C

I .

\A.(div u )I+ p.I \]u... + \]u->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
~ Clifll

· 2 L (dD,do)

171

ELLIPTIC BOUNDARY VALUE PROBLEMS

We now turn to a sketch of the proof of Lemma 3.1.7. \We will need the following unpublished real variable lemma of G. David ([10]). LEMMA

3 .1.1 0. Let F : R x Rn ... R be a function of two variables t

x = (x 1 ,. .. ,xn)

l

Rn. Assume that for each x, the function t

1-+

l

R,

F(t,x)

is Lipschitz, with Lipschitz constant less than or equal to M, and for each i, 1

~

i ~ n, the function xi

1-+

F(t,x) is Lipschitz, with Lipschitz

constant less than or equal to Mi, for any choice of the other variables. Givenaninterval IxJ =IxJtx .. ·xJn, where the J/s and I are

1 dimensional compact intervals, there exists a function G(t,x) · Rx Rn ... R with the following properties: (a) G(t,x) (b) If

~

F(t,x) on I xJ.

E = {(t,x) d

x

J: F(t,x)

=

G(t,x)l, then

lEI ~

i IIIIJ 1.

{c) For each i, the function G{t,x 1 ,x 2 , .. ·,xi-l'-, xi+l'"",Xn) is Lipschitz, with Lipschitz constant less than or equal to Mi, and one

of the following statements is true: Either for each x, -M < aG (t x) < 4M

-s·

-at·

for each x

or

4M aG , -S ~ at (t,x) :<: M.

The proof of this lemma is the same as in the 1 dimensional case, treating x as a parameter {see [9]). Before we proceed with the proof of Lemma 3.1. 7, we would like to point out that in the analogue of Lemma 3.1.7 for bounded domains, a normalization is necessary since if ii"{X) solves the systems of elastostatics, so does ii"(X) +

£: + BX,

where

£:

is a constant vector,

while B is any antisymmetric 3 x3 matrix. The right-hand side of the inequality in the lemma of course remains unchanged, while the left-hand side increases if B 'increases.' The most convenient normalization is that for some fixed point X* in the domain Vii"(X*)-Vii"(X*)t

=

0. This

gives uniqueness modulo constants to problem 3.1.2 in bounded domains.

172

CARLOS E. KENIG

We now need to introduce some definitions. Let 0 0 C R~ be a fixed, C

00

domain with \(x,O): \1\x\11 = max\xi\ ~ 1\C

l\lxll\~liCD 0 CI(x,y):O
aD 0 ,

l(x,y):O < y < 1,

If ¢:Rn-l .... R is

\IIV' ¢111 ~ M, we construct the mapping T¢: R~ .... Rn by

Lipschitz, with

T¢(x,y)=(x,cy+T/y*¢(x)) where TJ(CQ'(Rn-l) isradial, JTJ=l, and C = C(M) is chosen so that T¢(R~) C {(x,y): y > ¢(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


•

L


=

0 in D¢ and

~ C\IN,J.(A(div ii)I + 'fJ

Here N,J. is the non-tangential maximal operator 'fJ

corresponding to the domain D ¢. This lemma will be proved by a series of propositions. Before we proceed we need to introduce one more definition. We say that Proposition (M,e) holds if whenever ¢ is such that constant vector

i

with

!II aIll

IIIV'¢111 ~ M, and there exists a

~ M so that l!IV'¢ -i Ill ~ e, then, for all

Lipschitz functions ii on D¢ with ~t; + (A+If) IJ div ii

=

0 in D¢,

with Vii(A¢) = V'iit(A¢) we have

where C = C(M,e). Note that if Proposition (M,e) holds, then the corresponding estimates automatically hold for all translates, rotates or dilates of the domains D¢, when ¢ satisfies the conditions in Proposition (M,e). In the rest of this section, a coordinate chart will be a translate, rotate or dilate of a domain D¢. Thebottom B¢ of i!iJ¢ willbe T¢(aD 0 U(x,O):xfRn-l).

ELLIPTIC BOUNDARY VALUE PROBLEMS

173

PROPOSITION 3.1.12. Given M > 0, there exists e = e(M) so that

Proposition (M,e) holds. We will not give the proof of Proposition 3.1.12 here. We will just make a few remarks about its proof. First, in this case the stronger estimate IIN,.~,.(\lt;)ll 2 'fJ

L (aD ,dO')

~ CI!A(div t;)N +/LI\i'U' +V'~tiNII 2 L

(® ,du)

holds. This is because in this case, the domain ~cp is a small perturbation of the smooth domain D... • For the smooth domain D.... , we can aX ax solve problem 3.1.2 by the method of layer potentials (see [24], for example). If e is small, a perturbation analysis based on the theorem of Coifman-Mclntosh-Meyer ([2]) shows that this is still the case. This easily gives the estimate claimed above. PROPOSITION3.1.13. Forall M>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 <

t)

if!,

and

The non is then

the image under T.p of (x,O) + y, with b chosen as above, suitably truncated, and where Q

=

T.p((x,O)). Let now, to lighten notation,

m = N¢(\7~), iii= N¢(A(div ~)I+ f!l'V~+'V~tl). For t > 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

ELLIPTIC BOUNDARY VALUE PROBLEMS

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<

~

S.

t +

C8t. Since for Q cTr{I(FjnFj),

(3t, and f3 > 1 , if 8 is small enough, we see that we must have

Q)? m(Q).

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

177

ELLIPTIC BOUNDARY VALUE PROBLEMS

<J3:;_1 )t if 8 issmalland QfTr;,(FjnFj). Thus,applying (M,e) to Df, we see that CX:{{3-1) 2 t 2 a{Uj) ~

J

T)(FPFi)

l . . - [vir(A f> -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
+ p.v.

fan

o-,

div;

g

f

=

L 2 (ciD, du).

0 in D and

IH'(Q) r{P-Q)I g(Q)du(Q)

a

_.

fan aP.

r(P-Q)g(Q)du(Q)

1

.... +

1 ....

(HSg )-(P) ~- ± 2 g(P) + p.v.

(e)

fan

....

IH(P) r{P-Q)I g(Q)du(Q),

For the proof of this lemma in the case of smooth domains, see [25]. The proof of Theorem 3.2.2 (at least the existence part of it), reduces to the invertibility in L 2 (ciD,du) of the operator } I+K, where Kg(P) ~ - p.v.

fan

to show (3.2.4)

IH'(Q)r{P-Q)Ig(Q)du(Q). As in previous cases, it is enough

ELLIPTIC BOUNDARY VALUE PROBLEMS

179

This is shown by using the following two integral identities. LEMMA

flii

3.2.5. Let

= \lp, div

J=

suitably small at

LEMMA

h

be a constant vector in Rn, and suppose that

0 in D, and that ii,p and their derivatives are

Then,

oo.

-·

...

3.2.6. Let h,p and u be as in 3.2.5. Then,

The proofs of 3.2.5 and 3.2.6 are simple applications of the properties of ii,p, and the divergence theorem . .... Chaos ing h = e 3 , we see that, from 3.2 .6 we obtain COROLLARY

3.2.7. Let ii,p be as in 3.2.5. Then,

J an

p 2 do :S C

J

IVii 12 do,

an

where C depends only on M.

A consequence of Corollary 3.2.7 and Lemma 3.2.5, is that, if

at: ~ aNat: p·N, av

then we have

180

CARLOS E. KENIG

COROLLARY

->

•

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

ELLIPTIC BOUNDARY VALUE PROBLEMS

(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.

ELLIPTIC BOUNDARY VALUE PROBLEMS

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

<

£-.

dz > = <

Jz, dZ > = 1 ,

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

az

=

0.

189

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

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
a= 0\D(z,e). We apply Stokes' Theorem to the 1-form d' on the domain

=I' l u(')

C: 1(-zi <

=

a (note that u has smooth coefficients on a

'""' but not on all of 0 ). Thus, by (1.10), neighborhood of the closure of 0, (1.11)

'""' This last is 0 by (1.7). Thus, since 00

=an U aD(z,e)

(with suitable

orientations), we have

J J u(') =

an Parametrizing aD(z,e) by

as e ...

o+.

REMARK

u(') .

aD(z,E)

'=

z + eeitl, 0

~ 6 < 217, we obtain

This completes the proof. o

1. In the proof of the theorem, if we assume that f is smooth

but not necessarily holomorphic, then the right side of (1.11) does not vanish; instead it equals

,_z

1, n (at/a') d( Ad'. Thus the proof of the

theorem yields

(1.12)

This formula, valid for all f

£

C 1(0), will be valuable later on.

el

191

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

REMARK 2. As it stands, the method of Stokes' theorem will not general-

ize to en. Namely, we used the fact that ~~J is holomorphic with an isola ted singular point at z . In en , n ~ 2, holomorphic functions

never have isolated singularities. A more sophisticated approach will therefore be needed.

n ~ e is a smoothly bounded domain and if 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(()
sup

o<e<eo ane

(Here ds is the element of arc length.) That this definition is unambiguous is a technical matter (see [31, Ch. 8]). We shall need PROPOSITION ([31] ).

f

If

l

Each f

l

H 2 ({}) has associated to it a unique

L 2 (aO). The Poisson inte8ra 1 of f is f. A I so

11 E:

(}{}E

-> (}{}

is normal projection then (flan)

a

<"e)- 1 ---+ f

in L 2 (an).

E

Thus we may make H 2 into an inner product space by definin8



=

Jtida an

for f,g

l

H2 (0).

(1.13)

192

STEVEN G. KRANTZ

BASIC LEMMA. If K C

0 is compact then there is a constant C = C(K)

such that sup jf(z)l ~ Cllfll 2 , all f zlK H

Proof. Fix z

l

K. Let e1 <

fo (distance

l

H 2 (n).

(K, afl) ). Then

(Stokes)

(Schwartz)

~ C(el) .

I If(~') ds(~') 12

1 /2

ane 1 < C(K) 11£11 2 H

•

o

Now (1.13) and (1.14) imply that H2 (n) is a Hilbert space. Fix

z ( n and define the functional

f

1->

f(z) .

(1.14)

INTEGRAL FORMULAS Then the lemma, with K

=

IN

COMPLEX ANALYSIS

193

lz!, shows that ¢z is continuous. The

Riesz Representation Theorem yields a unique kz

£

H2 (0) such that

In other words,

f(z)

Jc(()~)ds((}, an

=

all f £ H2(0), z £0.

Formula (1.12) is called the Szego formula and k/0

(1.15)

=S(z,()

the Szego

kernel (see [31, Ch. 1] for further details). REMARK 1. Formula (1.15) has the advantage of working on any smoothly bounded domain (even in en ), and the disadvantage of being nonexplicit. As an exercise, check that when 0 = .1 s;_ e and z = 0 then S(z,()

=2~.

any z

£

Then use Mobius transformations to calculate S(z,() for

.1. You will rediscover the Cauchy formula!

§2. The Cauchy-Fantappie Formula Now we begin to consider integral formulas in en. For purposes of differential analysis we introduce the notation

a_!._ (:a

dZ- - 2 itx, J

i

J

a)

~

J

•

a)

_i__!_~_i_+i az - 2 ax. dy, j

J

J

•

It is easily checked that

!_ if"

z.=_l_z.=1,

j 1

azj

1

=<_l_,crz.>=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 .

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

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

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

If a 1 , a 2

f

1

197

then we define

B(a 1 , a 2 ) = det (a 1 ,
= I,

e (o) a~(l)

A

d(a;(2) .

ocs 2 We claim that B has three key properties: (1) B(a 1 ,a 2) does not depend on a 1 ; -

1

2

(2) a(B(a ,a) :0; (3) If a 1 ,a 2 ,{3 1 ,{3 2

(1

then B(a 1 ,a 2 ) - 8({3 1 ,{3 2 ) is "'J( exact.

Assuming (1 ), (2) and (3) for the moment, let us complete the proof (note that (1) is used only to prove (3)). Letting

we have (observe that B(a 1 ,a 2 ) = 1J(w))

J

f(() 'Tf(W) A w(()

=

an

J

f(() B(a 1 , a 2) A w(() .

an

Note that, by (2), d((f(()B(a 1 ,a 2 ) =

a,f(()

= 0+0

=

A

A

w(())

B(a 1 , a 2)

0.

A

w(() + f(()a,B(a 1 ,a 2 )

1\

w(()

(2 .5)

198

STEVEN G. KRANTZ

Hence, letting 0 < e < dist (z, ail), we have by Stokes' Theorem that (2.5)

J

f(()B(a 1 ,a 2)" w(()

dB(z,e)

J

f(()!B(a 1 ,a 2)-B(,8 1 ,,8 2 )1 Aw(()

(2.6)

dB(z,e)

+

f

B(,Bl, ,82) "w(().

dB(z,e)

But the first integral

f

f(() dA " w(O

dB(z,e)

(for some A , by (3))

f

d(f(() " A " w(())

aa(z,e)

(2.7)

by Stokes' Theorem, since aaB(z ,e) = 0. Thus (2.5)- (2 .7) give

J

f(() lJ(w) " w(() =

an which, by (2.4),

J aa(z,e)

f(() B(,8 1 , ,8 2) " w(()

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

1 e4

J

f((} fl((-z) "
oB(z,e)

1 e4

J

f(z)

rfl-z) "
oB(z,e)

(Stokes) =

.!.

e4

f(z)

J

2w((-z) " w((> + O(e)

B(z,e)

J =

2w(()" w((> + O(e)

f(z) · 2W(2) + O(e) .

Letting e -+ 0 yields the desired result. o We conclude this section by proving (1)-(3). For (1), we have

199

200

STEVEN G. KRANTZ

which, by adding row 2 to row 1,

This calculation is correct for ( 1 .f- z 1 , ( 2 .f- z 2 • The full result follows by continuity. For (2), imitate the proof ci (1). For (3), use (1) to write

Now an easy calculation, as in (1), shows that this last equals '"JA where

This completes our discussion of the Cauchy-Fantappie formula.

§3. Introduction to the

a problem

One of the principal problems in complex function theory is the construction of holomorphic functions with specified properties. In one dimension, there are a number of highly developed techniques: Runge and Mergelyan theorems, power series, infinite products, integral formulas, and so on. In several variables, these techniques are either unavailable, much less useful, or much less accessible.

201

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

The two most prevalent techniques for constructing functions in several complex variables are sheaf theory and the inhomogeneous Cauchy-Riemann equations. The latter interact strongly with the subject of integral formulas, and in any case are a more flexible technique than the former. To these we now turn our attention.

n

The setup for our study is that, for a given form f =

.~ j=l

seek a function u such that du =f. Notice that since 0

(ao +'da) Hence

+

all

f 1.(z) dz1., we

=

d2

=

"d 2 , linear independence considerations yield that ff 2 =

f necessitates df

=

a2 + =

0.

0.

A simple calculation shows that, for n ~ 2, this compatibility condiat. ar tion is equivalent to _J =___li, all j, k. Notice, however, that when

azk azi

n

=

1 the condition df = 0 is always vacuously satisfied. This differ-

ence can be explained in part by the fact that the equation

au = f

is

really n equations (:j = fj) in one unknown (namely u ). For n > 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

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

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¢<() -- - )' c

( -z

d(

A

d.,

205

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

~- 2;i

II

(&p/iJZt' +Z) d, Ad/;

c 1

= -

27Ti

If

(&p;at;>
D(O,R)

where D(O,R) is a disc which contains supp ¢. We apply Remark 1 of II in Section 1 to ¢ on D(O,R) to obtain that the last line

=

¢(z)-

2~i

J ~~}

d/;.

an(o,R) The integral vanishes since ¢ = 0 on aD(O,R), hence

all = f.

finally that u c ck by differentiation under the integral sign.

Observe

0

REMARK. In general, the u given by the lemma will not be compactly supported. Indeed if If¢ /. 0 and if u were compactly supported (say u

~

D(O,R) ) then a contradiction arises as follows:

an(O,R)

The supports of solutions to the

D(O,R)

a problem explain many phenomena in

one and several complex variables. We explore this theme later. Meanwhile, contrast the Lemma and Remark with the following result.

206

STEVEN G. KRANTZ

THEOREM.

Let n

>1

and let
C~(Cn).

on en and suppose each j t function

ui(z) • -

If',j

2~

= 1dz l

+ · ·· + ndz1 be J-closed

Then for any 1

~j ~n

the

(z " .... z;' ~~·z i+" ...•z.) d( • d(

c satisfies ui t

C~(Cn)

Proof. Fix 1

:S m :S n. We need to check that

and

~j

=


;j

=
zm

:5 m ~ n.

If

m = j then the result follows from the lemma. H m J j then use the compatibility condition

acpj

azm

=

acpm aZj

to write

~ ui(z) =-1~ m

217i

c

By the lemma, this last equals
eJ j,

then uj = 0 for zf large (since then j = 0 ). Also ui is

holomorphic for zf. large (since continuation, u

=0

all =
is then 0 ). So, by analytic

off a compact set. Next, uj - uf = 0 since it is

compactly supported and holomorphic. Finally, uj

f

C~Cn) by differen-

tiation under the integral sign. o REMARK.

The proof actually shows that u is zero on the unbounded n

component of c( j~l supp j). It also shows that there is at most one

INTEGRAL FORMULAS

IN

207

COMPLEX ANALYSIS

compactly supported solution to ~ =f. In the present case, one exists and is given by an integral formula. §4. The Harto!Js Extension phenomenon and more on the

a problem

We have cited the Hartogs extension phenomenon in the examples of Section 3. The reader will want to check that the proof of it that we now give is independent of those examples. THEOREM (The Hartogs Extension Phenomenon). Let

be a bounded, connected open set. Let K

s; 0

n s; en

I

n >1

I

be compact. Assume that

O\K is connected. If f is holomorphic on O\K then there is a holomorphic F on 0 such that F IO\K =f. Proof. Choose and

cp = 0

cp

f

C""(Q) such that

cp = 1

in a neighborhood of aG

in a neighborhood of K. Define

"' t

cp(z) · f(z)

F(z)

if

z ( 0\K

if

z

=

0

f

K.

Then F is a C"" extension of f to 0, but it is not in general holomorphic. We now seek v such that F + v is holomorphic on 0 (and "V

F

+vln\K =f).

Thus we seek

v

satisfying ,...., a(F+v)=O

or or (4.1) since f is holomorphic on supp

cp.

Now

cp = 1

near ail so (- &/>)

·f

is smooth and compactly supported in Q. The theorem of Section 3 now guarantees that there exists a v satisfying (4.1 ). Moreover, the remark following the theorem guarantees that v = 0 near

oO.

Thus F + v is

208

STEVEN G. KRANTZ

holomorphic and, near

an' F + v = F = ¢

. f =f. By analytic continua-

"' +V = f on U and the result follows. o tion, F Notice how the hypothesis n > 1 was used in the proof to control supp v. The Hartogs extension phenomenon has several interesting consequences: (i)

A holomorphie function f in en, n ?: 2, cannot have an isolated singularity. If it did, say at P, then f would be holomorphic on B(P,2e)\B(P,e) for e small hence, by the Hartogs phenomenon, on B(P,2e), and hence at P. That is a contradiction

(ii)

A holomorphic function f in en, n ?: 2, cannot have an isolated zero. If it did, say at P, then apply (i) to 1/f to obtain a contradiction.

(iii) If U s; en is open, E

s; U ,

f is holomorphic on U\E, and E is

a complex manifold of complex codimension at least 2, then f continues analytically to all of U. To see this, notice that for n = 2 the set E is discrete and the result follows from (i). For n > 2 , the result follows from the case n = 2 by considering fl
fC en. Now we return to discussion of the

lf

operator. There are essentially

four aspects to this matter: (1) Existence of solutions;

lf data and lf solutions;

(2) Support of

(3) Choosing a good solution, where "good" means smooth or bounded; (4) Estimates and regularity. Regarding (4), we have noted that ellipticity considerations imply that when (), = g then v is smooth wherever g is. As an exercise, use the theorem of Section 3 to give another proof of this assertion. (Hint: if g is smooth on B(P,r), then let ¢ B (P, ~). Let

lfr

t

c;(s

u = ¢ · v and f = 'J(¢ · v)

~· 1}} =

t

c;(B,(P,r)) satisfy ¢

satisfy

lfr

=1

=1

on

on B (P, ~}. Define

(j¢ · v +¢·g. ~hen d"u = f. Apply the theorem

209

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

of Section 3, decomposing f as f =

(1/1· d¢ · v) + (1-l/l)(ff¢ · v)+ ¢·g.)

This is all that we shall say about interior regularity. Topic (2) has been discussed vis

a vis the Hartogs phenomenon.

Topics (1) and (3) are more subtle. First note that if

a(u+h) = f

air = f

then also

for any holomorphic h. Given f, one cannot expect all

u + h to be nice (i.e. bounded, or L 2 , or C"" up to the boundary). How does one find a nice solution? An idea from Hodge theory is to study the solution u to

a,

is orthogonal to the kernel of

au

=

f which

i.e. which is orthogonal to holomorphic

functions. This solution has been studied by Kohn [25], [27], [28], Catlin [3], [4], Greiner-Stein [16), and others. It is often called the Kohn

solution or canonical solution to the

a equation.

We now briefly review some of what is known about the Kohn solution, and other solutions, to the

a problems on domains in

en. (In this

section we shall take "pseudoconvex" and "strongly pseudoconvex" as undefined terms. These terms will be discussed in detail in Section 5; for now, a pseudoconvex domain is a domain of existence for the

a

operator.) (a) If

n

is a bounded pseudoconvex domain in en and f = I.fjdzj is a

a-closed form with all fj

f

L 2 (fl), then there is a u

f

L 2 (n) with

du=f. Also lluiiL 2 :SC(n)TilfjiiL2 • See[20]. (Exercise: the canonical solution also satisfies this estimate.) (b) If

n~

en is smoothly bounded and pse~doconvex and f

a a-closed (0,1) form with all fj

f

=

I.fjdzj ~s

C (fl) then there is a u 00

f

C""(n)

satisfying au =f. See [26]. It is not known whether the canonical solution has this property. (c) If

n s;

c_:n is strongly pseudoconvex with c

2

boundary and if

f = If.dz. is a a-closed (0,1) form with bounded coefficients, then J

J

thereisa u satisfying au=f and llull "".::;ci!lf1.11 L

oo·

L

The

210

STEVEN G. KRANTZ

canonical solution has this property. See [11], [17], [22], [16], [351. Sibony [39] has shown that there are smooth pseudoconvex domains on which uniform estimates for

a do not hold.

It is not known on

which parameters the uniform estimates depend (however see [13]). Range [38}, Henkin [17}, and others have proved uniform estimates on certain weakly pseudoconvex domains. (d) Complete, and sharp, estimates have been computed on strongly pseudoconvex domains in Lipschitz, Sobolev, Besov and other norms. See [16], [30}. These estimates hold for the canonical solution. One feature of the theory is that the operator assigning the canonical solution Kf to a a-closed (0,1) form f is compact in these norms. This compactness is best exp:essed as a "subelliptic estimate" (see [27] ). Catlin [4] has announced a characterization of those domains on which

a satisfies a subelliptic estimate.

Many times an estimate tells us how to choose the right solution to ~ =f. We conclude this section with an example of how estimates can be

useful. DEFINITION. Let 0 ~ en be a domain and P £a{}. A holomorphic

function f: 0-+ C is called sin~ular at P if for every e > 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

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

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
::j

Then Tp(an) is the (real) tangent space to

(p). aj =

an

o!.

at P. If n

=

!p < Ol

we say that n is convex at p fan if

(5.1)

We say that n is strongly convex at P if strict inequality obtains in (5.1) when 0 I a £ Tp(an). A domain is convex (strongly convex) if each boundary point is. EXERCISES

(i)

(see [31 ]):

For a smoothly bounded domain, geometric convexity is equivalent to convexity.

(ii)

If n is smoothly bounded and convex, then n can be written

as an increasing union of strongly convex domains. (iii) The above definitions are independent of the choice of p. In order to understand the

~:ole

of convexity in complex analysis, we

need to discuss inner products. If z, w (en, we define the Hermitian

inner product

213

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

n

H

=I zjwj j=l

and if we identify

and likewise

then we define the real inner product 2n

Re

=I

tjsj .

j=l

Notice the following facts: (1) Re (2) If 0

=

Tp(aO) (3) With

=

Re (u);

lz ten: p~) < 0} has smooth boundary, P =

la t en: Re

n. p

=

tan,

then

Ol .

as in (2), let j"p(aO) = Ia fen: H

call 1p(an) the complex tangent space to

a t 1p(aQ) then ia

t

an

=

01.

We

at p. If

1p(a0). Also ~-p(aD) ~ Tp(an) and it is

the largest subspace of Tp(aQ) which is closed under multiplication by i. Now if 0 = lz

t

en: p(z) < 0} is smooth and convex and P

t

an

then

0 lies on one side of Tp(a{l). Thus if we define fp(z) = H then the zero set Z(f p) of f p lies in P + Tp(aD). In particular Z(fp)

n 0 ~an

(and if

n.

is strongly convex then Z(fp)

nan= I PI).

Thus 1/fp is singular at P. Also Re fp < 0 on 0 so we may choose 0
t

Z such that 1/(f p) 1 /N is holomorphic on

n

and in L 2 (0).

214

STEVEN G. KRANTZ

Thus each P

l

an

has an L 2 singular function. By the argument at the

end of Section 4, 0 supports an L 2 holomorphic function which cannot be analytically continued to any large open set. So any convex domain is

a domain of holomorphy. If we want to understand domains of holomorphy, convexity will not

tell the whole story. For convexity is not a biholomorphic invariant: consider ~:A ... C given by ~(z) = (z+3) 3 . What is needed is a new notion called pseudoconvexity: DEFINITION, If 0 = {z len: p{z) < 01 is smoothly bounded we say that

0 is (Levi) pseudoconvex at P

l

an

if

(5.2)

We call 0 stronAly pseudoconvex at P if strict inequality holds in (5.2) for all 0 /, w

l

j"p(OO). The domain is pseudoconvex (strongly pseudo-

convex) if each boundary point is. EXERCISE. If P

£

a0 is strongly pseudoconvex prove that if A > 0 is

sufficiently large and 'P(z)

=(eAp(z)_l)/A

then

n a2"' ~ _ _P_(P)w-wk>Ciwl 2 , VP laO,Vw lip(OO).

~

j,k=l

az.az J k

1

-

(5.3)

See [31] for details. The rather technical notion of pseudoconvexity is vindicated by the following deep theorem (see [31 ]): THEOREM. If 0 ~ en is smoothly bounded then the following are

equivalent :

0 is pseudoconvex (ii) 0 is a domain of holomorphy (iii) the equation ~ = f, f a a closed {p,q) form, is always (i)

solvable.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

215

This theaem means that pseudoconvex domains are the natural arena for complex function theory. Also (exercise, or see [31]) any pseudoconvex domain is the increasing union of smooth strongly pseudoconvex domains. So strongly pseudoconvex domains, in a certain sense, are generic. In order to unify and illustrate the ideas introduced so far in this section we prove LEMMA.

If 0 ~en is smoothly bounded and P faD is a point of

convexity then P is a point of pseudoconvexity. Proof. Let p be a defining function for 11. Let a ( j"p(an). Writing the definition of convexity in complex notation we have

cf2 I az.Iz n

j,k=l

1

a2 I __ P_(P)a.(ik n

(P)a-ak +

k

j,k=l azj

But a similar inequality also obtains for ia

1

azk

f

j"p(an). Adding the two

inequalities yields the result. o §6. Solutions for the

a problem

We briefly describe the Hilbert space setup for H
ff problem. We fix a smoothly bounded 0

~

en

and introduce the

notation

Ill

Ill

Ill

The operators T,S are of course unbounded, but they are densely defined

216

STEVEN

(since

c;

also SoT

=

=

KRANTZ

is dense in L 2 ). It is easy to check that T, S are closed;

0 so if F

=

ker S then Range T s;_ F .

An existence theorem for the Range T

G.

a equation amounts to proving that

F . Moreover, it is an exercise in functional analysis to

check that this is equivalent to proving an inequality of the form (6.1) See [20) for details. Rather than prove (6.1), it is more convenient to study the symmetric inequality

Notice that when y

l

F, (6.2) reduces to (6.1).

Unfortunately this program, as stated, fails. The difficulty is that the computation of T* gives rise to boundary terms which, in general, cannot be controlled. (However, in the strongly pseudoconvex case and on weakly pseudoconvex domains satisfying a non-degeneracy condition, there are delicate techniques for handling the boundary terms. See [25].) Hermander 's idea [20) was to work not in Euclidean L 2 but rather in L 2 of the measure space e- ~ dx . If

~ is chosen to have certain con-

vexity properties and to blow up rapidly at

an.

then the boundary is

effectively suppressed ( 0 becomes a complete Riemannian manifold) and the Hilbert space program outlined above works. After the existence problem is thus tamed, the weights can be eliminated (provided bounded) and one obtains an existence theorem in L 2((},dx).

n

is

A leisurely

exposition of all these ideas can be found in [31). We now formulate a version of HOrmander's result, which we will use freely in what follow: THEOREM (Hormander). If 0 ~ en is smoothly bounded and pseudo-

convex, and if f is a

a closed

cients, then there exists u

l

(0,1) from on 0 with L 2 (0,dx) coeffi-

L 2 (!l,dx) such that

au =f.

INTEGRAL FORMULAS IN COMPLEX ANALYSlS

217

REMARK. Hormander's theorem can be used to prove most of the theorems at the end of Section 5 characterizing domains of holomorphy. See [31] for details. Our next main goal is to obtain an integral formula for a solution to the

J

problem on a convex domain. We first need:

THEOREM (Bochner-Martinelli). Let 0 be a domain in en with C 1 boundary. Let f

f(z)

=

f

C 1 (0). Then for all z

Wl( ) { n n

f

0 we have

Jf(~)T/ ( 1(-z\ ~-z 2\~ "cu(~)

an

Proof. Apply Stokes' theorem to the form

on the domain 0\B(z,e). Imitate the proof of the Cauchy Integral Formula (or see [31] ). Now fix 0 a bounded, C 2 , convex domain. Choose e > 0 so small that

n=lz (en: dist (z, ll) <

E

I

is convex (hence pseudoconvex). Let f

be a smooth, a-closed (0,1) form on 0. By Hormander's theorem, there

is a smooth u on 0 such that

au =f.

We apply the Bochner-Martinelli

formula to u (which is certainly in C 1 (0) ). Thus

218

=

STEVEN G. KRANTZ

1 nW(n)

J( ( u( )7j

(-z \

\(-z

12)

ao The first term on the right is not useful, since it involves u, so we will remedy matters by subtracting an appropriate holomorphic function from the right side of (6.3) (see the discussion in Section 4 on choosing a good solution). The Cauchy-Fantappie formalism now comes into play:

If U =

!p < Ol,

let

r-~

w(z,()

(()

=~ ::. ()

-2£._ (')) ,

,

~~. ()

a

n

with
H(z) =

nW~n)

J

u(()7j(w(z, ()) "(<)(()

ao is well defined and holomorphic in z (because w is). Now let v(z)

=o

u(z) - h(z). Then certainly

Jv

= ~ =

f and, by (6.3),

219

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

-

Wl() Jf(() AT/ (

n n

(-z \ l(-z12}

"UJ(()"' I- II.

{}

(-z

r

r

Let G=Ox[O,l] anddefine g(z,.,,A)=(l--A)--+Aw(z,.,) tobea

l(-zl 2

form on G . Then I = n.

~(n)

I

u(()T/(g)

A

UJ(() •

ao By Stokes' theorem, applied on G, this

=

nW~n)

J

d(u(()T/(b)

A

UJ(()) .

G

But d(u((}T/(g)

A

UJ((}) = {]u

A

T/(g)

A

+ u(()d(T/(g)) ""f

A

71(g)

A

UJ((} A

UJ(()

w(()

(this last equality takes advantage of the special algebraic properties of Cauchy-Fantappie forms). So we finally obtain HENKIN'S INTEGRAL FORMULA.

If {}

=

lp < 01

is C 2 and convex and

f is a smooth, a-closed (0,1) form on a neighborhood of {} then the

function

220

STEVEN G. KRANTZ

v(z) = n·

~

f

.. )

!((} • '!(g) • ..,(()

( a!lx(O,l]

satisfies

a;,.

=

f on 0. Here

g(z, '· ..\.) = (1-..\) (

w(z, O

'-z \

IP-z1 2}

+ A.w(z, 0

,

~-~,1 <00 ' ... ' -~a::n <0) 0 0

=

ll>(z,

tl>(z,

and

The standard reference for Henkin's work is [18]; see also [311 Similar formulas were derived by Grauert-Lieb [11]. Kerzman [22], and fJvrelid [35]. Now we assume that 0 is strongly convex, and show how to use Henkin's formula to obtain uniform estimates for solutions to the

a

problem. For simplicity we work in C 2 only. So

f " 71(g) " w(O

= f "

[g 1

~ d..\ - g ~ dj 2

" d ' 1 "d ' 2 •

After some algebra, and integrating out A, the Henkin formula is

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

+

2 ~(2 )

221

J

f 1 (()8 1 (z, ()d(1 A d(2 A d(1 A d( 2

{}

+ 2 w~2 )

J

f 2 (()B 2 (z,(}d(1 Ad(2 Ad( 1 Ad( 2

•

{}

In order to prove an inequality of the farm

1\v\\ it suffices to check that

L

oo

~

Cl\£1\ L .,., '

(6.4)

222

STEVEN G. KRANTZ

JIAj(z,O\da(() 'S C,

1,2

(6.5)

i=1,2

(6.6)

j

=

an and

.J\Bi(z,()\dV(()'SC, {l

with the estimates uniform over z

En.

(Here da is area measure on

an.)

By symmetry we check only j = 1. For B 1 , choose R > 0 such that B(z, R) 2 0 for all z

E

0. Then

For A 1 , we must work harder. We need to know something about the degree to which the denominat"or (z, () 1(-z 12 of A1 vanishes. (This is where strong convexity plays a role.) Think of z ( 0 as fixed. Notice that, writing the Taylor expansion for p about ( in complex coordinates, we have

+ (quadratic terms) +(error terms)

The last

esti~te

is by strong convexity - see Section 5. Thus

223

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

Let

7TZ

be the normal projection of z to

nate in the complex normal direction at

ao.

Let tl be a coordi-

(that is, i times the real

TTZ

normal direction) in (){}. Recall that

and

It follows that

lm

(~ ap (0(z.-(.~ ~a(,. J J J=l

J

=

lm

(~ dp (z)(z ·-(·~ ~ az. J J j=l

+ O(iz-(i 2)

J

measures (essentially) the complex normal component of z- (; at

1TZ.

As a result

Introducing two additional coordinates t 2 , t 3 centered at span the complex tangential directions at i«l>(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


J }·

rdr = C · C'r 0 < oo

0

Likewise

•

INTEGRAL FORMULAS IN COMPLEX ANAJ..YSIS

225

J


C'r 0


J}

(ilog rl + llog Cj)rdr

0


The estimates for A 2 , B 2 follow by symmetry, as already noted. We have proved THEOREM. With 0 strongly convex and f, v as in Henkin's integral

formula for solutions of the

a problem,

REMARKS. 1) To eliminate the hypothesis that f is defined on a neighborhood of

n

we observe that for e > 0 small enough the domain

ne = lz dl: dist (z' aQ) > el

226

STEVEN G. KRANTZ

is smoothly bounded and strongly convex. Moreover, the estimates in the theorem on Oe depend boundedly on e {by a calculation). Thus estimates can be obtained for f a smooth form on 0 with bounded coefficients by applying a limiting argument to the solution u

oe.

e

of a(*)= f on

2) The results of the theorem actually hold on smoothly bounded strongly

pseudoconvex domains. This is most easily seen by using the following important result: THE FORNAESS IMBEDDING THEOREM [9}. Let 0 CC

en

be a

stron~ly pseudoconvex domain with C 2 boundary. Then there is a neigh-

borhood

n of n' a

k > 0' a

c 2 strongly convex domain u ~ cn+k'

and a holomorphic imbeddin~ F : 0 . . cn+k such that (i)

F(O) ~ U

(ii)

F(0\0) ~co

(iii) F(Ofl) ~

(iv)

ima~e

au

F is transversal to

au.

The upshot of this theorem is that the Henkin singular function

«<>

which

we know how to construct on U can be pulled back to 0. The construction of the Henkin solution to the

a equation and the uniform estimates

follow just as before. 3) The singular function «<> can be constructed more directly on a strongly pseudoconvex domain 0 as follows: first write 0 where

(see (5.3)). For ( tail fixed we define

=

lp < ol

227

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

The function L, called the Levi polynomial, has the property that there a neighborhood

u(

of ( such that

0 n U( n lz: L{z,O = Ol =I( I.

One

can modify L, by solving a (j problem (see [18] or [31]) to obtain cp -

such that !l

n lz: IP(z, 0 = Ol = 1(1

n

and IP(z, () = .l: Pj(z, ()(zr(j) j=l

with Pj holomorphic in z. Henkin's program may be carried out using this and w(z,()

=

(-P1(z,()/IP(z,(),···,-Pn(z,()/IP(z,()).

Notice that, by the discussion in the preceding paragraph, 1/L( · , () is a local singular function at (. This, together with the uniform estimates for the

a equation which we have obtained, completes the program

outlined in Section 4 to show that a strongly pseudoconvex domain is a domain of holomorphy. §7. Connections between various integral formulas and applications

In Section 1 we constructed the Szego kernel for domains in e

1•

How-

ever the construction goes through for domains in en once one has the basic lemma, and that follows in en from the Bochner-Martinelli formula. We leave the details of the basic Szego theory in en as an exercise. Recall that the Szego kernel for a domain !l is the reproducing kernel for H 2 (!l). Now fix z ( !l. By construction, S(z, ·) ( H 2(!l). By the reproducing property, it follows that

S(z,() =

J

S((,w)S(z,w)da(w)

an =

J

S(z,w)S((,w)da(w)

an =

S((,z)

=

S((,z) .

228

STEVEN G. KRANTZ

Thus the operator

has the following three properties: (a) S: L 2(a!l) ... H2(!l)

(b) S is self-adjoint (c) S is idempotent. Therefore S is the Hilbert space projection of L 2 (a!l) onto H 2(1l). It turns out that the Henkin operator on a strongly pseudoconvex

domain very nearly has properties (a)- (c). First, by a theory of nonisotropic singular integrals developed especially for boundaries of strongly pseudoconvex domains (see [8], [36]): the Henkin operator

H:f

1-+

nW~n)

J

£(() T/(w)

A

cu{()

(7.1)

ao (with the w produced from the Fornaess theorem as in Section 6) maps L 2(a!l) onto H 2(0). Also H is idempotent. Now H is not quite self-

adjoint, but it is nearly so. To see this, one needs to write (7.1) in the form _1_ If(() N(z,() do((}

nW(n)

n(z, ()

an where do is area measure on iKl. This is a straightforward but tedious calculation (see [23]). It turns out that N is real. It also turns out that (z, () - ((,z) vanishes to higher order at z = ( than does . (Try this when 0 is the ball to see how this works - details are in [23 ].)

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

229

As a result of the preceding observations, the kernel

is a kernel which is less sin§ular than the original Henkin kernel. Thus H - H*, rather than being a non-isotropic singular integral operator {as is H), is a smoothinB operator. This observation of Kerzman and Stein is now exploited as follows. Denote H* -H =A. The reproducing properties of S and H guarantee that {1) S = HS

and (2) H

~

SH.

Thus (3) S = S* ~ (HS)* - S*H* = SH* .

Subtracting (2) from (3) gives S-H~S(H*-H)=SA.

This is an operator equation on L 2 • We may resubstitute the equation into itself as follows:

S == H +SA ~

H + (H+SA)A

~ H + HA + SA2 H + HA + (H+SA)A2 - H + HA + HA2 + SA3

(7.2)

==

"' • •• ==

H + HA + · ·· HAk + SAk+l .

Now we know that each of the operators HA, HA2, ··· are smoothing. If we apply both sides of (7.2) to a sequence ¢j t c;(OO) such that ¢i .... 8(

230

STEVEN G. KRANTZ

in the weak-* topology on aG, we obtain an equation relating S(z, () and H(z, 0. In particular, H and S are equal modulo terms which are

less singular. From this fundamental result, many basic mapping properties of S can be determined (see [36] ). The basic construction of Kerzman and Stein can be used in other contexts. Let us turn now to one of these: the Bergman kernel. Fix a domain

n cc en

and define

(Notice that, for {} smoothly bounded, H2 (fl) is a proper subspace of A2({})

-

Exercise.) Define

= JfgdV

n 11£11 =

J

lfl 2 dV 1 12

n for f,g

E A2 (fl).

The basic lemma in this context, sup jf(z)l ~ CKIIfll K

for K CC {}, is easily derived from the mean value property for holomorphic functions. As in Section 1, the abstract Hilbert space theory yields a reproducing kernel for A2 which we call the Bergman kernel. Just like the Szego kernel, the Berman kernel (denoted by the letter K) satisfies K(z,O = K((,z). Thus the associated operator

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

B: f r--

231

J

f(') K(z, ()dV(()

{}

is self-adjoint. It maps onto A2 (by construction) and is idempotent. So B : L 2({}) .... A2 ({}) is Hilbert space projection. A remarkable construction of S. Bergman (see [31]) is as follows: note that, for z

f {}

fixed,

f

K(z,z) = K(z,w) K(w,z)dV(w)

=J1K(z,w)l 2 dV(w) > 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

<; n satisfy

Aut fl such that ¢j(P0)= Pj. Then the holomorphic sectional curvature tensor K for the Bergman metric l

satisfies K(P0)

=

K(¢j(P0)) = K(Pj)-> (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

<; en is smoothly bounded and if

is Coo sufficiently close to the unit ball B then either (i)

n~ B

or (ii) fl

i; B and Aut fl is compact and has a fixed point.

234

STEVEN G. KRANTZ

Proof. Step 1. If

0 i B then Aut 0 is compact. For if Aut 0 is not

compact then a normal families argument [12] implies that for P0 t 0 '3 cpj tAut 0 such that cpj(P0) ..., an. As in the proof of the preceding theorem, it follows that n ~ball. Step 2. Recall the following result of Cartan-Hadamard (see

[24]):

THEOREM. If M is a complete Riemannian manifold of non-positive curvature and if K is a compact group of isometries on M then K has a fixed point.

Now we prove the theorem by denying (i) and proving (ii). Step 3. By a calculation, the ball B has negative (bounded from zero)

Bergman metric curvature. But the stability result (/3) implies that this statement holds for domains 0 which are Coo sufficiently close to B. By Step 1, Aut 0 is compact. So the result follows from (y) and Step 2. o Now we turn to a conjecture of Lu Qi-Keng (see [34]): CONJECTURE. If n ~en is a bounded domain then the Bergman kernel K(z, ') never vanishes on n

X

n.

On the disc and the ball this conjecture is correct; for the ball in en one can calculate (see [31]) that K(z,,) = n! 77 n

1 (1-z·On+l

However it turns out that in C 1 the conjecture is true if and only if 0 is simply connected (see [40]). From this it follows that the conjecture is not always true in en either. lO See this, let e 2

2 {}="disc

X

annulus. Then the uniqueness of the Bergman kernel easily implies that the kernel for 0 is the product of those for the disc and annulus. Now n

=

unj where nj ~ nj+l and each nj is smooth and strictly pseudo-

convex (see [31]). By a theorem of Ramadanov [37], Kn ..... Kn normally. J

235

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

By Hurwitz's theorem [31 ], Kn. vanishes for j large enough. So there J exist smooth strictly pseudoconvex domains with vanishing Bergman kernels. Thus we have the MODIFIED LU QI-KENG CONJEC~URE. If n CC en is smoothly bounded and diffeomorphic to the ball, then Kn never vanishes. Some results about the modified conjecture may now be formulated. Let

us agree to topologize the collection of all smoothly bounded strictly pseudoconvex domains by equipping their defining functions with the C 00 topology. Then we have (see [12 ], [13]): (i)

If

(ii) If

e = tn: Kn 'U

never vanishes I then

=In: Kn is bounded from

Ol

e is closed.

then

'U

is open.

Statement (i) follows from Hurwitz's theorem. Statement (ii) follows from ({3). Facts (i) and (ii), together with the fact that

e,

'U are non-

empty (since both contain the ball), nearly provide a connectedness argument to verify the modified Lu Qi-Keng conjecture. The conjecture was recently resolved in the negative by Boas and by Catlin. Now we turn to a semi-continuity result for automorphism groups: THEOREM (Greene-Krantz (14] ). If n 0 CC en is a smoothly bounded

strongly pseudocorwex domain and it

n

is a sufficiently small smooth

perturbation of no then (i)

Aut n

s;

Aut n 0

subgroup

(ii) 3¢:

n . . no

a diffeomorphism such that Aut n

J

a

1-+

¢oao¢- 1 tAut n 0

is a univalent group homomorphism.

t

Sketch of proof. We may as well suppose that n 0 ball, else the result is straightforward. Then normal families arguments show that, for n

•

sufficiently near n 0 , n 1::, ball. Thus Aut n is compact and, by averaging the Euclidean metric, one can construct a new metric y, smooth

236

STEVEN G. KRANTZ

across all. which is invariant under Aut {}. By patching this metric with the Bergman metric, and modifying it near Isom(y) = and that

y

an,

we can arrange that

is a product metric near

an.

Finally, we construct the metric double M of {} equipped with y. Known theorems [5] about semi-continuity of isometry groups of deformations of a compact Riemannian manifold now give the result. o

-

We introduce our final result by recalling a corollary of the Uniformization Theorem (see [2]): THEOREM. If 0 CC e, P c 0, and the isotropy group lp of Aut{} is

infinite, then

n ~ A.

The generalization of this result to en would require new ideas since, in that context, there is no uniformization theorem. Also, on dimensional grounds, the infinitude of lp is an insufficient hypothesis when n > 1. Instead we have THEOREM (Greene-Krantz [15]). If M is any n dimensional, connected,

non-compact complex manifold, and if lp has a compact subgroup H which acts transitively on real tangent directions at P, then M is biholomorphic to either the ball or en. Idea of proof. First create an H-invariant metric on M by averaging over .H. By a continuity argument, we show that metric balls B(P,r) centered at P are biholomorphic to the unit ball in en. This last is the heart of the argument: It involves analysis of geodesics and equivariance properties of the Bergman the

a equation.

metri~

on B(P,r) and of the canonical solution to

Since (by inspection of the proof), the biholomorphisms

of B(P,r) to B vary continuously with r, and since the biholomorphisms match up as r increases, the conclusion follows. o Let us conclude by briefly reviewing the course we have come. We began by exploiting the many symmetries of the disc to derive an integral

237

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

reproducing formula on the disc. Since generic domains possess few symmetries, we developed two alternate techniques to find integral formulas - via Stokes' theorem and via Hilbert space theory. The latter method has the advantage of being canonical while the former is explicit. We used explicit integral formulas in several complex variables to establish a number of basic results in the theory. Then, using an idea of Kerzman-Stein, we were able to relate the explicit formulas to the canonical ones. Finally, we used this connection between explicit and canonical formulas to return to the question of symmetries of domains. We established results which explain how the automorphism group of a domain

!l'

depends on the geometry of

an.

There are still many open problems in the study of automorphism groups of domains. One of the most compelling is to decide which domains have non-eompact automorphism groups. Another is to relate the dimension of Aut (n) as a Lie group to the rank of the Levi form on

an.

I hope that

the survey presented here wiH inspire some new people to consider these questions. DEPARTMENT OF MATHEMATICS THE PENNSYLVANIA STATE UNIVERSITY UNIVERSITY PARK, PA. 16802

BIBLIOGRAPHY

[1J

L. Boutet de Monvel and J. Sjostrand, Sur Ia Singularite des noyaux de Bergman et Szego, Soc. Mat. de France Asterisque 34-35 (1976), 123-164.

[2]

R. Burckel, An Introduction to Cl~ssical Complex Analysis, Birkhauser, Basel, 1979.

[3]

D. Catlin, Necessary conditions for subellipticity of the J-Neumann problem, Ann. of Math. (2)117(1983), 147-172.

f4]

, Boundary invariants of pseudoconvex domains, to appear.

[5]

D. Ebin, The Manifold of Riemannian metrics, Proc. Symp. in Pure Math., Vol XV (Global Analysis), AMS (1970), 11-40.

[6]

C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65.

238

STEVEN G. KRANTZ

[7]

G. Folland and J. ]. Kohn, The Neumann Problem for the CauchyRiemann Complex, Princeton Univ. Press, Princeton, 1972.

[8]

G. Folland and E. M. Stein, Estimates for the Jb complex and analysis of the Heisenberg group, Comm. Pure Appl. Math. 27(1974), 429-522.

[9]

]. Fornaess, Strictly pseudoconvex domains in convex domains, Am. jour. Math. 98 (1976), 529-569.

[10] B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables, Translations of Mathematical Monographs, American Mathematical Society, Providence, 1963. [11] H. Grauert and I. Lieb, Das Ramirezche Integral und die Gleichung a£ = n im Bereich der Beschrankten Formen, Rice Univ. Studies 56 (197~), 29-50. [12] R. E. Greene and S.G. Krantz, Stability of the Bergman kernel and curvature properties of bounded domains, in Recent Developments in Several Complex Variables, J.E. Fornaess, ed., Princeton Univ. Press, Princeton, 1981. [13]

, Deformations of complex structures, estimates for the of the Bergman kernel, Adv. Math. 43 (1982), 1-86.

[14]

, The automorphism groups of strongly pseudoconvex domains, Math. Ann., 261 (1982), 425-446.

a equation, and stability

[15] ----,Characterization of complex manifolds by the isotropy subgroups of their automorphism groups, preprint. [16] P. Greiner and E. M. Stein, Estimates for the Cf-Neumann Problem, Princeton Univ. Press, 1977. [17] G. M. Henkin, A uniform estim.ate for the solution of the Cf-problem on a Weil region, Uspekhi Math. Nauk. 26 (1971), 221-212 (Russ.). [18]

, Integra I representation of functions holomorphic in strictly pseudoconvex domains and some applications, Mat. Sb. 78(120) (1969), 611-632; Math. U.S.S.R. Sbornik 7(1969), 597-616.

[19] G. M. Henkin and A. Romanov, Exact Holder estimates of solutions of the d equation, lzvestija· Akad. SSSR; Ser. Mat. (1971), 1171-1183, Math. U.S.S.R. Sb. 5 (1971), 1180-1192. [20] L. Hormander, L 2 estimates and existence theorems for the operator, Acta Math. 113 (1965), 89-152.

a

[21] T. Iwinski and M. Skwarczynski, The convergence of Bergman functions for a decreasing sequence of domains, in Approximation Theory, Reidel, Boston, 1972.

239

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

au

[22] N. Kerzman, Holder and LP estimates for solutions of = f on strongly pseudoconvex domains, Comm. Pure Appl. Math. XXIV (1971), 301-380. [23] N. Kerzman and E. M. Stein, The Szego kernel in terms of CauchyFantappiekernels, Duke Math.]. 45(1978), 85-93. [24] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, II, Interscience, New York, 1963, 1969. [25] J. J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds I, Ann. Math. 78(1963), 112-148; II. ibid 79(1964), 450-472.

a

[26] ------·Global regularity for on weakly pseudoconvex manifolds, Trans. Am. Math. Soc., 181 (1973), 273-292. [27]

, Sufficient conditions for subellipticity on weakly pseudoconvex domains, Proc. Nat. Acad. Sci. (USA) 74 (1977), 2214-2216.

[28]

, Subellipticity of the a:.Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math. 142(1979), 79-122.

[29]

, Methods of partial differential equations in complex analysis, Proc. Symp. Pure Math. 30, Part 2 (1977), 215-237.

[30] S. Krantz, Optimal Lipschitz and LP estimates for the equation ali= f on strongly pseudoconvex domains, Math. Ann. 219(1976), 233-260. [31]

, Function Theory of Several Complex Variables, John Wiley and Sons, New York, 1982.

L32]

, Characterization of smooth domains in C by their biholomorphic self-maps, Am. -Math. Monthly (1983), 555-557.

[33] E. Ligocka, The Holder continuity of the Bergman projectioo and proper holomorphic mappings, preprint.

[34] Lu Qi-Keng, On Kahler manifolds with constant curvature, Acta Math. Sinica 16 (1966), 269-281 [Chinese]; Chinese ]. Math. 9(1966), 283-298. [35] N. 0vrelid, Integral representation formulas and LP estimates for the a equation, Math. Scand. 29(1971), 137-160. [36] D. H. Phong and E. M. Stein, Estimates for the Bergman and Szego projections on strongly pseudoconvex domains, Duke Math. ]our. 44(1977). [371 I. Ramadanov, Sur une propriete de la fonction de Bergman, C. R. Acad. Bulgare des Sci. 20(1967), 759-762.

[38] R. M. Range, The Caratheodory metric and holomorphic maps on a class of weakly pseudoconvex domains, Pac. ]our. Math. 78(1978), 173-189.

240

STEVEN G. KRANTZ

[39] N. Sibony, Un exemple de domain pseudoconvexe regulier ou 1'equation u = f n 'ad met pas de solution bornee pur f bournee, Invent. Math. 62 (1980), 235-242. [40] N. Suita and A. Yamada, On the Lu Qi-Keng conjecture, Proc. Am. Math. Soc. 59 (1976), 222-224. [41] B. Wong, Characterizations of the ball in en by its automorphism group, Invent. Math. 41 (1977), 253-257.

VECTOR FIELDS AND NONISOTROPIC METRICS Alexander Nagel * The main object of this paper is to show how nonisotropic metrics constructed from vector fields play an important role in certain recent developments in partial differential equations and several complex variables. As we shall see, these metrics are useful in describing boundary behavior of holomorphic functions in pseudoconvex domains, in estimating the kernel of the Szeg6 projection in some of these domains, and in estimating the size of approximate fundamental solutions to certain nonelliptic, hypoelliptic partial diiferential operators. The exposition is divided into three parts. In the first, we set the stage by recalling certain classical theorems which are models for and which motivate the more recent results. In the second part, we outline the construction of metrics from a given family of vector fields. In the third part, we show how to apply this construction to some examples from several complex variables and partial differential equations, and obtain in this way analogues of the results sketched in part one. It is a pleasure to thank Professor M. T. Cheng, and my other hosts at the University of Peking for their invitation to participate in the Summer Symposium in Analysis in China. It was an honor and a privilege to attend the Symposium, and I am grateful for the very warm hospitality I received. It is also a pleasure to thank E. M. Stein for organizing and directing the

Symposium. All of his efforts are greatly appreciated.

*Research supported in part by an NSF grant at the University of Wisconsin, Madison.

241

242

ALEXANDER NAGEL

Much of this paper is an exposition of joint work with Eli Stein and Steve Wainger, and I am particularly grateful to them for many years of stimulation, encouragement, and collaboration. Part I. Some classical theorems and examples In order to motivate our later discussion, we begin by considering three examples of melrics: the standard Euclidean metric; a nonisotropic but translation invariant metric on Rn; and the translation invariant metric on the Heisenberg group. In the Euclidean case, we see how the balls and metric are involved in Fatou's theorem on nontangential limits of Poisson integrals, and in estimates for the Newtonian potential andrelated singular integral operators. In the other two examples, we see how analogous estimates can be made for kernels related to the heat operator and to the Kohn Laplacian, and how nonisotropic balls on the Heisenberg group are involved in Koranyi 's extension of Fatou's theorem to existence of admissible limits of holomorphic functions. In each of these settings, there is a natura \ly given family of first order linear homogeneous differential operators, or vector fields. In part II of this paper, we shall see that the general construction of metrics applied to these families of vector fields gives back the natural metric in these classical settings. The discussion of results in these examples will be very brief, but references are given for the complete proofs of all the results. §1. The isotropic Euclidean metric and the Laplace operator The standard metric on Rn is defined by

In this example, the important first order operators are just the partial derivatives with respect to the n variables

J:. 1 ,... , ~ n . The Laplace

VECTOR FIELDS AND NONISOTROPIC METRICS

243

operator

is of course just the sum of squares of these first order operators. We first study the role of the Euclidean metric in the solution of the Dirichlet problem for R~+l = {(x 1 , ... , xn,y) ~ (x,y)ly >01. We identify the boundary of R~+l with Rn = Rn x

IOI,

and, given a function f on Rn,

we want a function u(x,y) harmonic in R~+l such that u(x,y)-+ f(x 0) as (x,y)

-+

(x 0 , 0).

For continuous boundary data, the problem is completely solved by the Poisson integral formula. Thus suppose f is continuous on Rn, and f c L 1 (Rn) + L 00 (Rn) . Set

where n+l en=

r(n~l) /17

2

Then: (a) Pf is harmonic on R~+l. (b) Pf extends continuously to the boundary and takes on the boundary values f. (c)

f

Rn

IPf(x,y)IPdx ~ llfllp for 1 ~ p

'S

oo.

(d) Suppose u is harmonic on R~+l and

244

ALEXANDER NAGEL

Then if s > 0 and fs(x)

u(x,s),

=

P(fs)(x,y) = u(x,y+s) . Proofs of these assertions, along with many other of the results discussed here, can be found in Stein [16], Chapter III. Assertions (c) and (d) above suggest a generalization of the ·Dirichlet problem to certain classes of discontinuous boundary functions. For

1

~

p

~ oo,

let hp denote the space of functions u(x,y) harmonic on

R~+l which satisfy

sup y>O

J

lu(x,y)iPdx = llul/p p

h

< oo if p < oo

Rn sup lu(x,y)l = llull 00 Rn+l h

< oo if p ="".

+

Now we are interested in the boundary behavior (along y =0) of functions u in hp, and it is here that the Euclidean metric begins to play an important role for us. To study the boundary behavior, a fundamental concept is that of a nontangential approach region. For a> 0 and x 0

!

Rn

define:

Note that if B(x 0 , 8)

=

lx

t

Rnllx-x 0 1 < ol are the balls defined by the

standard Euclidean metric, then

Thus the nontangential approach regions in R~+l are really defined in terms of the projection 11(x,y) = x of R~+l onto the boundary, the "height function" h(x,y)

= y,

and the family of Euclidean balls on the

boundary. We shall later see that in other examples, natwal approach regions can be defined in essentially the same way.

VECTOR FIELDS AND NONISOTROPIC METRICS We say that a function u(x,y) has a nontangential limit at x 0

245

f

Rn

if and only if for all a > 0, lim u(x,y) exists as (x,y) approaches (x 0 ,0) and (x,y) t ra(xo). In 19~ Fatou [4] proved: THEOREM 1. For 1

~

p

~

"", if u t hp, then u has a nontangential

limit at almost every point of Rn. A standard modern approach to this theorem involves two main estimates. The first involves the Hardy-Littlewood maximal operator. Let f t Lf0 c(Rn) and set Mf(x 0) =sup \B\- 1

J

lf(y)j dy

B

where the supremum is taken over all Euclidean balls B which contain x 0 • The basic estimates for the maximal operator are given in: THEOREM 2 (Hardy and Littlewood). For 1

~

p < oo, there are constants

Ap < oo so that (i)

IIMflip ~ Aplifllp if 1 < p ~ ~

(ii) !lx t RniMf(x) >All ~ A 1A- 1 IIfll 1 if p

=

1.

The very definition of the maximal operator involves the family of Euclidean balls, and the proof of the crucial estimate (ii) depends on a covering lemma for these balls. The second basic estimate needed to prove Fatou's theorem involves the nontangential supremum of a function defined on R~+ 1 . Thus for any

a > 0 and any v(x,y) defined on R~+ 1 set Nav(xo)

=

sup lv(x,y)l. (x,y)tra<x 0 )

For Poisson integrals, this non-tangential supremum is point-wise dominated by the Hardy-Littlewood maximal function of the boundary data:

246

ALEXANDER NAGEL

THEOREM 3 (Hardy and Littlewood). For a > 0 there exists a constant Ca < "" so that if f ( L 1(Rn) + L ""(Rn) and if u(x,y) = Py for all x

l

* f(x),

then

Rn

These are the two quantitative estimates which underlay the qualitative statement of Fatou's theorem. Complete proofs d these results can be found for example in Stein [16], Chapters I and III. However, since we shall appeal to this kind of argument again, we now recall how Fatou's theorem follows fro!D these two theorems. Let p <"" and let u f hp. If s > 0 and if we let fs(x) = u(x,s), then sup

\u(x,y+s)\ = Na[P(fs)](x 0)

(x,y)fra<xo>

Therefore if .\ > 0 \lx 0 fRn\

sup

\u(x,y+s)\ >.\I\

(x,y)lra<x 0 )

1 -< [C a,\ - \\M(fs )\\ p ]P 1 < - [C a Ap.\- \\fs !I p ]P

Since s > 0 was arbitrary, we obtain for any u f hp (1)

VECTOR FIELDS AND NONISOTROPIC METRICS

Now let u

t:

247

hp be real valued, and set Oau(xo) "'lim sup u(x,y) -lim inf u(x,y)

where the limits are taken as (x,y) approaches (x 0 ,0) and (x,y) ( 1a(x 0 ). Then the following facts are easy to verify: (a) Oau(x)

«; 2Nau(x)

(b) Oa(u+V )(x) 'S_ Oau(x) + Oav(x) (c) Oau(x 0 ) = 0 if and only if u has a limit within la(xo) (d) Oau(x)

=0

if u = Pf and f is continuous.

Now let un(x,y) = u (x,y +

=

~)

• Then

0 a (u--u n) < - 2N a (u-u n)

Hence, by inequality (1), we have: llx t: RniOaulx) >..\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
*N

is of class C 2 and all second derivatives again

249

VECTOR FIELDS AND NONISOTROPIC METRICS

satisfy a Holder continuity condition of order a. (See Bers, john, and Schechter [1], page 232.) Proofs of these continuity properties of the operata f ... N

*f

depend on size estimates of the kernel N(x,y) =

N(x-y) which can be written:

(4)

where

o = lx-yl.

Written in this way, these inequalities again make clear

the important role played by the Euclidean metric and the Euclidean balls. This importance can also be seen when we consider certain singular integral operators. We claimed earlier that N

*f

is two orders smoother

than f. Using formula (2), this means that for all i,j,

a2cp

ax.ax. 1

as smooth as carries

!'!.cp

this.

!'!.cp. Thus a2cp

should be

J

we are led to the study of the operator which

to ~ . There are two ways in which we can think of 1 J

Formally differentiating equation (2) we see that

a2cp (x) ~ 1

Y·Y·

J

=

f

(5)

kij(x-y)!'!.cp(y)dy

where kij(y) =en _!_.1 IYI-n for an appropriate constant en IYI 2

i 0. We note

that the kernel kij is not locally integrable at 0, so we must study the integral in equation (5) in the principal value sense. Now the kernel k· ·(x,y) = k··(x-y) satisfies the following estimates in terms of the 1J

1J

Euclidean metric:

250

where 8

ALEXANDER NAGEL

=

\x-yl. It is well known how to combine estimates of this type

with the Calderon-Zygmund decomposition of L 1 functions to prove LP(Rn)

boundedn~ss of the operator A.cp

-+

d2 cp provided that one

~ 1 J

knows that this operator is bounded on L 2 (Rn) (see for example, Stein (16], Chapter 2). Bounded ness on L 2(Rn) follows from the cancellation property

J ~j(x)dx

=

0,

a
¢<~) =

I

e-21Tix·~ cp(x)dx

Rn so that

cp(x)

=

I

e2"ix·~ ~(~)d~

Rn if, for example,

cp

f

C~(Rn). Then:

(A.cp) ~(~)

=

-417 2 1~1 2 ~0

a2r:p A.cp-+ ax.ax. 1

J

in

251

VECTOR FIELDS AND NONISOTROPIC METRICS so

(

::12..1.. 'fJ

u

)

diidXj

~· f 1~1 2

~

~<0 =-l_J (!lrp} (~}.

Our operator is thus given on the Fourier transform side by multiplication by a bounded function. It follows from Plancherel's theorem that the operator is bounded on L 2 (Rn), Thus we have THEOREM 4. For 1 < p < oo there are constants Ap < oo so that if

¢

f

C~(Rn)

§2. Spaces of

hom~eneous type

In our discussion of the Laplace operator, we emphasized the role of the Hardy-Littlewood maximal operator and the Calderon-Zygmund decomposition of L 1 functions. These basic tools of Euclidean harmonic analysis can be used in much more general settings, and can be applied to a variety of interesting non-Euclidean examples. What is really necessary for the theory to work is a measure space together with an appropriate family of balls, and many people have been involved in these generalizations; see for example Koranyi and Vagi [10], Riviere [14], and Coifman and Weiss [3]. Here I want to briefly sketch the approach of Coifman and Weiss to what they call "spaces of homogeneous type.'' A locally compact space X is a space of is a continuous map p: X x X d~t

-4

homo~eneous type

[0, oo) and a non-negative Borel measure

on X so that:

(1} p is a pseudometric; i.e., for all x,y,z (a) p(x,y)

=

if there

0 if and only if x = y

(b) p(x,y) = p(y,x} (c) p(x,z) ~ K~(x,y}+p(y,z}].

f

X,

252

ALEXANDER NAGEL

(2) The measure p. has the "doubling property" relative to the family of falls B(x,B) = !y£X\p(x,y)<8): There is a constant A so that for all

XfX, 5>0

~t(B(x ,25))

:S Ap.(B(x, 8))

.

In order to emphasize the crucial importance of properties (lc) and (2), we now recall the proof of part (ii) of Theorem 2 in this general setting. Thus if f f

Lf c(K,dp.), define 0

Mf(x 0) =sup IBI- 1

J\f(y)\d~t(Y) 8

where the supremum is taken over all "balls" B which contain x 0 • We prove: THEOREM 5.

There is a constant A 1 so that if f

£

L 1(X,dp.),

A> 0, let E = IMf >AI, and let ICE be any compact subset. If x £ I, since Mf(K) > ,\ there is a ball Bx containing x so that Proof. For

IBx 1- 1

J

\f(y)\ dp.(y)

B

> ,\ .

X

The balls lBx lxfi cover :I, and since :I is compact we can find a subcover, which we call B 1 , ···, BN. Suppose Bj = B(xj,8j) so Bj has "radius" 8j. Choose Bi 1 so that inductively choose Bi , ···, Bi 1

(1) B·

1k

k

for all j. We can then

so that

is disjoint from Bi , ···, Bi 1

l\ ~ 5j

k-1

(2) 8. > 8. for all j such that Bl. is disjoint from lk- J In this way we get a finite subsequence Bi , ... , Bi 1

m

which are disjoint.

VECTOR FIELDS AND NONISOTROPIC METRICS

253

Suppose Bj is a ball from our original finite sequence which does not appear in the subsequence. Then there is a first k so that Bik n B; .f

0.

By property (2), oik ~ oj • Let xik be the center rJ. 1\k and xi the center of Bj. Then if z ( Bik

n Bj,

p(xik'xj) ~ K[p(xik'z) + p(z,x;)l < 2Ko 1. • k If y ( Bj, then p(x;,y) < oj, and so p(xik'y) ~ K[p(xik'xj) + p(xj,y)] < K(2K +l)o 1• k Let B: k

=

•

B(xi ,K(2k+l)8i ) . Then B. C B~ , and so 1k k k J

!,

c

N

m

j=l

k=l

U B; c U s;k .

m

Hence I!(!)< :I p.(B7 ) . But by property (2) of spaces of homogeneous - k=l

type

k

*

tt(B 1·

) < A h -

=

l+log 2 K(2K+l)

p.(B-1 ) k

•

A tll(Bik) .

Thus

~-t(!,)~Al ~ #L(Bik)
f B.

lk

lf(y)idy<;A 1i\- 1 lifll 1

254

ALEXANDER NAGEL

since the balls IBi I are disjoint. Since I was an arbitrary compact k subset of E, we obtain the same estimate for the measure of E. §3. The heat operator and a nonisotropic metric We turn now to an example which, though elementary, involves truly nonisotropic phenomena. We consider the heat operator

a2

n

a

L=g-~ - = - - 6 . Ul 4. ax~ at x J

j=l

on Rn+l, where we use coordinates (x 1 ,· .. ,xn,t)

=

(x,t). Unlike the

Laplace operator 6. on Rn, L is not elliptic. Nevertheless there is a remarkable fundamental solution for L. Define n

(4nt)

-:ze-lx \2 /4t

if

t

>0

if

t

<0 .

E(x,t) 0

Then E is C"" on Rn+ 1 \I(O,O)I, and LE = tions. (See Folland [5], Chapter 4.) Thus if

JJ oo

¢(x,t) =

L =- ~ -

f/J E CQ'(R~+ 1 )

n

(471S)-2 e-IYI 2 14s 'L¢<x-y ,t-s)dyds

0

where

o in the sense of distribu-

Rn

f'...x is the formal adjoint of L.

In order to study the operator f .... E

* f,

we would like to obtain size

estimates on the kernel E and its derivatives analogous to those for the

VECTOR FIELDS AND NONISOTROPIC METRICS

255

Newtonian potential N in equation (4). However, if B((x,t), 8) denotes the ball in the standard Euclidean metric, the estimate IE(x,t)l

s c o 21B((x,t), o)i- 1 ~ 0-n+l

with {) = (ixl 2 +t 2)1 12 is false. To see this, let t = lxl 2 with lxl small, so E(x,t) = Clxl-n, while = (lxl 2 + lxl 4) 112 ~ lxl, so

o

5-n+l ~ lxl-n+l. Thus we carinot obtain the appropriate kind of estimate with the Euclidean metric. To remedy this situation, we make the crucial observation that, like the Newtonian potential N, the fundamental solution E possesses a certain homogeneity, though now this homogeneity is non isotropic. For

,\ > 0 define It is easy to check that

We associate to the family of dilations a pseudometric p((x,t),(y,s)) =
The corresponding family of balls Bp((x, t), 5) = l(y ,s)

f

ftn+ljp((x,t), (y ,s)) < ol

are now ellipsoids of size 5 in the directions of x 1, · · ·, xn, and of size 5 2 in the direction of t. Thus

256

ALEXANDER NAGEL

Moreover, it follows from the homogeneity of E that we now have: (a) jE(x,t)j ~ C o 2 jBP((x,t), o)j- 1 (b) I'VxE(x,t)l <:; CoiBP((x,t),o)j- 1

(c)

~~

(7) (x,t)l

:S C \Bp((x,t), o)\- 1

l

a2E . (d) raxi C'&j (x,t)l ~ CjBP((x,t),o)j- 1 where

o = p((O,O), (x,t)).

Thus we obtain estimates for the fundamental solution E(x,t) which are exactly analogous to those we have for N(x), provided we view the operator

£

as acting like a second order operator, so in equation (7c)

we loose two powers of

o rather than one.

We can now use the general theory of spaces of homogeneous type to show that L satisfies certain subelliptic estimates analogous to the elliptic estimates for N given in Theorem 4. For example, one can prove: THEOREM

6. For 1

< p < oo there are constants Ap < oo so that if

¢ ( C 0 (Rn+ 1) then

The operator L is the sum of squares of the first order operators

£:. ···, ~n

minus the operator

~,

which we shall count as an operator

1

of order two. We shall later see how the appropriately weighted vector

fields

£-., .. ·,-£=n , and 1

~ give hack the non isotropic metric

p.

257

VECTOR FIELDS AND NONISOTROPIC METRICS

§4. The

Sie~el upper

half space and the Heisenberg group

Nonisotropic balls and metrics play an important role in the theory of boundary behavior of holomorphic functions in strongly pseudoconvex domains. This is discussed for example in the monograph by Stein [17]. Here we recall what happens in the case of a model strictly pseudoconvex domain, the generalized upper half space, and its boundary, the Heisenberg group. We let n=l(zl,···,zn,zn+l) =

n

(z,zn+l) ( cn+lllmzn+l >

~

lzjl 2 = lzl 2

Recall that n is the image of the unit ball B =

!

l(w 1 ,···,wn+l)l~tlwl<1l J=l

under the biholomorphic map

1

The boundary of n is the set an= l(z,zn+l) ( cn+lllmzn+l = lzl 2 1 =

Thus "Ye can identify

l(z,t+ilzl 2 )1z ( cn,t ( Rl.

an

with en

X

R by associating to the point

(z,t+ilzl 2 )£an thepoint (z,t)£CnxR. Wecanmake cnxR intoa group Hn, the Heisenberg group, by defining (z ,t) · (w ,s) = (z+w, t+s+21m) n

where

=

.~ zjwj. This definition can be motivated and the group

J=l

\

258

ALEXANDER NAGEL

properties verified by the following considerations: to each point h = (w,s) (en

X

R we associate a holomorphic map Th: cn+l .... cn+l by

the formula

Th is clearly holomorphic. Moreover, if hj

= (wj,sj),

j

= 1, 2,

then a

simple calculation shows that

where

From this it follows that Hn is a group, and

Let p(z,zn+l) = Imzn+l- lzl 2 be the "height function" for that (z,zn+l) (

n

if and only if p(z,zn+l)

and only if p(z ,zn+l) = 0. If h = (w ,s)

f

> 0' and (z,zn+l)

n

(an

so if

Hn we compute

p(Th(z,zn+l)) = lm(zn+l +S +ilwl 2 +2i)- lz+wi 2 """ Imzn+l - lz 12

Thus T h carries

n

to itself and dO to itself. If h = (w ,s) ( H0

have identified (w,s) with the point

(w,s+ilwi 2)

(an,

£an

and if h

=

(w,s)

f

Hn, then

we

and this is the

same as Th(O,O). Thus under the identification, Hn acts on follows: if (z,t)

,

an

as

VECTOR FIELDS AND NONISOTROPIC METRICS

Th(z,t)

'=

ThT(z,t)(O,O)

'=

Th·(z,t)(O,O)

'=

(w,s) · (z,t)

=

(z+w, t+s-2Im< z ,w >) .

259

We now make Hn (or an) into a space of homogeneous type. Define d :Hn xHn ... [0, 0<>) by d((z,t); (w ,s)) == [lz-w 14 +It-s +21mCz ,w >! 2 ] 1 14 , or, if we identify Hn with

an,

The function d is invariant under the action of Hn . Thus if h = (u,r)

f

Hn, easy algebraic manipulations show that d(Th(z ,t), Th(w ,s)) = d((z,t), (w ,s)) .

If we define the balls B((w ,s), o) = I (z ,t)jd((z ,t), (w ,s)) < ol then this invariance property means that B((w,s),o) = T(w,s)(B((O,O), o)). We claim that d is a pseudometric. For if t' = (z,t), w = (w,s) and

U' "'

(u,r) are in Hn with d(t',w) < o,d(w,u)<

o

then

o !w-ul < o

!z-w! <

lt-s+2lml


o

ls-r +2Im<w ,u >I < 2 •

260

ALEXANDER NAGEL

Hence \z-ul ~ lz-wl + lw-u\ <28, and lt-r+21m! ~It-s +21m\ + \s-r+2Im<w,u>l + \2Im(--<w,u>)l

~ 28 2 + 21ImI

Therefore d(Z,1i) ~ 2max (d(z,Vi), d( w,1i)) ~ 2[d+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
pair of inequalities: lz-wl < ay 112 lt-s+21ml < ay. As above, lt-s+21ml is essentially the distance from (z,t+ilzl 2 ) to the complex part of the tangent space at w. Thus Aa(w) allows tangential approach to VI of order two in the complex directions at VI,

262

ALEXANDER NAGEL

but requires nontangential approach in the complementary real direction, and so the sets lA a< w)l are essentially the admissible approach regions introduced by Koranyi [9]. The in variance of the balls B( w, o) under the action of Hn , together with the fact that the height function p is invariant under the mappings Th, imply that the approach regions Aa(w) are carried into each other under the action of Hn. Thus it is easy to check that if h = (w ,s) ( Hn, then (z,z 0 +1 ) ( Aa((O,O)) if and only if Th(z,zn+l) ( Aa(w,s+ilwl 2 ). Let F be a function defined on 0, and for ' (dO, set NaF(, 0) =

sup

IF(z,zn+l)l .

(z ,z:n+l )£Aa<(o>

If f

£

L{0 c(afl),

define

Mf('o) =sup IBI- 1

I

lf(,)jda(()

B

where the supremum is taken over all nonisotropic balls B containing

'0,

and da is surface area measure on

an.

We now have the following analogues of Theorems 2 and 3, which were used to prove Fatou's theorem. THEOREM

7.

For

1

:5 p ~

:5 Ap II flip

oo

there are constants Ap < oo so that 1

(i)

IIMfllp

(ii)

II' (ani Mf( ')>>-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

VECTOR FIELDS AND NONISOTROPIC METRICS

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~

<

oo

if

p <

oo

p

an

For 1

COROLLARY.

so that if F (i)

~ p ~""

and a> 0. There are constants Ap,a <""

E Hp(fi)

l!N~ll p<:_Apal\FliH

for l
p (ii) \1, tdU\NaF{,)>AI\ <:_ A1,i- 1\\Fl\H 1 L

'

if

p = 1.

Proof. Put Fe (z ,zn~ 1) = F(z,zn+l +ie). Then FE is continuous on and pleurisubharmonic on 11, so by Theorem 8

Since llFell p :':: l\FliH L

and

p

for 1 ::: p ~"" we see that

n.

264

ALEXANDER NAGEL

I NaF >AI=

U lNaFE>AI

e
so the corollary follows by the monotone convergence theorem and the regularity of the measure da. We can now apply exactly the same argument we used for Fatou's theorem to prove_ THEOREM 9 (Koranyi). If F ( Hp(Q), then F has admissible limits at

almost every point of

an.

It remains to prove Theorem 8. For simplicity, we deal wit~ the case

n =1 . Since the approach regions A. a({.) are carried into each other by the action of Hn , it suffices to prove the estimate at the origin (0,0). We first obtain an estimate on the radial maximal function:

J

271

\u(yteiO,s+it)\dOds dt

0

3y

y

J'" J J 2

0

--y2

2

Iu({te i9,s+it)l dt ds d9 .

y

-

2

Letting r = yt, and interchanging the order of integration, we get

VECTOR FIELDS AND NONISOTROPIC METRICS

f

265

lu(()ido(()

J

B((O, 0), 2

iu(()idu(() .

VY)

Thus, by translation invariance, we see there is a constant A so that for (z,zn+l)

l

n iu(z,zn+l)l

~ AIBl- 1

J

lu(Oldo(()

B

where B =B(77(z,zn+l),2p(z,zn+l) 1 12 ). Now let (z ,zn+l)

l

Aa((O,O)). Then 77(z ,zn+1)

t

B((O,O), ap(z,zn+l ) 1 12 ),

so

But now using the doubling property of the ba Us, it follows that there is a constant A so that if (z,z 0 +1)

lu(z,z 0 +1)1

t

Aa(O,O)

~ A'IBi-l

I B

lu(()ida(()

(8)

266

ALEXANDER NAGEL

where B =B((0,0),(2a-t4)p(z,zn+ 1) 112), and this last average is certainly dominated by A'Mu(O,O). We remark that estimates of type (7) are actually false for Poisson integrals of harmonic functions. The nonisotropic ba Us on iJO are also useful in studying singular integrals and fundamental solution_s. For example, the Cauchy-Szego projection of L 2(ofi) onto H 2(fi) is given by the operator

Sf(z,zn+ 1) = Jf(w,wn+ 1)S((z ,zn+l), (w,wn+l))do

an where

with en,;, 2n- 1 nl/~+ 1 (see Nagel and Stein [11], page 23). In particular, for z = (Z,t) and w ~ (w,s) on

an

we have,

. -( n+1)[(s-t-2Im)-1"\ z-w \2]-(n+1) . S(z,w) = cn(1) Thus \S(z,w)\ ~ \B(z,o)\- 1 where

o = d(z,w).

It is also easy to check

that derivatives of S yield estimates with corresponding negative powers of

o.

Thus S behaves like a singular integral kernel relative to this

family of nonisotropic balls, and from this follows LP and Lipschitz estimates for the operator f ->Sf (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

VECTOR FIELDS AND NONISOTROPIC METRICS

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 <:n. In this J case we shall recover the standard Euclidean metric on Rn.

(A) Let N =-q =n, let

Y;

let dj = 1 for 1 ~ j :Sn and J let dn+l = 2. In this case we shall obtain the nonisotropic metric

(B) Let N =q =n+1, let Yj

=~-,and

on Rn+l appropriate for the study of the heat operator

VECTOR FIELDS AND NONISOTROPIC METRICS

269

n (J2 _a__ ~-.

axn+l

(C) Let N = q

.J-

=

c1xf

2n 4 l and let

Y· = + 2y. J ox, J J

4

l

Ql

~j~n with d. J

a - 2xj ata

=

1

l ~j ~n

y n+j = ay. J y2n+l

i=l

= [Yj,Yn+j] = - 4

!t

with d 2 n+l = 2 .

In this case of course we are dealing with the Heisenberg group, and we shall recover the invariant metric on Hn defined earlier. (D) (An example of Grushin type). Let N = 2, q = 3, and let

with d 1

=

d 2 = 1 and d 3 = 2. This example leads to a metric

appropriate for studying the hypoelliptic operator See Grushin [6a].

£ 2 +X~ a22 • ax 1

ax 2

(E) (This example generalizes (A), (C), and (D).) Let X 1 ,···,Xp be Coo

real vector fields on fi. Let x

=

X <2 >=

IX 1 ,···,XP! I· ··, [X 1'. XJ·] , · · · I

x< 3 > = I· .. , [Xi, [Xj,xk]),

···I, etc.

so that xCk) is a vector whose components are all the commutators of X 1 ,···, Xp, of length k. We say that the vector fields _X 1, .. ·,Xp are of Hormander type m or finite type m if at each X ( n , the components of x span RN.

270

ALEXANDER NAGEL

Let Y 1 ,···,Yq be some enumeration of the components of x
as in (E) of finite type, but this time let d(Xj) = 1 for 1

~

j ~n

while d(Xo) = 2, with appropriate weighting of higher commutators. This example leads to a metric appropriate for studying the generalization of the heat operator X~+Xf +··· + X~:-X 0 • Our object now is to construct natural metrics out of the family of vector fields Y 1, .. ·,Yq. I want to distinguish between two general approaches to this problem. On the one hand, one can define a metric in terms of a "global definition." Here, we let the metric be given as the infimum d some functional over a large class of curves. Then the defining properties of a metric, such as the triangle inequality, are relatively easy to verify, but the local geometry of the corresponding family of balls is hard to understand. On the other hand, we can define a metric in terms of a "local definition''; here we want the metric to be given in terms of an exponential mapping. Now the local geometry is clearer, but it is much more difficult to verify that one really has a metric. We now discuss each of these approaches, and then try to sketch why the two definitions are in fact equivalent.

§5. Global definitions DEFINITION

1. Let x 0 ,x 1 £0, and say p(x 0 ,x 1) < 6 if and only if

there is an absolutely continuous map ¢: [0,1] .... j

=

0, 1 , so that for almost all t c (0,1)

n

with ¢(j) = xj,

271

VECTOR FIELDS AND NONISOTROPIC METRICS q

<{J'(t)

=I, a/t)Y;(¢(t)) j=l

where

Ia; (t)\ < 8

PROPOSITION.

d(Y.) 1 •

~ CC

p is a metric on fi. For every compact set

there are constants C 1 , C 2 so that if x 0 ,x 1

i

fi

~

1

C 1 \xo-x 1 \ where

\x 0 -x 1 \

S p(x 0 ,x 1) S C 2 \x 0 -x 1 \iii.

is the standard Euclidean metric.

We sketch the proof: (1) That p(x,y) = 0 if and only if x = y is clear. (2) p(x,y)

=

p(y,x) since we can replace cp(t) by ¢(1-t).

is a smooth curve joining them, so p(x,y) < oo. (4) The triangle inequality: Suppose x,y,z l fi. Given e > 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) !
= z and (J'(t) =

q

.2 c;(t)Y;(8(t)) where J=1

lc;COI

d(Y .) 1 ~ (d(x,y)+d(y,z)+2e)

•

This shows, since e

~

is arbitrary, that d(x,z)

d(x,y)+d(y,z).

Now let I C Q be compact. Then there is a constant C that if x ,y

t

=

CI so

I, there is a smooth curve ¢ joining x to y with

l¢'(t)\ ~C\x-y\.

Since Y 1 , .. ,Yq span RN at every point, we can

q

write ¢'(t)

=

I c .(t) Y · with \cJ·(t)\ ~ C\¢'(t)\. But then: J J

j=1

- \c/t)\ ~ C\x-y\ = C(\x-y\

1 d(Y .) d(Y .) l

1

J

1

and so p(x,y)

:S C\x-y\m.

Conversely, if x,y £I and p(x,y) =

o,

there is a curve ¢ joining

q

x to y with ¢'(t) = _I aj(t)Yj(¢(t)) and \a/t)\

:S (28)

d(Y .) J

almost

J=1

everywhere. But then

J'

lx-yl "

'(t)dt

0


l

Q and say p 2 (x 0 ,x 1 ) <

there is a Coo curve ¢ : [0 ,1] .. Q with ¢(j) q

¢'(t)

=

o if and only if

xj , j

=

0,1, and

d(Y .)

=_I ajYj(¢(t)) where aj £Rand \aj I < o

l .

Note that in this

J=1

definition, we have somewhat restricted the class of curves over which we take an infimum. However, in general it is not true that p 2 (x,y) is finite for any two points of 0. For example, if 0 is not convex, N

=

q

=n, Yj

=

£:, and 1

dj

=

1, 1 :S j

:S n,

then p 2 (x,y) < oo if and

only if X and y can be joined by a straight line c;ontained in 11

SO

p2

is not even a pseudometric in this case. On the other hand, it is clear that at least in this example, the metric p 2 is locally equivalent to the standard Euclidean metric. Now suppose we are in the case of vector fields IX 1,···,Xpl of finite type m . DEFINITICiN 3. Let Xo,Xl (

n

and say PlXo,xl)

< 0 if and only if

there is an absolutely continuous map ¢: [0,1] -~ n with ¢(j) = xj, q

j

=

0,1, with ¢'(t)

=

_I aj(t)X/¢(t)) and laj(t)l < o almost everywhere. J=l

Note that in this definition, we only allow the derivative ¢'(t) to belong to the span of IX 1(¢(t)),···,Xp(¢(t))l, which may not be all of RN. It is again easy to verify that p 3 satisfies the triangle inequality, but it is not a priori clear that p(x,y) is finite for any two points of {}. This is in fact a consequence of the finite type hypothesis, and was first proved by Caratheodory in 1909 [2]. To get some feeling for why p3 is finite, and to begin to understand the role of commutators, let us consider the following example of Grushin type in R 2 . Let the coordinates be X and y and let xl = x2 = xk

t·

!,

Note that these vectors fail to span

[Xl,X2] = kxk-1

t,

[Xl [Xl'X211

=

R2

along X = 0, but

k(K-1)xk-2

t ,··· ,

and

where the commutator is of length k + 1, so IX 11 X 2 1 are of finite type k + 1.

What points belong to the ball centered at (x 0 ,0) of radius shall consider two special cases: x 0 = 1 and x 0 = 0.

o?

We

274

ALEXANDER NAGEL

Case 1. If we let

and

then this show-s that the points (1 +0,0) and (1, o) belong to the p3 ball centered at (1,0) of radius Euclidean ball of radius

o.

In fact, this ball is essentially the

o centered at

(1,0).

Case 2. Again, starting at (0,0) we can go to (o,O) by using the vector

Jx.

field

but it is not immediately clear how to get from (0,0) to other

points on the y axis. To do this, we need to use a curve which is only piecewise smooth. Let

¢> 1(t)

=

(ot,O)

¢~ 2 = (o, sk+ 1t)

¢> 3 (t)

=

(o-<St, ok+ 1 )

O
so

¢>i = oxt

O
so

¢12 = oxz

0
so

¢13

=

-oxt.

This shows that the p3 distance from (0,0) to (0, sk+l) is at most 3o. In fact the p 3 ball with center (0,0) is an "ellipsoid" of size the

X

o in

direction and size Ok+l in the y direction.

Thus we have given three

possi~le

definitions of a metric or pseudo-

metric in terms of the family of vector fields Y 1,· .. ,Yq. It is also clear from the definitions that we have the following inequalities:

p(x,y)

~

p 3 {x,y) (when p 3 is defined).

In fact, these three quantities are locally equivalent. One can prove that for every x 0

f

0 there is a neighborhood U of x 0 and constants Cl'C 2

sothatforall x,yf0, x/.y, and j=2,3.

275

VECTOR FIELDS AND NONISOTROPIC METRICS

Pj(x,y) 0

< c 1 < p(x,y ) < c 2 < oo

§6. Local definitions In order to begin to see why these pseudometrics are locally equivalent, we need to consider a local definition of metric, and this in turn relies on the notion of an exponential map. Given a point p c {}, we let TPO denote the tangent space to 0 at p. Suppose we are given C 00 yector fields Sl'···, SN defined near p which form a basis for RN at p. Then we can construct a map from a neighborhood of 0 c TPO to a neighborhood of p in 0 as follows: N

every tangent vector -; at p can be uniquely written as N

with (a 1 ,···, aN)

£

RN, and so

I a.S .(p) "' -;

j=l. J J

_I a1.s1. is a smooth vector field defined

J=l

near p. We can flow along the integral curve of this vector field for unit time if IJajl is sufficiently small, and the result is by definition exp

C¥

1 sj) (p), the exponential map of -;.

Given vector fields S 1 ,···,SN we can Klentify TP{} with RN via N

(a 1, ... , aN) ....... I

aJ.SJ.(p), and then the exponential mapping

J=l

introduces a coordinate system centered at p, the so-called canonical coordinates relative to the vecta fields S 1,·· ·,SN. The Jacobian of this exponential map at 0

£

RN is just det (S 1, ... ,SN), the volume of the

"parallelopiped" spanned by S 1' .. ·, SN. It is important to remember however that this exponential map from T PO to 0 depends on the choice of vector fields S 1 ,···,SN. Now we return to the general situation of vector fields Y 1 ,· .. ,Yq in n

c RN.

On each tangent space T xn there is a natural notion of length.

276

If ~

ALEXANDER NAGEL

t

T x{} we say

if and only if q

~ = ~ a.Y.(x) ~ j=l

where Ia j I <

d(Y .)

o

ll

...

J •

Of course this representation of v need not be

unique. Nevertheless questions about the set

are presumably just problems in elementary linear algebra. In giving a local definition of metric, we want to transfer these "balls" in the tangent space at

X

to genuine balls in {1, and this suggests the USe of

an exponential map. The question is: how does one choose an appropriate N-tuple of vector fields in order to construct such a map? To motivate the answer, we first ask a simple question: what is the volume in T xn of the set

I ~INx(v) < o\? This amounts to the following

problem. Let Y 1' ... ,Y q be vectors in RN which span, and consider the map

q

9(al''",aq)

=

~

ajYj.

i=l

What is the volume of the image under 6 of the box

Forany N-tuple l=(i 1 .-··,iN) let d(l)=d(Y. )+ .. ·+d(Y.). and let lt

A1 =det(Yi•'",Yi ). 1

N

lN

VECTOR FIELDS AND NONISOTROPIC METRICS LEMMA.

277

There are universal constants Cl'C 2 so that 0 < C 1 '::

19(Qa)l/; IX 11£ld(I) ~ C 2 < oo. (Here the sum is taken over all N-tuples I.) Proof. For each N-tuple I, the image 9(Qa) contains all vectors of N

the form

! ajYi. where la1.1 < 5

j=ol

d(Y i.>

J

J ,

and this is just the image under

a linear map 'from RN to RN with determinant A1 • Thus

We must now prove the reverse inequality. Pick an N-tuple 10 so that

for all N-tuples

J.

By renumbering, we may assume 10 = (l,···,N).

Since AI -j 0, we can write 0

and using Cramer's rule, we see that

where Ijk is obtained from 10 by replacing Yk by Yj. From our choice of 10 , it now follows that

278

ALEXANDER NAGEL

Now let v l0(Q8 ), so v

q

la-1

~ a.v. with

=

J J

j=l

J

<8

d(Y.)

J.

Then

and

I
q

j=l

d(Y .) d(Y k)-d(Y .)

o

Jo

J

=qo

d(Y k)

j=l

~ (qo)

d(Y )

k

•

Hence

I8CQ8 >1 < q

d(I ) 0 IA 1

~ qmN

I

0

lo

d(I ) 0

IA-11 od(l)

.

I

This lemma suggests the following further definitions. For each x

l

{l

and each N-tuple I = (i 1 , ···,iN) let

Clearly for every

Hence U B1(x,o)C B 2 (x,o). We also have I

since p(x,y) ':: p 2 (x,y).

279

VECTOR FIELDS AND NONISOTROPIC METRlCS

It is now reasonable to conjecture that for each x

l

Q

and each 5 > 0,

if we choose an N-tuple 10 so that

j.\ 1 (x)l8 d(Io> 0·

~ l.\1 (x)! 5d(J)

for all

J,

then for this particular 10

•

B(x, 8) C B1 (x, C8) 0

where C is a constant independent of x and 8. This would of course show that p and p 2 are locally equivalent, and also show that the local and global definitions of metric are equivalent. We give a very brief sketch of a proof. Suppose y there is ~: [0,1] _. 0 with ~(0)

=

£

B(x, 8). Then

x, ~(1) = y and

q

~'(t)

=I bj(t)Yj(~(t)) j=l

with lbj(t)j ~ 8

d(Y .) 1

almost everywhere. Now assume without loss that

10 = (1, .. · ,N) and let (ul' ... ,uN) denote canonical coordinates near x relative to the exponential map using IYl'· .. ,YNI. Then the curve ~(t) is given in canonical coordinates by (u 1 (t), .. ·,uN(t)) and our object is to show that there is a uniform constant C with luj(1)1 < (C8)

J4 1

u-(1) J

=

u-(1)- u-(0) = 1 J

ut

[u-(t)]dt J

0

J~ bk(t)Yk(~(t))(uj)(t)dt. 1

=

q

0

Now since IY l ' ... ,y Nl span near x we can write

d(Y .) 1 •

But

280

ALEXANDER NAGEL

where akf

l

Coo . Th us q

I

bk(t)a~(c/>(t))(Yfuj)(t) dt

.

k=l We now need to make two kinds of estimates: (1) lat(Cj)(t))l (2) IYeuj(t)l

~ Co d(Y frd(Y k>

~ c8d-d.

Combined with the estimate lbk(t)l ~ 8

d(Y ) d(Y .) 1 k , this gives luj(l)l ~Co

which is what we want. Now estimates (1) are obtained by using Cramer's rule to write at(y) =A 1(y)/A 1 (y) as in our earlier lemma. At x, our choice of 10 0 d(Yf)-d(Y k) shows that JA./x>I/IA- 1 (x)l ~ 8 , and in order to obtain a 0

similar estimate for nearby points y, we expand at in a Taylor series in canonical coordinates about x. The proof of (2) is more complicated, and involves the CampbellHausdorff formula. However, note that at x, Yfuj = ojf, which is the right estimate there. Complete proofs can be found in [13). As a corollary of our analysis of the metric p, we can estimate the volume of the balls B(x, 8). For every compact

Icc fl

there are con-

stants cl and c2 so that

In particular, the balls B(x, 8) satisfy the doubling property, and so fl with metric p and Lebesgue measure is a space of homogeneous type.

VECTOR FIELDS AND NONISOTROPIC METRICS

281

Part III. Applications In this part of the paper, we show how the constructions outlined in part II can be applied in partial differential equations and several complex variables.

§7. Estimates for approximate fundamental solutions We can t&e the metrics constructed from vector fields to obtain estimates for the integral kernels of parametricies of certain hypoelliptic differential operators. We briefly describe the setting. In 1967 Hormander [7] obtained a far reaching generalization of Kohn's result on the hypoellipticity of Db. Suppose Xl'··· ,Xp are

n c RN

coo

real vector fields on

of finite type m. Hormander showed that the second order opera-

tors Xi+···+ X~ or X 1 +X~+··· +X~ are hypoelliptic. As in the work of Folland and Stein [6] on Db, Rothschild and Stein [15] want to construct parametricies for these more general operators by inverting model operators on appropriate nilpotent groups. There is in general no nilpotent group of dimension N which works, but Rothschild and Stein overcome this difficulty by proving their "lifting theorem." Given Xl'···,Xp on

n c RN

(with coordinates (xl'···,xN)) of type m, they show that one

can find additional variables (tl'···,ts)

f

Rs and form new vector fields

These new vector fields will again be of type m on a neighborhood of

n X I ol c n X Rs ,

and in addition they are free up to step m' so that one

can model Xl'···,Xp by vector fields on a free nilpotent group of step m with p generators. Rothschild and Stein are able to construct a parametrix for Xi+···+ X~ on 0 x Rs, given by a kernel k((x,t), (y ,s)) which comes from a homogeneous kernel on the nilpotent group. In particular, this kernel satisfies

282

ALEXANDER NAGEL

5 CS 2 \El((x,t),S)\- 1

\k((x,t),(y,s))\ where

C>

=p((x,t),(y,s)), and

p

is the metric constructed from the vector

fields xl,···,Xp. They then define a restriction operator

Rk(x,y) =

J

k((x,O), (y ,s))r,b(s)ds

Rs where r,b

f

C 00 (Rs) is supported near s

=

0, and if k((x, t), (y ,s )) is the

parametrix for X~+···+ X~, then D(x,y) = Rk(x,y) is a parametrix for

X~+ · · · + X~ . Using the properties of the metric

p

constructed from

Xl'···,Xp one can now prove, for example: THEOREM 11. Let D(x,y) be the Rothschild-Stein parametriK for Xi+···+ X~. Then, if N _? 3

and if N _? 2,

\Xl'X 1 ,···,X 1 _D(x,y)l

5 cs 2 -i\B(x,S)r 1

J

where 8

= p(x,y).

Thus we get estimates analogous to those for the Newtonian potential and Kohn Laplacian discussed in part I. These estimates have also been obtained by Sanchez-Calle [1Sa]. One also gets estimates of this type for the Rothschild-Stein parametrix of the operator X1 +X~+··· +X~, in analogy with the estimates in part I for the heat operator. For details of the argument. see [13].

§8. Nonisotropic metrics on domains of finite type We now apply the theory of me tries to boundaries of domains in en+ 1 . (Some of these results were announced in [12] .) Thus let p c C00 (Cn+l),

283

VECTOR FIELDS AND NONISOTROPIC METRICS

with dp f. 0 when p

so that

n

=

0 and let

is a domain with smooth boundary. If we fix a point

'0

l

an'

then near ( 0 we can find n linearly independent tangential holomorphic ap , vector fielfls Ll'···, Ln. For example, if ~ ((0 ) r 0, we can take azn+l

We write Lj

=

~ (Xr iXn+j), 1 ~ j ~ n, so that Xl'···, X2 n are Coo

real vector fields defined near ( 0 on an . Since the real dimension of an is 2n + 1 ' near we can find an additional real tangential vector

'0

field T, so that {X 1 ,···,X 2 n,TI is a basis for the real tangent space to

an

near ( 0 • Again if dzap (( 0 ) /. 0, we can let n+l

The notion of type of a point was introduced by Kohn [8]. If (

an

l

is near ( 0 , ( is of type !!! if every commutator of Xl'···,X 2 n of length at most m - 1 at ( lies in the span of IX 1' ··· ,X 2 n I, while some commutator of length m does not lie in this span, and hence has nonzero T component. One easily checks that the type of a point does not depend on the particular choice of the vector fields L 1 ,···,Ln and T, but is really a biholomorphic invariant. We say that the domain {} is of finite type m if every point (

l

an

is of type ~ m. In this case of course, the vector fields 1Xl'···,X 2 nl, viewed as being defined on an open set in R 2 n+l , are of finite type as defined earlier. In particular, we then have a metric defined on

an,

various equivalent descriptions of the associated family of balls.

and

284

ALEXANDER NAGEL

Our first object is to show that we can define an equivalent family of balls on

an

in terms of a single exponential map using the vector fields

{Xl'···,X 2 n,TI. For each finite sequence i1' .. ·,ik of integers with 1

'S ij :S 2n, we can write the commutator

where ,\i 1, ... , ik ( C""'(!l) (near ( 0 ), since !Xl'···,X 2 n,TI is a basis for the tangent spaces near ( 0 • Let ~k be the ideal in C""{!l) generated by the functions {,\il'··· ,iel with

e::: k.

It is easy to check that this

ideal does not depend on the choice of the vector fields L 1 , · · ·, Ln or T. Also

so in particular, Xik noting that X1. [Xik

+1

+1

.

(,\i ···ik) t ~k+ 1 . One can verify this claim by

1····· 1 k+l

1

=,\. l····,ln+l . T' 1

and also X 1·

1'

•••

i

'k+l

, Xi ... ik], and then expanding this commutator. 1'

,

DEFINITION. For (

E

0!} near ( 0 , set

and m

A((, B)=

I

Aj(()Bj

j=2

where the first sum is over all the generators of gk. Note that Ak(() is a function whose size measures how much T component the commutators of Xl'···,X 2 n of length :S k can have. In particular, since aG is of type m, then Am(() ~ llo > 0.

VECTOR FIELDS AND NONISOTROPIC METRICS

285

We are now in a position to define balls in terms of a single exponential map. For ( (

an

8((,8)

set

·{o•a!llo • ex{~a;X;+yTJ ((), whe"

la;l < 8,

1$ j $ 2n, and

IYI < A((,8)}.

Thus B((, 5) is the image of a box of size () in the "complex directions" given by Xl'···,X 2 n, and of size A{(,()) in the complementary real direction. Our first result is then: THEOREM 12.

The "balls" B((, B) are equivalent to the balls 8((, 5)

defined in part II in terms of the vector fields 1Xl'···,X 2 nl; i.e., there is a constant C so that B((,5)C 8{(,CB) and 8((,5)C B((,C5). We sketch only part of the proof. Let 11 ( B{(,S), so that

m

with lajl <

.

o, IYI < A((,o) = ~ Aj(()ol.

Since Aj{()

=

Il\ 1 , ... ,if(()i,

it follows that

where lbjl < ()j and Ij is a j-tuple of integers. Hence

where

f3 1. f J'

( R and X1. is the jth order commutator defined earlier. J

Hence we can join ( to 11 with a curve ¢(t) (the integral curve of 2n

q

I a-X-+yT) with ¢'(t) = j~l ajYj and lajl
J J

., ( 8((, co).

d(Y .) 1 •

Thus

286

ALEXANDER NAGEL

The opposite inclusion is more technical, involving ideas used in the proof of the equivalence of the families of balls introduced in part

n

and we omit the details. A major ingredient is the use of Taylor series expansions of functions in canonical coordinates. If f is a smooth function defined near ( , and if

~(a1,···,a 2 n,y) f(exp(~ajxj+YT) (()) =

then g has a formal Taylor series at the origin, and it is given by: 00

g(al'···,a 2n•Y)"-

~ ~!(~ajXj+YT)kfl 0 k=O

where (IajXj+yT)k is a kth order differential operator. (See Rothschild-Stein [15] for further details.) To give an idea of how this formula is used, we can easily see that the function 11 ... A('q,o) is essentially constant on B((,o); i.e. there is a constant

c

so that for ., ( a((, a),

A typical term in IA(fl, o)l is okiAi ... ik(7J)I. We can expand 1,

,

Ai ... ik (n) ., about "'"' in canonical coordinates. Any term involving 1, , applying T to this Ai 1, ... , ik is certainly of the right size since IYI
1,

•••

,

ik with one

of the vector fields x1 ,···, x2n, this gives us an element of gk+l, so again we get the correct estimate.

§9. Balls in terms of a polarization We now want to find a description of the balls on the boundary of a domain 0

c cn+l

directly in terms of inequalities involving the defining

function for D. As before, let

287

VECTOR FIELDS AND NONISOTROPIC METRICS

0 == lz c en+ 1 jp(z) < Ol where p: cn+l ... R is coo and dp p is a function R : cn+l

(a) R(z,z)

=

X

cn+l ... c

-I 0 when p == 0. A polarization of of cia ss

c""

satisfying:

p(z)

(b) dzR(z,w) vanishes to infinite order on the diagonal z = w. (c) R(z,w)"- R(w,z) vanishes to infinite order on the diagonal z

=

w.

Polarizations always exist, and are unique up to functions vanishing to infinite order on z == w. When p is a polynomial or is real analytic, it is easy to construct a polarization. H

then

is a polarization which is holomorphic in z and antiholomorphic in w , and in all that follows, we shall be dealing with polynomial p. We now define a new family of balls on the boundary. For w c a1l and

o> 0

set B#(w,o)

=

{z

can llz-wl
can, Vw == lzccn+ 1 jR(z,w)==OI is a holomorphic hypersurface tangent to an at w . Thus B# (w, o) is essentially the set of points z c an within Euclidean distance o of w, and within Euclidean Note that for w

distance A(w, o) of Vw. Let us quickly calculate what all this means for the Siegel upper half space discussed in part I, where

p(z) ==

.I

J==l

lzjl 2 - Imzn+l. Then

n

R(z,w) ==I zjwj- ;i (zn+l-wn+l).

j==l

288

ALEXANDER NAGEL

Also, on the Heisenberg group [Xj, Xn+j] = -4T so A(w, 5) ~5 2 for all w

l

afi . Thus in this case B 1 (w ,8)

o{ z «lllllz-wl <8,1 ~ •;"; -1, (z_, -Wn+lll <8'}

and it is easy to check that this defines essentially the same balls as we did earlier in part I. We now want to show that the balls B'(w,S) are equivalent to the balls B(w, S) defined in terms of the exponential mapping. For simplicity we will do this only in the special case

where

cJ>: C ... R

is a polynomial of degree m. Here the defining

function is

It is easy to check that if

-

a

.acJ>

a

L = - - 2 t - ( z 1) iJZ1 iJZ1 iJZ2

then

L globally spans the space of tangential antiholomorphic vector

fields. We also set

We want to investigate the geometry of the balls near a fixed boundary point, say (0, ic/>(0)). Since all our families of balls are invariant under biholomorphic mappings, we can choose a special coordinate system to study the geometry. Let 1/J(z) = c/>(z) - c/>(0) - 2 Re

I,-,\J. -. aicJ> (0) zj' azJ

( m

j=t

VECTOR FIELDS AND NONISOTROPIC METRICS

289

If we make the change of variables:

z; = z 2

[¢(0)-t- 2

- i

I~ &~ (O)zjl

J

j=l J. azl

•

our original domain now has the form

and moreover 1/J now satisfies

~1/J

aj.p

=-.

1/J(O) = - . (0)

azl

azJ

(0) = 0

We shall work in this coordinate system near (0,0) We identify

an

with

cXR

l

a1l.

so that (z' t+i!/J(x)) corresponds to (z ,t)

0

Then our basic vector fields are: L

a 1-~<>a =dZ+ dz" z CJt'

-L

a 1.(jjl<>a =azaz z CJt

We can calculate the various functions ,\ 1. I ... 1-k using the vector fields L and L (instead of Re L, Im L ). Thus:

[L

so the ideal

'

g2 is generated

[J = -2i ft a aza'Z" (1t a2.p

by - - or

azaz

tl.p.

Thus

290

ALEXANDER NAGEL

Ak(zlt)

=

I, a+/31

Iikaaz--/3 aly,

(z)l·

To investigate the exponential map exp (aL+aL+yT) (0) with a y £

R,

£

C

I

we must find a cuiVe

with

And so

exp(aLML+yT)(O) +,y+i

f (•:

(as)-

a~ (as~ d~

0

We can estimate the integral by expanding a(NI and fN, about 0. Since

.

z

~ (0) =

ozl

()z

0 1 < j < m all the terms will be dominated by the correspond-

-

1

ing Aj(O), so

J(a: a:: (as) 1

(as)-

ds <S I2 A 2 (0) + ··• + lalmAm(O)

0 = A(O.Ial).

291

VECTOR FIELDS AND NONISOTROPIC METRICS

In particular, we see that the image under the exponential map of the box

l(a,y) l C x Rllal < 8, jy) < A(0,8)! is essentially

l(z,t) l CxRjjz) < 8, It)< A(O,S)! .

•

On the other hand, the polarization of our defining function is

and

are equivalent to the inequalities

lti
§10. Estimating pleurisubharmonic functions We continue our study of domains of the form

but we now make the additional hypothesis that

A¢~

0; i.e. ¢

is a

subharmonic polynomial of degree m, which is not harmonic. Our object is to obtain an analogue of Theorem 8 in part I. We will prove:

292

ALEXANDER NAGEL

THEOREM.

Suppose u is continuous on {} and pleurisubharmonic on

{}. Let ": {} .... dO be the projection onto the boundary. Then if (zl,z2)

l

n iu(z 1 ,z 2 )1

~ CIBI-1

J

lu
B

where B

= B(77(z ~ ,z 2), A8)

and A(77(Z 1 ,z 2 ), 8) = Im z 2 - ¢(z 1). Here A, C

are constants independent of u and (z 1 ,z 2 ). As before, by making a holomorphic change of variables, we can a¢ iJ¢ assume ¢{0) = ~ (0) = (0) = 0, 1 :S j ~ m, and it is enough to

--=uz j

j

estimate u at the point (0, iy). We begin as we did in part 1:

J

lu(O,iy)l < ....i.. rry2

iu(O,s+it)l ds dt .

{8)

Is 12+1 t-y l 2< 0 and a function G((), continuous for

1(1 :S 1 ,

holomorphic on

1(1 < 1 ,

so that the holomorphic map

( _, (8(,G(())

has the required properties. This means we want

and G{O) = s +it .

Let ¢e;(ei()) ~ ¢{8ei0). Let P[¢al be the Poisson integral of ¢a, and let Q[¢al be the conjugate Poisson integral, with Q[¢al (0) = 0. Then

VECTOR FIELDS AND NONISOTROPIC METRICS

293

has the required properties provided that 8 is chosen so that

(9)

But by Green's theorem

so the function t(8)

=

1 2 271 f0 " 1/l(S eiO)dO is a monotone increasing func-

tion of 8. Thus given t, there is a unique 8 = B(t) so that equation (9) holds. We thus obtain

f" J J

y/2

lu(O),iy)l

<-

2-

"2Y2

-y /2

3y/2

°

iu(S(t)eiiJ,

y /2

We want to make the change of variables r = 8(t). Let

A(S) =

JI

~2

x2+y2~()2

so that t'(8) = ~ BA(S). We get:

!11/J(x,y)dxdy

294

ALEXANDER NAGEL

y/2 lu(O,iy)j

~ ~2

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 <"" o< c 1 ':: A(o, o);cS 2 A(5) ~ c2 < ""

(iii) IQrbs<ei 6)i

':: C2 t(l>)

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<-Cya(y)2

JJ J \sl
0

lu(rei6,s+irb(rei6)\rdrdOds.

0

But finally, rdr clOds is essentially surface area measure, and on the surface af! we are integrating over a region centered at (0,0) of size y in the "real" s direction, and of size a(y) in the "complex" directions. But this is exactly our nonisotropic ball B(O, l>) where A(O, 5)

= y

and

since \B(O,o)\ ~ yo(y) 2 , we have shown: there are constants A and C so that

VECTOR FIELDS AND NONISOTROPIC METRICS

J

~ CjB(O,A8)r 1

ju(O,iy)\

295

\u((}jda((}

B(O,A8) where i\{0, 8) = y . Thus we have managed to estimate u along the "radius" from {0,0). Just as in part I, we now easily extend this result to obtain tlte theorem. We finally turn to the proof of the lemma. The inequalities t(8)

S

CA(O, 8) and 8 2 A(8) ~ CA(O, 8) follow easily by expanding c/J or !'!c/J in a Taylor series about 0. The main content of the lemma is contained in the opposite inequalities, and follows from a homogeneity and compactness argument. We claim for every integer m 2' 2 there is a constant Am so that if cf>(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

VECTOR FIELDS AND NONISOTROPIC METRICS

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

VECTOR FIELDS AND NONISOTROPIC METRICS

and

"'L dXa .[dYa b ,(x) dta] +

=

+

1

it is easy to check that L(f o
ker~el

where b'"(x)

of

> 0.

m

.

Our finite type hypothesis is now _I lb(J)(x)l ~ #Lo

> 0,

J=2

and we want to obtain estimates in terms of balls defined by the vector fields

!

If u

and

~ + b'(x) ~ in R3.

= u(x,y ,t) is a reasonable function we define partial Fourier

transforms by

~u

=

u(x.71.r) =

ff

e- 21Ti(y7f+-tr>u(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)\r
301

VECTOR FIELDS AND NONISOTROPIC METRICS

L2(e4TT(7Jx+Tb(x))dx) contains the constants. Let

P7J,T

of L2(e4TT(7Jx+rb(x))dx) onto the constants, if (77,r) (

be the projection

2, and let

P77 ,r = 0

otherwise. Let

be defined ~y

where ~. 7 (x) = g(x,7J,r). It is then easy to check that

P

is an orthogonal

projection whose range is precisely the null space of 3x on L 2(e4TT(J7x+rb(x)) dx d77 dr). Thus if we set

then P is the orthogonal projection of L 2 (dxdydt) onto the null space of L. Now if (71,r) ( 2

p

- < g,l >1

TJ,Tg- <1,1>

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

VECTOR FIELDS AND NONISOTROPIC METRICS

Then replacing r by r +}, and 'T/ by 'T/ + rb'(~} it follows that F(.\+it,r) =

J

00

t :+.

-~

.

e21Ti'Tiid'T/

~•'(})-<• {<+})+"(})]'

-00

Let G(r)

= rrb'(~}

- rb {r+}} + rb(~} . Then G'(r) = rb'(~)-

rb'{r+~}

Since G(O) = G'(O) = 0, we have

Hence 2171 F(A+it,r) = e

GJ~) + itb'{~)~

L-,2

2

~

foo e21Ti'T/td'T/ -oo Joo e 41Tf.,r-r l I~J. ben(~) rj] dr -----------

2

2

--00

Now choose 11 = 11(.\,r) so that

mlb(j) {}}rlljl 2 _

I

j=2

.,

-1

J.

and in the last integral, make the change of variables r

->

11r, 'T/

-+

~ 'T/.

304

ALEXANDER NAGEL

Then

F(A.+it, r) = e

2TTT

r2b(~) 2 + itb{~)] 2 p.-2

t

J

00

27Ti7J(t)ay 41rf71r- I a.ri] dr e L 2

f" e

-oo

J

-00

m

where a· ~! b(j) {~} rllj and hence l\a .\ 2 J j! 2 , 2 J

=

1.

We now make two observations. First, in terms of size,

This is clear from the definition of 11.· Second, the collection of functions

m

~

.

where a = (a 2 , ... , am), !\aj\ 2 = 1, and r -• ~a/ is convex, is a compact set of functions in the Schwartz class S(R). Thus 2TTT

F(A+it,r) = e

[2b(~) + itb'(~\1 2 2/J

IJ.(A, r)-l

oi(;}).

From this, one can make estimates on the size of F(A+it, r) and its derivatives. Finally we have

J 00

S((x,y,t); (r,s,u)) =

0

e- 2 TTT[b(x)+b{r)+i(t-u)]F(x+r+i(y-s),r)dr

VECTOR FIELDS AND NONISOTROPIC METRICS

305

and we can use the estimates on F to estimate S. A consequence is, for example: IS((x,y,t); (r,s,u))i ~ CIB((x,y,t),B)I- 1 where 8 is the nonisotropic distance between (x,y,t) and (r,s,u). ALEXANDER NAGEL DEPARTWNT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WISCONSIN

REFERENCES

[1]

Bers, L., John, F., and Schechter, M., Partial Differential Equations, Interscience Publishers, John Wiley and Sons, Inc., New York 1964.

[2]

Caratheodory, C., "Untersuchungen iiber die Grundlagen der Thermodynamik," Math. Ann. 67 (1909), 355-386.

[3]

Coifman, R. R., and Weiss, G., Analyse harmonique non-commutative sur certains espaces homog'enes, Lecture Notes in Math. tt242, Springer-Verlag, 1971.

[4]

Fatou, P., "Series trigonometriques et series de Taylor," Acta Math. 30 (1906), 335-400.

[4a] Fefferman, C., and Phong, D. H., "Subelliptic eigenvalue problems" in Proceedings of the Conference on Harmonic Analysis in Honor of Antoni Zygmund, 590-606, Wadsworth Math. Series, 1981. [5]

Folland, G. B., Introduction to Partial Differential Equations, Mathematical Notes Series, 1117, Princeton University Press, Princeton, N. J. 1976.

[Sa] Folland, G., and Hung, H. T., "Non-isotropic Lipschitz spaces" in Harmonic Analysis in Euclidean Spaces, Part 2, 391-394; Amer. Math. Soc., Providence, 1979. [6]

Folland, G. B., and Stein, E. M., "Estimates for the ~ complex and analysis on the Heisenberg group," Comm. Pure Appl. "Math. 27 (1974), 429-522.

[6a] Grush in, V. V ., "On a class of hypoelliptic pseudo-differential operators degenerate on a sub-manifold," Math. USSR Sbornik 13 (1971 ), 155-185. [7]

H~rmander, L., "Hypoelliptic second order differential equations," Acta Math. 119(1967), 147-171.

[8]

Kohn, J. J ., "Boundary behavior of on weakly pseudoconvex manifolds of dimension two," J. Diff. Geom. 6(1972), 523-542.

a

306

ALEXANDER NAGEL

[Sa] Kohn, J.J., "Boundaries of Complex Manifolds," in Proceedings of the Conference on Complex Analysis, MiMeapolis, 1964; 81-94; Springer-Verlag, New York, 1965. [9}

Koranyi, A., "Harmonic functions on Hermetian hyperbolic space," Trans. Am. Math. Soc. 135 (1969), 507-516.

[10] Koranyi, A., and Vagi, S., "Singular integrals in homogeneous SJ:IQces and some problems of classical analysis," Ann. Scuola Norm. Sup. Pisa 25 (1971 ), 575-648. [10a]Krantz, S., Fun~tion theory of several complex variables, John Wiley and Sons, New York, 1982. [11} Nagel, A., and Stein, E. M., Lectures on Pseudo-differential Operators, Mathematical Notes Series, 1124, Princeton University Press, Princeton, N. J. 1979. [121 Nagel, A., Stein, E. M., and Wainger, S., "Boundary behavior of functions holomorphic in domains of finite type," Proc. Natl. Acad. Sci. USA, 78(1981), 6596-6599. [131

"Balls and metrics defined by vector fields I: Basic properties" to appear in Acta Math.

[14] Riviere, N., "Singular integrals and multiplier operators," Ark. for Mat. 9 (1971 ), 243-278. [15] Rothschild, L. P., and Stein, E. M., "Hypoelliptic differential operators and nilpotent groups," Acta Math. 137 (1976), 247-320. [1Sa1Sanchez-Calle, A., "Fundamental solutions and geometry of the sum of squares of vector fields," lnventiones Math. 78, 143-160 (1984). [161 Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J. 1970. [171

, Boundary behavior of holomorphic functions of several complex variables, Mathematical Notes Series, ltll, Princeton Univ. Press, Princeton, N.J. 1972.

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS E. M. Stein

• Introduction Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject. Besides the obvious fact that the Fourier transform is itself an oscillatory integral par excellence, one needs only bear in mind the occurrence of Bessel functions in the original work of Fourier (1822), the study of asymptotics related to such functions in the early works of Airy (1838), Stokes (1850), and Lipschitz (1859), Riemann's use in 1854* of the method of "stationary phase" in finding the asymptotics of certain Fourier transforms, and the application of all these ideas to number theory, initiated in the first quarter of our century by Voronoi (1904), Hardy (1915), van der Corput (1922) and others. Given this long history it is an interesting fact that only relatively recently (1967) did one realize the possibility of restriction theorems for the Fourier transform, and that the relation of the above asymptotics to differentiation theory had to wait another ten years to come to light! The purpose of these lectures is to survey part of this theory and at the same time to describe some new results. We have found it convenient to divide our discussion into oscillatory integrals of the "first kind," and those of the "second kind." The main difference between the two is that for the first kind we are studying the behavior of only one function as the parameter increases to infinity, while for the second kind we are dealing

*In Section XIII of his paper on trigonometric series.

307

308

E. M. STEIN

with the boundedness properties of an operator which carries an oscillatory factor in its kernel. However this distinction need not be taken literally since sometimes these different types merge. We begin by considering the more-or-less standard facts about oscillatory integrals of the first kind, first in one dimension and then in n dimensions. Next as a first application we deal with some estimates of the Fourier transform of smooth surface-carried measures in Rn. This leads us naturally to restriction theorems. (Differentiation theorems, which are another application, are not dealt with here; but these are the subject of Wainger's lectures [3].) Next we discuss oscillatory integrals (of the first kind) arising in the theory of Hilbert transform along curves and their generalizations. We then tum to oscillatory integrals of the second kind suggested by twisted convolution on the Heisenberg group and the theory of Radon singular inte&rals. Finally we return to restriction theorems and the oscillatory integrals of the second kind they give rise to, which operators are closely related to Bochner-Riesz summability. *

1. Oscitlatocy integrals of the first kind, n

~

1

We are interested in the behavior for large positive X of the integral

J b

I(X)

=

eiA
a

where

t/1

is

complex-valued and smooth; often, but not always, one assumes that

t/1

has compact support in (a,b).

*The reader will note that there are several related topics not touched on in this survey. Chief among them is the subject of oscillatory integra Is arising in the solution of hyperbolic equations and their generalizations -the class of "Fourier integral operators." For an elegant introduction to that subject see [1], Chapter 4.

309

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

The basic facts about I(.\) can be presented in terms of three principles. (a) Localization: The asymptotic behavior of I(.\) is determined by those points where rp'(x) = 0 1 (assuming that r/J has compact support in (a,b) ). More precisely,

PROPOS~TION 1. Suppose r/J Then I(.\) = O(>.. -N), as >.. ...

oo

£

C';(a,b) 1 and rp'(x)

I 0 for

x m [a,b].

for every N ~ 0.

The proof is very simple. Let D denote the differential operator Df = - -1- dd i)..rp '(x) x

1

and let

!o

denote its transpose,

.!or=,:. (-f-). ax i)..rp'

Then clearly DN(ei>..rp) = ei>..rp for every N, and integration by parts shows that

Thus clearly II(>..)!

:S AN>..-N, and the proposition is proved.

(b) Scaling: Suppose we only know that and we wish to obtain an estimate for

Jab

dkrp~x)' ? 1

for some fixed k, dx eiArp(x)dx which is independent

of a and b . Then a simple scaling argument shows that the only possible estimate for the integral is 0(.\ - 1 /k). That this is indeed the case goes back to van der Corput. PROPOSITION

2. Suppose rp is real-valued and smooth in [a b]. If

lrf> (k)(x)\ 2': 1 , then (1.1)

1

f a

holds when

b

ei.Arp(x)dx :Sck>..-1/k

310

E. M. STEIN

(i)

k~2

(ii) or k = 1 , if in addition it is assumed that ¢'(x) is monotonic.

Proof. Let us show (ii) first. We have

The boundary terms are majorized by 2/A, while b

J b

J

eiA¢t_D(1)dx =

a

eiA¢

J-iA dx _! (_l) dx ¢'

a

11


-A

a

=}

f

b

d~ (~-) dx

a

by the montonicity of ¢'. The last expression equals

tl¢'~b)- ¢'~a)

I,

which is dominated by 2/A. This gives the desired conclusion with c 1 = 4. We now prove (i) by induction on k. Let us suppose thaf the case k is known, and assume (taking complex conjugates if necessary) that ¢(k+l>(x)?: 1. Let x = c be the point in [a,b] where lf/>(k)(x)l takes its minimum value. If ¢ (k)(c)

=

0, then outside the interval (c- 8, c+ 5),

we have that lr/>(k)(x)l ~ 8, (and of course ¢'(x) is monotonic when k = 1 ). Write

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

311

By the previous case

f

c~

eiA¢> dx

'S ck/(.\o)l /k

.

'S ck/(.\0)1 /k

.

a

Similarly

•

f

b

!

eiA¢> dxl'

c-tO Clearly however

Thus

2ck - (.\o)l /k

<---+20.

If q,(k)(c)

I 0, and so c

is one of the end-points of [a,b], a similar

o is an upper bound to the integral. follows by taking o = ,\ -l /k+l , which

argument shows that ck/(.\0) 1 /k + either situation the case k + 1

proves (1.1) with ck+l = 2ck+2. COROLLARY.

Under the assumptions on ¢> in Proposition 2, we can

conclude that

J b

(1.2)

a

eiAif>(x)r/l(x)dx

ln

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E. M. STEIN

This follows from (1.1) by integrating by parts an estimate of the form

J x

eiAcf>(x)dx <S c0-l/k, for a

<S_

x <S_ b.

a

(c) Asymptotics: We already know that the behavior of

fab eiAcf>t/Jdx

is

determined by those points x 0 , where cf>'(x 0) = 0, (the "critical" points of cf> ). Assuming that the support of t/1 is so small that it contains only one critical point of cf> , the character of the asymptotic expansion then depends on the smallest k so that cf>(k)(x 0 ) I 0, and is given in terms of powers of A in a way which is consistent with Proposition 2. PROPOSITION

3. Suppose cf> is real and smooth, k? 2, and

cf>(x 0 ) = cf>'(x 0)··· = ct>(k)(x 0 ) I o. If t/1 ( c~ and the support of 1/J is a sufficiently small neighborhood of x0 , then

(1.3)

I(.\)=

J

ei.\.cf>(x)I/J(x)dx "'A-l/k

00

~ aj.\.-j/k,

in the sense that lor any non-negative integers N and r

REMARK.

When k is even, then aj = 0 for j odd.

We shall give a proof of the proposition when k

=

2. There are three

steps.

Step 1. This is the observation that 00

(1.4)

I

-00

00

eiAx2/e-x2dx "- A-l/2-V2 Icsf)A-j j=O

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

where

e. is a non-negative integer; in the case

e is

313

odd the integral

vanishes. In fact the left-side of (1.4) is ("'

e-(I-iA)x 2 xedx which by

a change of variables equals (1-iA)-'h-V 2

e-x 2xedx. However, when

-oo

("' -00

..\ > 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

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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.

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

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-
for some

\a I = k,

there is a unit vector .;

=

.;(x 0 ), so that

Moreover since we can assume that the ck+l norm of ~ is bounded we can also conclude that

\(g, \7:)~(x)l ? akf2

whenever x c B(x 0 ), where

B is the ball centered at x of fixed radius. (The radius of B can be taken to be a small multiple of the ck+l norm of ~ .) Next choose an

318

E. M. STEIN

appropriate covering of Rn by such balls of fixed radius, and a corresponding partition of unity, 1 = I 1/j(x), with 0 ~ 7lj ~ 1 , ~ l\771j I :S bk , and each 71j supported in one of our balls. So

JeiAcP.pdx

=

~ Jei>..
To estimate feiA
JeiA

e determined as aboye, choose a

coordinate system so that x 1 lies along

e.

Then

For the inner integral we invoke (1.2) giving us an estimate of the form

A final integration in the other variables then leads to (2.1). REMARK.

Let us note that in R2 if cf,(x)

=

x 1x2

,

the above proposition

gives no better than a decrease of order A- 1 12 , while the asymptotics of the proposition below shows that the true order is A- 1 • Let us go back for the moment to the case of one dimension. If 1/J has a critical point at x 0 , and 1/J' does not vanish of infinite order at x 0 , then after a smooth change of variables simple canonical form

'¢,

with ;f;(x)

= ± xk


no analogue of this in higher dimensions, except for k = 1 and in a special case corresponding to k = 2. To the asymptotics of the latter situation we now turn.

319

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

Suppose cb has a critical point at x 0 . If the symmetric n x n matrix

~a~~:)~

is invertible, then the critical point is said to be non-

degenerate. It is an easy matter to see by the use of Taylor's expansion that if x 0 is a non-degenerate critical point, then in fact it is an isolated critical point.

6. Suppose cb(x 0 ) = 0, and cf> 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;~<;xo~ ~ .

The proof of the proposition follows closely the same pattern as that of Proposition 3. First, let Q(x) denote the unit quadratic form given by Q(x)=x 21 +x 22 + .. ·+x 2m -x 2m+l -· .. -x 2n' where O<_m<_n, with m fixed. The analogue of (1.4) is

f

(2.3) J

Rn

ei.\Q(x)e-lxl2xfdx "".\-n;z-lf\!2

i ~0

cj(m,f)A-j

320

E. M. STEIN

e=
1 ···,

zero. To prove (2 .3) write it as a product

-'12--f·/2

n

TI (1 /A+ i)

and expand the function

J

, (for linge A ) in a power

series in 1/A. The analogue of (1.5) is the statement that

J

~At. -n/ 2 -lfl 12 ;

eii.Q(x)xf71(x) dx

(2.4)

if 71

l

C~(Rn) .

Rn To prove it we con.sider the two-sided cones rj defined by rj

'=

{xllx;l 2

Thensince

~in lxl 2 }

1

and the smaller cases

.u r 19=Rn, wecanfindfunctions

J=l

rj = {xllxl ~ klxl 2} 0 1., j=1

1 "· 1

n, each

homogeneous of degree 0 1 and C00 away from the origin, so that n

1 "' ~ UJ·(X) I X = 0 I with i=l

u.

supported in rJ. . Then we can Write

J

Jeii.Q(x)xf-q(x)dx =

~ JeiAQ(x)xf71(x)Oj(x)dx. J

In the cone rj one uses integration by parts via DJ.ei>.Q(x) = eiAQ(x)

"th

Wl

This, together with the fact lx·l > 1 J

v2n

D (f) j'

l af =~ ~... Ili\X j

OJ<

lxl in rJ.' and

J

I (tDJ.)NOJ.(x)l ~

•

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

C:

321

lx!- 2 N, allows one to conclude the proof of (2.4) in analogy with that

.\

of (1.5). A similar argument also show that whenever ~

l

S

and ~ vanishes

near the origin, then

(2.5)

Jei.\Q(x)f(x)dx

=

0(.\-N), for every N?: 0.

We then combine (2.3), (2.4) and (2.5) as before to obtain the asymptotic formula (2.2) in the special case when ¢(x) = Q(x). To pass to the general case one is then fortunate to be able to appeal to the change of variables guaranteed by Morse's lemma: Since ¢(x~ = 0, ('V¢ )(x 0 ) = 0, and the critical point is assumed to be non-degenerate, there exists a diffeomorphism of a small neighbOrhood of x 0 in the x-space to the y-space, under which ¢

is transformed Yi +Y~ +·" + y~­

y~+l- .. ·- y~. Observe that the index m is the same as that of the

form corresponding to

~~ a~~:?f

.

References. For a proof of Morse's lemma see Milnor

[20], §2.

§3. Fourier transforms of surface-carried measures Let S denote a smooth m-dimensional sub-manifold of Rn (not necessarily closed). We let da denote the measure on S induced by the Lebesgue measure on Rn, and fix a function now the finite Borel measure dJl

=

t/1 in

C~(Rn). Consider

ljF(x) da on Rn , which is of course

carried on S. The problem we wish to deal with is that of finding estimates at infinity of the Fourier transform of Jl, i.e. d~(t). We shall consider two cases of this problem. (1) Suppose first dim S

= n -1,

and S has non-zero Gaussian curvature

at each point. By this we mean the following: Let x 0 be any point of S,

322

E. M. STEIN

and consider a rotation and translation of the underlying Rn so that the point x 0 is moved to the origin, and the tangent plane of S at x 0 becomes the hyperplane xn"' 0. Then near the origin (i.e. near x0 ) the surface S can be given as a graph xn and r/>(0) = 0, (Vr/>)(0)

~ ~ '~~)~k -. J

~

==

= cp(x 1 ,··· ,xn-t)

C 00

f

,

0. Now consider the (n-1) x (n-1) matrix

Its eigenvalues llt, ... , lln-l are called the princi-

- s at

t<J,k
pal curvatures of

(

= det

x 0 , and their product 0

.

Gaussian curvature at x . THEOREM

with rf>

a t ax.ax k

2r/>(O))

is the

J

1. Suppose S is a smooth hypersurface in Rn, with non-

vanishing Gaussian curvature at each point, and let djJ.

= rpda

as above.

Then (3.1)

Proof. It would be convenient (in applying Proposition 6 above) to change notation momentarily by taking n to be n + 1 . Now by the compactness of the support of r/1 (and since when the surface is given as a graph xn+l

= cp(x 1 , ... , xn),

then da

=

J1 + IVci>\ 2 dx1' ... , dxn) we can reduce

matters to showing that

f

(3.2)

eiA<{>(x,1J)if/(x)dx

< AA-n/2

Rn where <{>(x,rr) = x 1 rr 1 +X 271 2 +· .. + Xn71n + rf>(x 1 , .. ·,xn)71n+l, with n+l = .\(711' .. ·, 71n+l), .I 71~ = 1 , and A > 0; also (0) == (V )(0)

e

det

(£c2~:}t 0,

J=l

=

J

and the support of

if/

0,

is a sufficiently small neighbor-

hood of the origin. We divide consideration of the unit vector 71 into three cases.

323

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

1° when 71 is sufficiently close to the "north pole" 71N = (0,···,0,1), 2° when Tf is sufficiently close to the "south pole" 71N

=

(0, .. ·,0,-1);

3° the complementary set on the unit sphere. Let us consider 1° first. The function cf,(x, 71) has the property that f.Yx¢)(0, TIN)= 0. We want to see that for each Tl sufficiently close to TfN there is a (unique) x ~ x(TI), so that ('ilx)rf>(x(TI). 71)

=

0. The latter

is a series of n equations and one can find the desired solution by the implicit function theorem, which requires that we check that the Jacobian determinant det (('Vx'Yxc/J) (0, TIN))

=I 0; but this of course is our assump-

tion of non-vanishing curvature. Notice that if the .,.,-neighborhood of TIN is sufficiently small, then det (

a 2~w~ l1))-:/ 0,

so we can invoke

Proposition 6 (with x 0 = x(71) ), as long as the support of

if;

is suffi-

ciently small. This proves (3. 2) when 71 is in region 1°. The proof when Tl is in 2° is the same. So we now come to the region 3°. Since c/J(x, Tl) ~ n

I XjTfj + ¢(x)71n+l, then 'ilx¢(x,71) = (l11'"'•l1n) + l1n+t('il¢)(x). How-

i=l

ever (TJ~ + ... + 71;) 1 12 ~ c > 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

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

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

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

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
'S Allfll

L

2 , all real t ,

An application of a known convexity property of operators (see [28]) then shows that IIT0(f)ll

p'

Lo

~ A!lfll

P

Lo

, with p 0 =

2n + 2 , and the proof of n+3

Theorem 3 is complete. REMARKS:

(i)

For hypersurfaces with non-zero Gaussian curvature this theorem is the best possible, only insofar as it is of the form (4.1) with

q:::: 2. If q is not required to be 2 or greater, then it may be conjectured that a restriction theorem holds for such hypersurfaces in the wider range 1

~

p < 2n/(n + 1). This is known to be true

when n = 2 (see also §7 below). (ii)

For hypersurfaces for which only k principal curvatures are nonvanishing, Greenleaf [12] has shown that then the corresponding results hold with 1 ~ p ~

2lk: ~), giving an extension of

Theorem 3. (iii) In the case of dim (S) = 1 (i.e. in the case of a curve) there are a series of results extending our knowledge of the case n

=

alluded to above. For further details one should consult the references cited below.

2

329

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

It would of course be of interest to know what are the exponents p

and q (if any) for which the restriction holds if we are dealing with a given sub-manifold S. This problem is highlighted by the fact quite general sub-manifolds S (those which are of finite type in the sense described in §3) have the restriction property: THEOREM

4. Suppose S is a smooth m-dimensional sub-manifold of Rn

of finite type. Then there exists a p 0 ~ p 0(S), 1 < p 0 , so that S has the LP restriction property (4.1) with q = 2, and 1 ~ p ~ p 0 . (In fact if the type of S is k, we can take p 0 COROLLARY.

"'

2nk/(2nk-1 .)

Suppose S is real-analytic and does not lie in any affine

hyperplane. Then S has the LP restriction property lor 1 ~ p ~ p 0 , for some p0 > 1 .

Proof. As we saw above, it suffices to prove (4.3). However Tf = f * K, and K(x) "'d~(-x), Theorem 2 tells us that IK(x)l ~Aixl-llk. So according to the theorem of fractional integration, (see [26], Chapter V), we therefore get (4.3) with ~

Po

=

p1 -

o

ff-,

relation among exponents is the same as

Further

biblio~raphic

where a "' n - 1/k, and this 2nk Po "' 2 nk _1 ·

Q.E.D.

remarks. The initial restriction theorem dates from

1967 but was unpublished. The sharp result for n = 2 was observed by C. Fefferman and the author and can be found essentially in [9]; see also Zygmund [33]. Further results are in Thomas [30], [31], Strichartz [29], Prestini [24], Christ [4], and Drury [7]. 5.

Oscillatory

inte~rals

of the first kind related to

sin~ular inte~rals

A key oscillatory integral used in the theory of Hilbert transforms along curves is the following: 00

(5.1)

p.v.

J

-00

e

iPa(t)

dt

-, t

330

E. M. STEIN d

.

where Pa(t) is a real polynomial in t of degree d, Pa(t) = _I a 1-tJ. It J=O

was proved by Wainger and the author in [27], that the integral is bounded with a bound depending only on the degree d and independent of the coefficients a 0 ,al'···,ad. The relevance of such integrals can be better understood by consulting Wainger's lectures [32]. We shall be i_nterested here in giving an n-:.dimensional generalization of this result. We formulate it as follows. Let K(x) be a homogeneous function of degree -n; suppose also that IK(x)l ~ Alxl-n (i.e. K is bounded on the unit sphere); moreover, we assume the usual cancellation property: fJx'l=l K(x')du(x')=O. We let P(x) =

THEOREM

I a xa be any real polynomial of degree d. lal~d a

5: p.v.

(5.2)

I

eiP(x)K(x)dx

~Ad

Rn with the bound Ad that depends only on K and d, and not on the coefficients aa.

Nagel and Wainger observed that if K were odd, one could prove (5.2) from the one-dimensiona 1 form (5 .1) by the method of rotations (passage to polar coordinates). To deal with the general case we need two lemmas. d

.

Let Pa(t) = _I a 1.tl denote a real polynomial on R 1 , and write also J=l

d

.

Pb(t)= I b/. j=l

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

331

LEMMA 1: 00

(5.3)

J

(e

iPaCt)

-e

iPb(t) dt )t

e

LEMMA 2. Let P(x)

=

I aaxa be a homogeneous polynomial of ]a]=d

degree d on Rn. Write

mp

=I

jP(x')] du(x') .

]x'\=1 Then,

(5.4)

J

Ilog (I~[) Idu(x') ~ Bd ,

\x'\=1 with Bd independent of P. * One can if one wishes give an elementary (but complicated) proof of Lemma 2. It may however be more interesting to obtain it as a consequence of a general property of polynomials in Rn related to the class of functions of bounded mean oscillation. This property can be stated as follows. It is very well known that the function log lxl is in B.M.O., and this is usually the first example discussed in that theory. It is surprising therefore that the following natural generalization seems to be been overlooked.

*Of course mp> 0.

in writing (5.4) we assume that P is not identically zero, i.e.

332

E. M. STEIN

THEOREM

6. Let P(x) be any polynomial of degree

< d in Rn. Then

log IP(x)l is in B.M.O. and in addition

:S Bel ,

II log IP(x)III 8 M 0

where Bel depends only on d, and not otherwise on P. The proofs of Theorem 6 and Lemma 2 will be given in an appendix. We now pass to the proof of Lemma 1. We prove it by induction on d. The case d

=1, i.e. the estimate

is classical. Let us now assume (5.3) for polynomials of degree d- 1, and observe that the estimates (5.3) we wish to prove are unchanged if we replace t by 8t, 8

J 0. Thus we may assume that bd

=

1 and lad I :S 1.

Now write

I

~

fJ

00

(e iP a(t) -e iPb(t) ) -dt t

~

e'

+

e'

1

and we treat these two integrals separately. (If e' > 1 , we have only

fE": ,

and that integral is estimate like {"' . ) Let us consider the second 1

integral. It equals

J~ eiPa(t) ~tt- J~ e iPb(t) dtt ' 1

1

Now since bd = 1, we see that (d/dt)d Pb(t)

=

d!, and hence

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

f

iPb(t) dt

oo

t

e

333

~ cd,

1

by the corollary of Proposition 2 in §1. Next, by a change of variables t ... \ad\- 1 ;d t, the integral

J oo

e

iPa(t) dt

-

t

becomes

where PaCt) is a polynomial of degree d , with td having coefficient one. Again

1

while

since bd = 1. Next

334

E. M. STEIN

with d-1

d-1

Qa(t) = I ajtj , Qb(t) = I j=1

bjtj,

j=1

since IPa(t)-Qa(t)l ~ ltl and \Pb(t)-Qb(t)\ ~ \t\. However, by induction hypothesis (using (5.3) for e' = e , and e =1 , and d - 1 ),

·

Gathering all these terms together then proves (5.3). Armed with Lemmas 1 and 2 we can now prove Theorem 5. We may

I a xa has no constant term, and using polar la\~d a d . coordinates x = tx', t > 0, lx'l = 1 , we write P(x) = I P 1-(x') t1 , assume that P(x) =

j=1

where Pj(x') are restrictions to the unit sphere of homogeneous polynomials of degree j. Let us also set mj = Ijx'l= 1 IPj(x')l du(x), and write K(x)

= cnO(x'),

with 0 bounded and Ijx'l= 1 0(x')du(x') = O.

Then to prove (5.2) it suffices to show that

J

eiP(x)K(x)dx

~ Ad ,

e 1 ~1x\~e 2

with Ad independent of e1 , e2 and P • The above integral can be written as

I, I (

lx

1=1

~

e e I"IP j(x ' ) tJ. dtt O(x')du(x') . 2

e1

Since 0 has vanishing mean-value this integral may be rewritten as

335

OSCIJ.. LATORY INTEGRALS IN FOURIER ANALYSIS

However by Lemma 1 the inner integral is bounded by d Ad ( 1 + ~

j'

log

lp (x')l'I) jmj

and so an appeal to Lemma 2 shows that

proving (5.2) and the theorem.

6. Oscillatory integrals of the second kind: an example related to the

.

Heisenberg group To motivate the interest in this example we recall the definition of the Heisenberg group Hm. The underlying space of Hm is em Hm

= l(z,t)l,

X

R' i.e.

with z £em, t £ R; the multiplication here is (z,t)(w,s) =

j;l zjwj' with m

(Z+w, t+s+), where = 2 Im

z = (zj), w

= (wj).

Now on the Heisenberg group one can consider two types of dilations and their corresponding quasi-distances. The first are the usual dilations (z,t) ... (pz,pt), p > 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

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

(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<.\= B( · , ·) , and F = f. Then the operators T A have the form

(Tf)(x) =

(6.4)

J

K(x-y)eiB(x,y)f(y)dy.

Rn We shall suppose B is a real bilinear form, but we shall not suppose that B is necessarily anti-symmetric nor that K is homogeneous of degree -n. THEOREM

7. Suppose K is homogeneous of degree

-IJ.,

0 :S

IJ.

:S n ,

smooth away from the origin, and with vanishing mean-value when IJ. = n. (a) If B is non-degenerate, then the operator T given by (6.4) is

bounded on L 2 (Rn) to itself, for 0 :S IJ. tor is bounded on LP(Rn) to itself if

:S n; when 1 < p < oo, the opera-

I!_-!.I < .!!:.._ • 2 P - 2n

(b) If we drop the assumption that B is non-degenerate but require that IJ. =

n, then T is bounded on LP(Rn) to itself for 1 < p < oo. The

bound of T can then be taken to be independent of B. We shall give only the highlights of the proof, leaving the details, further variants, and applications to the papers cited below. Let us consider first the L 2 part of assertion (a) when n/2 < IJ.

:S n. Suppose

T/

*For further details see Mauceri, Picardello and Ricci [19] and Geller and Stein [1 o].

E. M. STEIN

338 is a C~ function, with 17(x)

=

1 for lxl ~ 1/2 1 and 17(x) = 0, for

lxl ~ 1. We write T = T0 +T 00 , where T0 is defined as in (6.4), but with K replaced by K 0 = 71K 1 and T 00 with K replaced by K = (1-11) K. Observe first that since K 0 (x-y) is supported where lx-yl ~ 1, estimating T0 (£ )(x) in the ball lxl

~

1 involves only f(y) in the ball

IYI ~ 2. We claim

I

(6.5)

IT0 (f)(x)l 2 dx

~A

\xl9

I

if(y)\ 2 dy.

IYI9

In fact when lxl ~ 11

'S c

J

lx-yi-IL+llf(y)\ dy

I

lx-yl
n ) proves (6.5). While operators of the type (6.4) are not translation invariant they do

satisfy

with rh(f )(x) = f(x-h). Applying this to T0 gives the following generalization of (6.5)

I lx-h\9

IT0 (f)(x)i 2 dx

~A

I \y-h\<2

\f(y)\ 2 dy,

OSCILL/a'ORY INTEGRALS IN FOURIER ANALYSIS

339

and an integration in h shows that as a consequence

J

\T0 (f)(x)\ 2 dx-:; A2n

Rn

J

\f(y)\ 2 dy.

Rn

We now turn to the proof of

J

\T.,i(x)\ 2 dx-:; A

Rn

J

\f(x)\ 2 dx .

Rn

This will be done by proving the corresponding result for the operator T!,T"". The kernel L of this operator is given by

L(x,y) =Je-iB(z,x-y)K 00(z-x)K 00(z-y)dz.

Now since K"" is in L 2 (Rn) (here the assumption n/2 < IJ. is used), Schwarz's inequality implies \L(x,y)\

'S A .

We next integrate by parts in the definition of L(x,y), using the fact that (Dz)Ne-iB(z,x-y) = e-iB(z,x-y), where Dz = i(a,\]z)/\x-y\, with a=

s- 1 (

x-y), and B denotes the matrix so that B(x,y) = (Bx,y).

lx-y\

The result is \L(x,y)\

'S ANix-y\-N, for every N ~ 0, and hence

(6.7)

This shows that L is the kernel of a bounded operator on L 2 proving the boundedness of T!, T"" and thus of T"" . The proofs of the L 2 boundedness when 0 'S IJ.

'S n/2 (in part (a) of the theorem), and the

L 2 boundedness when IJ. = n but when B is not assumed to be nondegenerate, are refinements of the above argument.

340

E. M. STEIN

Let us now describe the main idea in proving the LP inequalities stated in (a) and (b) above. We shall need a generalization of BMO (and of H 1 ) which may be of interest in its own right. Suppose E = leQI is a mapping from the collection of cubes Q in Rn to complex-valued functions on Rn so that

where X' Q denotes the characteristic function of the cube Q . Let us define on "E-atom" to be a function a so that for some cube Q (i)

a is supported in Q

(ii)

\a(x)l :S

1/IQI

(iii) Ja(x)eQ(x)dx = 0. The space

Hk

and I l.\jl <

is then given by lf\f =I ,\jaj, with each aj an E atom,

ool.

In a similar vein the function f~ will be defined as

f~(x) =sup ...!...

(6.8)

xtQ

IQI

J

If-

f~l dx ,

Q

where

and we take BMOE = lfif~

L ""1.

f

Some of the basic facts about the standard H 1 and BMO spaces* go through for

Hk

and BMOE, and sometimes these come free of charge.

One such case is the following assertion: Suppose f

< "" , and

f~

f

LP . Then f

f

f

Lp 0 , 1

LP and

"'The standard situation arises of course when eQ"'X'Q• a 11 Q .

~ p :S

Po

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

341

(6.9) To prove this we need only observe that (if!)# ~ 2f~, and use the result (see [10]) for the standard tl function. The point of all of this is that for operators of the form (6.4), there is a naturally associated H~ and BMOE theory, and it is given by choosing (6.10) where cQ is the center of the cube Q. The basic step in the LP theory (besides an appropriate interpolation which goes via (6.9)), is the proof that when 11. = n our operator T maps L"" to BMOE. Let us give the proof in the case (a). We may assume that l!fll "" ~ 1 , and L

suppose first that Q is a cube centered at the origin. Then we have to show that there exists a constant YQ, so that

(6.11)

The corresponding inequality for a cube centered at another point, say cQ, then follows from the translation formula (6.6), (and this is the reason for defining eQ as we do). Turning to (6.11), the argument is not exactly the same as in the standard case (see e.g. Coifman's lectures [5] or [9] ), since we must split f into three parts to take into account the oscillations of eiB(x,y). Suppose Q = Q8 , has side-lenghts 8, then write f = f 1 + f 2 + f 3 , where f 1 = f in Q 28 , f 1 = 0 elsewhere, f2

=

f in ~Q28

n Q8-1'

f3

=

f in ~ 2 8

n cQ8 _ 1 ,

f2

=

f3

0 elsewhere, =

0 elsewhere.

342

E. M. STEIN

(Note that f 2 occurs only when 5 S v?./2 . ) We have F = T(f) = F1 + F2 + F 3 , where Fj = T(fj). For F1 we make the usual estimate, using the fact that T is bounded on L 2 . Next observe that IK(x-y)eiB(x,y)_ K(-y)J S cS

if x

£

Qs and y £ ~ 28 . Thus if

{f

YQ

=

[--1- _1_]' jyjn+1

1Yin+1

~28

jyjn-1

{K(-y)f 2(y)dy, we get that for

-dy - +8

jF2 (x)-yQJ
+

f

-dy- }
Jyjn- 1

-

Qs- 1

Finally

-- Fl3+ F23' For F~ we again make the standard estimates, and for F~ we use Plancherel's formula (which we may since we have assumed that B(x,y) is non-degenerate). The result is

JjF~(x)j 2 dx J IF~(x)j 2 dx I S

Q

=A

\K(-y)f/y)J 2dy S A1

I

Rn

Combining these estimates proves (6.11), and hence the fact that T takes L oo to BMOE .

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

343

We shall now state a generalization of this theorem which also indudes the oscillatory integral result given in Theorem 5 (in §5). Suppose P(x,y) is a real polynomial on RnxRn of total degree d. Consider the operator (Tf)(x)

(6.12)

=

p.v.

J

eiP(x,y)K(x-y)f(y)dy

Rn where K is homogeneous of degree - n, smooth away from the origin and with vanishing mean-value. THEOREM

8. The operator

given by (6.12) is bounded on L 2 (Rn) to

T

itself, with a bound that can be taken to depend only on K and the degree d of P , and is otherwise independent of P .

This is a recent result obtained jointly with F. Ricci. The proof is based in part on a combination of ideas used in the proof Theorems 5 and 7. This result has also many variants, and we now state some of these: (i)

One may also show that the operators (6.12) are bounded on LP,

1
Given p, with 1 < p < oo, then there is an e =e(p,d), so that if K is homogeneous of degree -fl., n - e ~ fl.

~

n , T is still bounded on

LP. However now the bounds may depend on P, and in addition one must assume that P(x,y) is not of the term P(x,y) = P0 (x) + P1(y). (iii) One can replace K(x-y) in (6.12) by a more general "CalderonZygmund kernel" K(x,y), a distribution for which the operator when P

=0

is bounded in L 2

,

and which in addition is a function (when

x ~ y) which satisfies jK(x,y)j ~ Alx-yj-n, l'ilxK(x,y)j + l'ilyK(x,y)j

'S Alx-yj-n-1. References. For the detailed proof of Theorem 7. other variants, and applications to the author [22], [23].

aNeumann problem see the papers of Phong and the

344

E. M. STEIN

7. Further oscillatory integrals related to restriction theorems and Bochner-Riesz summability

We have seen that if S is a hypersurface in Rn with non-vanishing Gaussian curvature, then

Je~x·e-

t/J(x) du(x)

=

ocle-r
e-

00

s whenever

t/1 £ C;', and this is a typical oscillatory integral of the first

kind. We may pass to an oscillatory integral of the second kind when we replace the C;' function t/J by an L r function f, and consider the resulting linear operator on f. The resulting operator, as is easy to observe, is in fact the dual to the restriction operator considered in §4. Hence by Theorem 3 we can state that operator is in fact bounded from L 2 (da) to LP,(Rn), where p' is the dual exponent to 2n + 2 . We shall n+3 now describe the sharper result in this setting that can be obtained for n = 2.

:S t :S 1, lying in R 2 , with and having non-vanishing curvature, i.e. IY2(t)\ ~ c > 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

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

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,

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

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:

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

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

~

<_S

p

3, if 1 ~ p

<4 ~

2.

REMARKS:

(1) When cl>(t,O=t 1 g 1 +t 2 g 2 +···+tn_ 1 gn_ 1 +¢(t 1 ,···,tn- 1)·gn, and rvci>) (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
(TA)(F)(t);

f

e-iA
t t Rn- 1 •

Rn We can calculate -

I

T~(F)T~(F)dt

Rn-1 and write as

I

KA(~.71)F(0F(71)d~d71 ,

RnxRn with

(7 .15)

KA(~,71)

o:

J

eiA(
Rn-1 It suffices therefore to see that KA is the kernel of a bounded operator

from Lr(Rn) to e(Rn), with norm not exceeding M- 2 n/r'. Because of our assumptions on
~ on Rn x Rn so that the following holds: we will write x (t,xn) with t = (x 1 ,···,xn_ 1 )

f

f

Rn, as

Rn-l. The ~ we can construct will

satisfy: (i) ~(x,O ~ ~(t,O +
OSCILLATORY JJ.IlTEGRALS IN FOURIER ANALYSIS

351

In fact 9 t 9 ~«<> already has rank n -1 by assumption (7 .12a), and so we need only choose «1> 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
(8.1)

(

_1 IQ\

J

IP(x)l 2

d~

1/2


Q holds for all P The space

l

I

IP(x)l dx

Q

Pd, and all cubes Q.

Pd

is invariant under translations and dilations, and so a

moment's reflection shows that to prove (8.1) for all P to prove it for Q

~

l

Pd, it suffices

Q 0 , the unit cube centered at the origin. However

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

(f.

Qo

353

IP(x)l 2 dx) 1 12 and f. IP(x)ldx are two (equivalent) norms on the Qo

finite-dimensional space Pd, so (8.1) holds for Q = Q 0 , and then for general Q. Now it is well known (see e.g. [6]) that a function which satisfies a "reverse Holder" inequality belongs to the weight space A00 • Examining the proof of this fact one obtains an r = r(d), 0

< r < oo, and a constant

cd, so that

•f' ~

~~I ~ IP(xJI •J· (i~l ~ IP(xW'

(8 .2)

cd

for a 11 cubes Q . From (8.2) and Jensen's inequality, Theorem 6 follows easily. Let us now assume that P is homogeneous of degree d. Observe also that since (8.2) holds, if we normalize P by the condition that mp =

J,

IP(x)l da(x) = 1, we can conclude that

lxl=l

f

(8.3)

IP(x)l-r da(x)

~ cd .

lxl=l However, when u > 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<x
= M8

r

f(x) = !. 2 [f(x+r) + f(x-r)] .

..1.1

near x = 0, have compact support, lx and be in C"" away from the origin, we would have a function f in So if we take f(x) to be log

every LP class such that »>Qf(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.

AVERAGES AND SINGULAR INTEGRALS

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

AVERAGES AND SINGULAR INTEGRALS

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

AVERAGES AND SINGULAR INTEGRALS

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

AVERAGES AND SINGULAR INTEGRALS

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.

AVERAGES AND SINGULAR INTEGRALS

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

AVERAGES AND SINGULAR INTEGRALS

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

AVERAGES AND SINGULAR INTEGRALS

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

AVERAGES AND SINGULAR INTEGRALS

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
J

exp (if t + .•• tif tn-1) dt 1 n-1 t

e<) t)
:f

tn dtt

:=: C(n) .

e:=:t<S1

Hence we have reduced the proof 30) for n to proving 30) for n - 1. Also the case of n

~

1 is easy. So we are done by induction.

We now tum to the LP theory. We wish to emphasize how curvature and Fourier Transform are joining together to help us. So we shall cornpare the case of the parabola (t,t 2 ) to the straight line (t,t). In the case of the parabola we are studying

34) where Dp is a distribution. For a test function ¢

376

STEPHEN WAINGER

35)

In the case of a straight line we are studying

36) where

37)

DL¢ =

J

¢(t,t) dt T.

"" 38) -oo

and 39)

""

J

-oo

We can calculate DL explicitly, and we find 40) Notice that DL is discontinuous along the line

g = -71.

We shall

show in contrast that Dp(g,.,) is continuous away from the origin. It is very easy to see that Dp(g,.,) is C"" away from the line 71 = 0 by complex integration. If for example 71 > 0, we think of t as a complex . t2 variable and integrate along the line lmt = Ret. Then the factor e 171 decays as fast as e-c.,lt1 2 , and one can easily justify differentiation under the integral sign as long as 71

~

e, for some positive e.

377

AVERAGES AND SINGULAR INTEGRALS

We shall now show that Dp(~.71) is continuous near 1~1

=

1. What we

must show is that lim Dp( ~.'1/) exists for ~ near ± 1 . We shall show 71-->0

J

00

DO

41)

r 71~0

J -DO

Assume for simplicity that 71 > 0. Of course 1

;(73 42)

lim 71->0

J

DO

f

ei~t dt ~

__1_

ei~t dt t

-00

711/3 and

43)

1

711/3

~ 211

f

tdt :S. 11 1 1 3

--.

0

as 71 .... 0 .

0

In view of 42) and 43) we can show 41) by showing

44)

J

71- 1 13:Sitl

e~tei71t 2 dt .... 0 . t

378

STEPHEN WAINGER

We shall prove 44) by using VanDer Dorput's lemma. By using VanDer Corput's lemma with j = 2, we see

f T/

(Let h(s) = ~s + 71s 2

-2/3

,

then h"(s) = 271 .) So an integration by parts

shows 00

f

45)

T/ -2/3-

t<.,- 2 13

Notethatif

So

is bounded if ~ is close to minus one. Hence an integration by parts shows

1

-;rn 46)

f e~teiT/t2

1 T/1/3

f :; 00

dtt


1

T/1/3

+ T/ 1/3

~ c 711/3 •

379

AVERAGES AND SINGULAR INTEGRALS

45) and 46) together with similar estimates for negative

prove 44)

and hence the continuity of Dp{~,77) away from the origin. If one is a little more careful in the above argument, one can prove that Dp{~,77) satisfies a Lipschitz condition away from the origin. One may then use Riviere's [R] version of Hermander's multiplier theorem [H] to obtain some LP results for p

1- 2.

In fact if y(t)

=

(t,t 2 ) one obtains

47) For quite a while we tried to p~ove the range of p, ~ < p < 4, in 47) was optimal- with no success. Also, there was a suspicion that the use of the Hormander Riviere theorem lost something. In the Hormander argument, one wishes to estimate an expression of the form

J=

I

where K is a kernel, in terms of

JK(X+h)- K(x)Jdx

K.

One does this by using Schwartz's

inequality and Plancherel's theorem.

J=

J

_1_ · JxlaJK(X+h)- K{x)Jdx lxla

R:SJxJ:S2R

where the sum is over all a 'th derivatives of

K.

380

STEPHEN WAINGER

Now there was the feeling that a use of Schwartz inequality like that above lost too much, and that more careful estimates for

J

for particular

kernels might lead to better results. To get an idea of what to do we calculated D(~.71) very precisely by the method of steepest descents. We found

48)

+ Better terms ,

where r.{l is a C~ function on R 1 which is one near the origin. Hence, the crux of the matter was to study the transformation Tf given by

49) where

50)

This suggests introducing an analytic family of operators in the sense of [SI] as follows:

51) where

52) Tzf is clearly bounded on L 2 if Re z

=

-1/2. So if we could

prove that the kernel Kz corresponding to T z for Re z positive satisfied a condition of the Calderon -Z ygmund type, we could prove T z was bounded in each LP if Re z > 0. Hence by Stein's interpolation theorem we would know that T 0 = T was bounded in all Lp, p S 2.

AVERAGES AND SINGULAR INTEGRALS

381

Then by a duality argument T would be bounded in all LP , 1 < p < oo. It turns out that one can show by a messy calculation that T z is of Calderon-Zygmund type if Re z

> 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

AVERAGES AND SINGULAR INTEGRALS

J•l•

jK 0 (,, 6)- K 0 (<, 6')1 d6 • 1 .

0

Let us consider now what happens with Ke. We should expect no help from the difference K(x,y)- K(x,y-1) when y But this can only happen if y

~

0, if y-1 'S_ 0.

'S 1 or 0 < 1/r. But over this set Ke(x,y)

is integrable at infinity

J J

1 /r

oo

r;

5

0

J J1 /r

oo

S

1-dOrdr ol-e

1.. r2

5

'S

Ke(r, O)dOrdr

0

Joo - 1-

dr .

rl+e

5

It is not difficult to complete the argument and to show 55).

After a laborious calculation one could prove that the kernel Kz corresponding to Tz of 51) was for Re z > 0 an operator of CalderonZygmund type. This proved that HY was bounded in LP 1 < p <

oo if

y = (t,t 2 ). However it would be extremely difficult to carry over this

proof to a three dimensional curve. For example it would be hard to derive an analogue d 48) for the curve (t,t 2 ,t 3). Essentially one needed a way to define a suitable analytic family T z without using the asymptotic formula 48). Recall that

56)

Hi

=

Dp

*f

where

384

STEPHEN WAINGER

So one might be tempted to define

57) where 58)

It turns out that 58) is not a good idea for a very important reason. By changing variables in formula 56) we see that DP(Ae, A2 Tf) = Dp(e,.,.,), for any A> 0. Note that also the function mz(e,.,.,) defined in 52) also has this type of homogeneity, namely mz(Ae,A-2 .,.,) Now experience has shown that

=

mz(e,.,.,), for A.> 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.

AVERAGES AND SINGULAR INTEGRALS

=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


385

386

STEPHEN WAINGER

One may finally deal with treated 1 :S t

oo

< t :S -1 in the same manner that one

< ""· Hence 62) is proved.

Let us calculate the kernel, Kz, corresponding to H~ if z = e > 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

AVERAGES AND SINGULAR INTEGRALS

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) .

AVERAGES AND SINGULAR INTEGRALS

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 <:: j

S n,

such that

So n

I ·

J=O

.

dJ+l

a·-.-77(u) 1 duJ-H

=

0,

and

In other words 77'(u) · ( satisfies an nth order constant coefficient differential equation. So if for some u and dj ' c - . n (u) · s = 0 dul 77'(u) .

e= 0

for all u. Thus 77(U) .

g

j = 1 ,2 ,. .. ,n-1

g is a constant.

But 17(U) .

e

>

0

390

STEPHEN WAINGER

as u • -

oo

since the eigenvalues of A have positive real part. Hence

17(u) is in the subspace of Rn orthogonal to ~. We shall conclude this section with the statement of some theorems that follow from the reasoning discussed above. THEOREM

1. Let y(t) satisfy y'{t)

=

~ y(t) .

Suppose the span in Rn of y(t) for positive t and the span in Rn of y(t) for negative t agree. Then

l
I

djy{t) dyj(t) t=O,

j

=

1,2, ...

span Rn. It turns out that well curved curves can be approximated by homogeneous curves. We can then prove THEOREM

2. Let y(t) be well curved then,

J' -1

f(x-y(t))

~

'S Ap,y llfJJLp,

l
LP

A general theorem in L 2 for curves which are approximately homogeneous was obtained by Weinberg [We]. III. Maximal functions and g-functions We turn now to a discussion of maximal functions. We are especially interested in how the Fourier transforms and g-functions may be used as a tool to relate our maximal functions to more classical ones. The story

AVERAGES AND SINGULAR INTEGRALS

391

began with the study of maximal functions along the curve (t,t 2). Thus we wish to consider averages

Mhf(x ,y)-

70)

J h

k

f(x-t,y-t 2) dt .

0

After much frustration it was decided to take Fourier Transforms and try to see if anything could be learned. It is easy to see that

where

"'k

mh(t".7J)

I

h

eitteit21Jdt.

0

Now mh(t,1J) cannot be evaluated explicitly. Hone hopes to gain some insight by staring at a formula, one should have a formula that is as explicit as possible. Now a similar situation arose in the path integral approach to Quantum Mechanics. See [FH). Feynman and Hibbs wished to have an explicit expression for the probability amplitude that a particle lies in a sphere of radius t. In essence, they had to consider an integral in Rn of the form

J

exp Q(x)dx

lxi~F.

where Q(x) was a quadratic function of x . Instead they considered

!.e

f R

lxl2

--+Q(x) e E2

dx

STEPHEN WAINGER

392

which could be calculated explicitly. This suggests to consider instead of 70) 71)

vh

* f(x,y) =

k

J ""'

exp(-

~:)f(x-t,y-t2)dt.

-oo

Then

72) where v is the measure considered in 23). In particular

i)(h~,h277) =

f

""' e-t2 eihbeih277t2 dt ,

-oo

and this integral can be computed explicitly by completing the square. We find 73)

v(h,h 2) =(nice smoothly decaying function). exp i h 4 ~ 2 77 . 1+h4772 One might guess that the appearance of the oscillatory factor

exp i h 4( 277 is a reflection of the fact that l'h is a singular measure. 1+h4772 On the other hand we see that if h 2 77 is large

which is independent of h. Thus one might try to write (from 72)

One might now hope that if one defines a measure

vh

by the formula

AVERAGES AND SINGULAR INTEGRALS

393

vh could be dealt with by classical arguments, while

where g is another L 2 function having the same norm as f. So one could hope that 74)

\I sup llh

* fll L 2 =

llsup

:S Cjjgjj

vh * gil L 2 L

2 =

Ci!fl!

L

2 .

Roughly speaking this works out. See [NRWM]. The proof of 74) was a hint on how to proceed. However, it depended (as had happened before) on very special computations. What was needed was a way to compare averages like vh to more classical averages by using only the decay of ~h and not so much the explicit expression ci

vh

as was used above. Stein [Ssp] and [SH] succeeded in doing this by introducting appropri-

ate g-functions. Stein's first argument with g-functions dealt with the averages M8 f of equation 4). Recall r

75)

M8 /(x)

=I B

f(x-y)dt-tr(Y) r

where Br is the ball of radius r centered at the origin, and dt-tr is the unit rotationally invariant measure on Br. We set, as before, 76)

mf(x) = sup r>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 .

AVERAGES AND SINGULAR INTEGRALS

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<- C Idm{r) dr

1

n-2 r 2 r 1/2

1 =C--, r

n-1 2

and ~ is bounded. See [SWE]. Thus

sJ.,., t

J\g(f)(x)\ 2 d>..

0

J~

j lm(t\e!) \f(g)j

J J lf
Rn

:t

t 1 m(t 1e1)2 dt

0

Thus to obtain 79) we need to show

f

00

t

1ft m(t\eif dt ~c.

0

First since ~ is bounded,

11\el

11\el

J

dt < 1e1 2

0

Next, since \m'(t)\

J 0

~ n~l t 2

,

dg

Rn 00

~

f

tdt < c

.

AVERAGES AND SINGULAR INTEGRALS

J

397

00

t 1ft m(t1(1)1 2 dt

1 !l(l

f

00

s ltl 2

t

1 dt (tl(l)n- 1

sc .

1 ;1(1

if n>4. Let us be more precise about the counterexample in 2-dimensions. We take

1 lxllog f(x)

l~l

x very near 0

=

in C""

away from 0.

0

We can disprove 77) for p = 2, n = 2 by showing f(x) =

MB

80)

oo

lxl for all small x. Because of rotational symmetry it suffices to prove 80) for points (a ,0) with a small. In that case

" Je

dO

-e IOiln

l~l

= oo.

398

STEPHEN WAINGER

A similar argument shows 81)

sup 1MB f(x)l = l~rS2

oo

r

a.e. for an appropriate f. We wish now to prove a theorem indicating how bad 81) fails in

L 2 (R 2 ). We set • 82)

where f is in L 2 (R2). We shall show

nm1111

83)

L

2

~ ck\lfll

L

2 .

To prove 83), it suffices to show that for each function j(x) taking the values 0,1,2,· .. ,2 k , 84)

\\Ms

l+j(X)2

< Ck\lf\1

-k f(x)ll

L

2 ,

with C independent of the function j(x). We show 85) by induction on k. That is given a j(x) taking the values O,l,2,. .. ,2k, we shall define a function j*(x) so that j*(x) takes values in the set 0,1,2, .. ·,2k-l and 85)

1\Ms

l+j(x)2

-k f-

MB

l+i*(x)2

-k+l f\1

L

2

~ Cllf\1

L

2 •

85) provides the inductive step to prove 86). If j(x) is given define j(x)

2

if j(x) is even

j*(x) = j(x)-1

-2-

if j(x) is odd .

399

AVERAGES AND SINGULAR INTEGRALS

Then, if we set

gf(x)=f£\M

86)

j=O

.f(x)-M 5

5

1+2_ k

J

. f(x)i 2 1 J-1

112

+2k

2

we see

IM5

l+j(x)2

-k f-

M5

-k+l

l+j*(x)2

fl

s;

gf(x) .

Thus to prove 86) and hence 85) it suffices to prove 87)

\lg(f )\l 2

L

s; Cl\fl\

L

2 .

We shall prove 87) by using the Fourier Transform.

where m(r) = j 0 (r). So to prove 87) it suffices to sh0111

,

400

STEPHEN WAINGER

But 88) follows because for s positive and r positive

IJ 0(r)) ~ C/yr

89) and 90)

To prove 88) we use 89) if

1~1 ~ 2k and 90) if 1~1 ~ 2k.

Finally we will show how Stein [SH] proved the maximal function along the parabola (t,t 2 ) is bounded by using g-functions. We start with the measure d#L defined in 24)

J 2

d#L(
cp(t,t 2 )dt.

1

We set 2

91)

d#Lh(cp)

"'I

cp(ht.ht 2 )dt.

1

Then

J 2

d#Lh

* f(x,y) =

f(x-ht,y-ht2)dt ,

1

or

92)

d#Lh

J

2h

* f(x,y) =k

f(x-t,y-t 2)dt .

h ~

We choose a function 1/J(x,y)

l

C~(R 2 ) with 1/J(O) =1. We set

401

AVERAGES AND SINGULAR INTEGRALS

93)

VIb(x,y) = : 3

{~

and

94)

~

g(f) (x ,y) •

iff(~' :2)

'

ldl'h • f(x,y)- .Ph • f(x ,yll 2

d:

r

Let us first assume 95)

1\g(£)11 2 L

:5

C\lfil 2 • L

Note that

sup~ e>o

I

e

{.I

ldiLh *f(x,y)-1/!h *f(x,y)\dh

0

:S sup - 1-

e>O ,112

<

{. ~

0

ldl'h •f(x,y)-,Ph •f(x,y)i 2

:S g(f )(x,y)

d:

r

.

So

96)

sup t>O

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<(•.,)12/f((.71)12 d( d71 0

J 00

=

J ~ ~h(( .,)-~h(( 00

1£((.71)1 2

0

•

•.,)l 2 d(d.,.

0

So to prove 96) it suffices to prove

97)

The integral on the left side of 97) is

J ~ l~(h(,h 2 71)-~(h(,h2 71)1 2 00

98)

.

0

Thus by replacing h by ,\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
1 < p <S

oo.

0

THEOREM S. Let y(t) be a curve in Rn. If the vectors y'(O),y"'(O),y< 3 )(0)··· span Rn,

II

sup o
~I 0

h

'S

\f(x-y(t))j dtl\ LP

Cp\lfi!Lp

1
AVERAGES AND SINGULAR INTEGRALS

405

IV. Vector field problems We turn now to problems III and III'. To study problems III and III' it is convenient to make a change of variables. It is possible to make a change of variables so that the integral curves of the vector field v(x) become lines parallel to the x-axis. Under this change of variables the vectors v(x) transform into curves y which vary from point to point. So we are led to studying problems IV and IV'. We facilitate the statement of these problems with 3 definitions. If y{x,t) is for each x a smooth curve in t with y(x,O) = 0, let

102)

H/(x) =

J'

f(x-y(t,x)) dtt ,

-1

let

103)

J h

M~f(x) = k

f(x-y{t,x))dt ,

0

and 104)

We are now ready for the statement of problem IV and IV'. Problem IV: When do we have

Problem IV': When do we have

The change of variables described above preserves tangency and curvature conditions. So for example

~ 't=O

will be parallel to the x

STEPHEN WAINGER

406

axis, and if the curvature of the integral curves of v(x) never vanishes

a2yl at 2

will not be zero. It turns out that one can prove the following t=O

theorems: THEOREM 6. Let y(t,x)

= (t,r(t,x))

at'ct,x) 1 = 0 at t=O

be a smooth curve in R2 satisfying

and

Then

105)

IIHyfll 2

2 L (R )

~ C\\f\1

2 2 , L (R )

and 106)

nmY£1\ L 2 (R 2 ) ~ cH£11 L 2 (R 2 ) ·

A consequence of Theorem 6 for vector fields will then be

THEOREM 7. If v(x) is a smooth vector field in R 2 such that the

integral curves of v have nowhere vanishing curvature, 1\Hvfl\ 2 2 ~ \\fl\ 2 2 L (R ) L (R )

and

kJ f(x-tv(x))dt h

lim h->0

=

a.e.

f(x)

0

These theorems are announced in NSWV. Here we shall give some discussion of the ideas. Because y(t,x) depends on x, the Fourier Transform no longer seems like a good tool to study Hy and

my.

We

must find a different way to employ orthogonality. Here we're motivated

AVERAGES AND SINGULAR INTEGRALS

407

by work of Kolmogorov and Silveristov [Z] on the partial sums of Fourier Series and an approach to the Poisson Integral by Paley [P]. The idea is to consider Hy · H; and M~(x) · (M~(x))* where · denotes composition of operators and * signifies Hilbert space adjoint. In order to gain some insight into this method we shall just discuss it in the case y(t) = (t,t 2 ) where, of course, we already know the results by a different method. Let us consider K = Hy · H;. The kernel of K will have support on l(u,v)!u=t-s,v=t 2 -s 2 l. Hence one might hope that K would have a much smoother kernel than Hy. (Of course if K were bounded on L 2 so would Hy.) One might hope then, that K would be a Calderon-Zygmund operator. Let us see if this could be the case. One can see that

and hence that

Let u = s- t, v = s 2 - t 2 , and we find

HYH;f(x,y) =

Jf

f(x-u,y-v) k(u,v)dudv

where

c

k(u,v) = IUI (U-U v) (:U+u v)" Now k(u,v) does have its support spread out. However the singularities of k across the curves v = ± u 2 are not locally integrable away from the

408

STEPHEN WAINGER

ongm. Hence HYH; cannot be a Calderon-Zygmund operator. Let us try to see what goes wrong on the level of the Fourier Transform. Recall

where 00

I

-00

We know from 48) that

+ better terms.

The multiplier for Hy · H; would be essentially 1Qp1 2 • Notice that 1Qp1 2 doesn't look any nicer than Qp. However IQp(~,77)-sgn~P is much nicer than Qp(~,77) or Qp(~,77)- sgn~. Now sgn~ corresponds to the operator Lf =Jf(x-t,y) dtt .

L is known to be bounded in L 2 . This suggests that we try to consider

If M is bounded in L 2 so will be L- Hy . Hence so will be Hy. This

actually works and is the basic idea in the proof of Theorem 4. We turn now to the idea of the proof of Theorem 5. Paley showed that

where Ph is the Poisson Kernel. It follows from 107) that

AVERAGES AND SINGULAR INTEGRALS

409

and hence

Since the function h(x) is arbitrar¥ we have llsup Ph *fll h

L

2

:S Cllfll

L

2 •

The fact that we had to modify Hy suggests that we should not expect 107) to hold, but we might expect a variant to hold - perhaps involving an operator Rh(x,y)f(x,y)

=

h(x~y)

J

h(x,y)

f(x-t,y)dt .

0

(We know

We might hope to prove

where Bh(x) is some bounded operator. Even 107) is not quite right. We refer the reader to NSWV for the correct technical modification of 107). This concludes our discussion of the vector field problem.

V. Recent developments The positive results of Theorem 2 and Theorem 5 assume that the curve y(t) has some curvature at the origin. There have been a number of papers trying to understand what happens when this curvature condition is dropped. See [C], [CNVWW], [NVWW] 1), 2), 2), and [NE]. Let us note that we don't have positive results for all C"" curves.

410

STEPHEN WAINGER

Suppose for example y(t) is odd and y(t) = (t,r(t)) where

r(t) = )

for 0 ~ t ~ 1

0

for t > 1

{ t-1 Then

where 00

I

-oo

which is easily seen to be unbounded if ~ = -71. In fact it is easy to see the following: Suppose y(t) is odd and linear on a sequence of intervals ai

~

t

~

bi

~

1 and assume that the linear extensiog,of y on

[ai'bi] does not pass through the origin. Then if the ratios .....!. are ai unbounded, lim

e...o

J

f(x-y(t)) dt t

\t\~e

cannot exist in the L 2 sense for every f in L 2 • It is somewhat more difficult to produce counterexamples for the

operator :my. For example in the case of the two dimensional curve (t,r(t)) described above, it is easy to see that :my is bounded in every LP, because it is dominated by

411

AVERAGES AND SINGULAR INTEGRALS

s~p ~

J h

lf(x-t,y)\ dt +

J h

s~p ~

0

f(x-t,y-t+t)dt .

0

Both of these operators are bounded by the classical theory. We refer to [SW] for an example to show that the maximal function along a C 00 curve has no non-trivial positive results. The above authors have been investigating the behavior of the Hilbert Transform and maximal functions related to convex curves. Let y(t) = (t, f'(t)) a plane curve with f'(t) convex for positive t. Let h(t) = tf''(t)- f'(t). -h(t) represents the y-intercept of the line tangent to the curve y at y(t). We then have THEOREM 8. Let y(t)

=

(t, f'(t)) with f'(t) convex and increasing for

t > 0. If f'(t) is even IIHyfl\ 2 ~ Cl\fl\ 2 L

L

if and only if t

for some C

>0

> 0.

If y(t) is odd

IIHyfl\

L

2

~ Cl\fl\

L

2

if and only if h(Ct) for some C

~

2h(t) ,

t>O

>0.

There are generalizations of Theorem 8 to higher dimensions, and an investigation of the Lp theory has begun. We know that

412

STEPHEN WAINGER

II:IRyfll L 2 ~ Cilfll L2 if • holds for some C > 0, however the maximal functions can be bounded on L 2 (and in fact in LP for any p curves even if

*

>1

) for some convex

fails.

Recently Phong and Stein [PS] introduced a general problem of which our problems are special cases. They consider at each point P in Rn a submanifold Mp of dimension say f and an f dimensional CalderonZygmund kernel K(P,Q). Then they consider

Tf(P) =Jf(Q) K(P ,Q)dm(Q)

where dm(Q) is a measure on Mp. They show that if n

k

=

~

3 and

n-1

if Mp satisfies a kind of generalized curvature condition. See [PS]. Various authors have also considered multiple parameter problems which are essentially multiple Hilbert transforms on surfaces and multiparameter maximal functions on surfaces. See [NW2], [V], [STRl], and [CSS]. Appendix 1. An introduction to the method of steepest descents. Here we shall try to give an explanation of the main ideas of the method of steepest descent. The interested reader can find a more detailed description in [B]. Let us first consider the behavior of the integral DO

A-1)

I(X) =

J

-DO

413

AVERAGES AND SINGULAR INTEGRALS

for large ,\. Of course we can make a change of variables t

= xJX

and observe A-2)

I(.\)=

B JX

where

B=

A-3)

""

I

2

e-t dt .

-00

The point we wish to make here is that if ,\ is large most of the contribution to the integral I(,\) comes from a small neighborhood of the origin. In fact

J

1ltl>,\2 /5

e-.\t2 dt +

I

,\

t2

e-2-2dt

t>l

\ 2 < e-1\\ l /5 + e-1\1

.... 0 very fast as ,\ .... "" . Thus the main contribution to the integral I(A) comes from the small interval - _1_ < t < - 1-. Now if we perturb the integrand in I(.\), we )1.15-

-,\2/5

can expand the integrand in a power series in that little interval. For example we might consider

J(A) =

f -oo

00

e-.\h(t)dt

414

STEPHEN WAINGER

where h(t) ~ t 2 for large t, h(t) = t 2 + <9(t 3 ) for small t and h(t) > 0 for t > 0. Then one can easily see that

is exponentially small as before. Now

f

I

e-Ah(t)dt =

lti
e-At2. (1+0(A.t3)

lti
B

=

JA. +

0

(e

->..1 /5

)+

f){

1 ) \>..1/2+1/5 ·

In the method of steepest descents we try to choose a contour of integration so that on the new contour the situation would be essentially that of

J. Thus for example, K=

if we had

f

"" eiAt2+AP(t)dt

-oo

and P(t) were very negative at infinity and 0(t 3) near t = 0 we would try to write t =a+ ir and integrate on the line o = r for It!< 1/>..2 15 , and we would expand eAP(t) in a power series in this small interval.

415

AVERAGES AND SINGULAR INTEGRALS

More, generally, if we were

~oncerned

with the asymptotic behavior

for large values of ,\ of an integral of the form

J

g(z)e,\h(z)dz

g(z) and h(z) are holomorphic, we would try to choose a contour on which Re h(z) had only a finite number of maxima, and argue that the main contribution to the integral should come from a small neighborhood of the largest maximum or perhaps an endpoint and we then expand g and h in a Taylor series at such points. At such a maximum,

e,

h'(') = 0.

Thus the main contribution to our integral should come from a point where h'(') = 0 or an endpoint. This is also reflected in VanDer Corput's lemma. A variant of this principle for non-analytic functions is called the principle of stationary phase and is discussed in Professor Stein's lectures in these proceedings. Appendix 2. The method of stationary plase and quantum meclanics The method of stationary phase lends itself to a formulation of quantum mechanics that is very appealing to at least some mathematicians. The principle of stationary phase asserts that the integral

with f(t) real gets most of its contributions for large ,\, near a, b, or a zero of f'(t). If we had no endpoints for example if f were periodic with period b- a or if the interval of integration was from

-oo

to

oo

and f oscil-

lated very rapidly for large t, we would expect the main contribution to the integral to come from small neighborhoods of a zero of f'.

416

STEPHEN WAINGER

Let us now turn to quantum mechanics. In particular let us consider a particle moving from a point xa to xb as time evolves from time ta to time tb. According to classical physics, the particle will follow a path for which the classical action is stationary. If x(t) is any path, and our particle has mass m and is moving under the influence of a potential V(x,t), the classical Lagrangian is defined by L=

f [x(t)]2 - V(x(t),t) .

The action along a path x is defined by

J

tb

S(x) =

L(x(t))dt .

ta

The path on which the classical particle moves will be a path x(t) such that x(ta) = xa, x(tb) = xb, and such that

One of the most important principles of quantum mechanics asserts that motion on a classical scale must be essentially described by the laws of classical mechanics. In our example it means the only paths that are important are paths near the path x of

*.

This principle is expressed in

terms of a small number h. The principle says that the only classical paths that should be important are paths which differ from

x

only on an

h scale of measuring. Now the Feynman path integral formulation is in terms of probability amplitudes of events. In our case the probability amplitude of passing from xa at time ta to xb at time tb is given by an integral

fi

F =

eh

S(x)

1lx

AVERAGES AND SINGULAR INTEGRALS

417

where the integration is an integral over all paths x(t) such that x(ta)

= xa and x(tb) = xb. Leaving aside the question of how such an

integral can be precisely defined, let us try to guess what paths contribute the most to the integral F . Since h is very small, the principle of stationary phase would indicate that the main contribution to the integral F should come from a small neighborhood of a path of which some kind of a derivative of S was zero. Thus we might expect the main contribu-

tion to the integral F to come from paths which are very close (on a scale of h ) to the path Y defined by

*.

There have been many papers in tbe mathematical and physical literature dealing with the problem of making sense out of the definition F. See for example lCS] and references cited there. However, the original definition in [FH] serves the purpose of making many formal calculations. We may imagine dividing the t interval into 2j subintervals, Ij, of equal length

We consider only paths which are linear on Ik. Such paths are determined by xk =X

k

=

~a + ~ [tb-ta])

1,2,···,2L1. We then take

F = _lim Aj J->00

R

2 1-1

where S(x 1' • · • ,x 2 j_ 1) is the action a long the polygonal path determined by xl'···,x 2 j_ 1 , and Aj is a normalizing factor. For more details see [FH]. We would like to make one final remark about the book [FH]. It's

418

STEPHEN WAINGER

great. It explains quantum mechanics in terms of mechanics and does not use notions of atomic physics as many of the standard books do. Thus it is accessible to many more mathematicians than standard quantum mechanics texts. The book also elucidates the differences between the nature of physicists and mathematicians. If you don't want to know h to 3 significant figures in ergs/sec, whatever they are, you probably would rather be a mathematician than a physicist. Finally, many mathematicians could probably learn a great deal to improve themselves as mathematicians by reading the book. STEPHEN WAINGER DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WISCONSIN

REFERENCES [B]

N. DeBruijn, Asymptotic Methods in Analysis, North Holland Publishing Co., Amsterdam, 1958.

[BF]

H. Busemann and W. Feller, "Zur Differentiation des Lebesguesche Integrale," Fund. Math, Vol. 22, 1934, pp. 226-256.

[CS]

R. Cameron and D. Storvick, A simple definition of the Feynman integral with applications, Amer. Math. Soc., Providence, 1983.

[CSS]

H. Carlsson, P. Sjogren, and J. Stromberg, "Multipara meter maximal functions along dilation-invariant hypersurfaces" to appear in Trans. of the A.M.S.

[CW]

H. Carlsson and S. Wainger, "Maximal functions related to convex polygonal lines," to appear.

[C]

M. Christ, preprint.

[CS]

J. L. Clerc and E. M. Stein,

"LP multipliers for non-compact symmetric spaces," Proc. Nat. Acad. Sci., U.S.A., Vol. 71, 1974, pp. 3911-3912.

[COR1]

A. Cordoba, "The Kekeya maximal function and the spherical summation multipliers," Amer. J. of Math., Vol. 99, 1977, p. 1-22.

[COR2]

"Maximal functions, covering lemmas and Fourier multipliers," Proc. Symp. in Pure Math., Vol. XXXV, Part I, 1979, pp. 29-50.

AVERAGES AND SINGULAR INTEGRALS

419

[CF1]

A. Cordoba and R. Fefferman, "A geometric proof of the strong maximal theorem," Annals of Math., Vol. 102, 1975, pp. 95-100.

[CF2]

- - - - , "On differentiation of integrals," Proc. Nat. Acad. Sci., U.S.A., Vol. 74, 1977, pp. 2211-2213.

[CF3]

- - - - , "On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier Analysis," Proc. Nat. Acad. Sci., U.S.A., Vol. 74, 1977, pp. 423-425.

[CNVWW] A. Cordoba, A. Nagel, J. Vance, S. Wainger, and D. Weinberg, "LP bounds for Hilbert Transforms along convex curves," preprint.

[F]

E. B. Fabes, "Singular integrals and partial differential equations of parabolic type," Studia Math., Vol. 28, 1966, pp. 81-131.

[FEF]

R. Fefferman, "Covering lemmas, maximal functions, and multiplier operators in Fourier Analysis," Proc. Symp. in Pure Math., Vol. XXXV, Part 1, pp. 51-60.

[FH]

R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York.

[H]

L. H6rmander, "Estimates for translation invariant operators in LP spaces," Acta Math., Vol. 104, 1960, pp. 93-139.

[KS]

R. Kunze and E. Stein, "Uniformly bounded representations and harmonic analysis of the 2 x 2 unimodular group," Amer. of Math., Vol. 82, 1960, pp. 1-62.

J.

[NRW]

A. Nagel, N. Riviere, and S. Wainger, "On Hilbert transforms along curves, II, "Amer. J. Math., Vol. 98, 1976, pp. 395-403.

(NRWM]

A. Nagel, N. Riviere and S. Wainger, "A maximal function associated to the curve (t,t 2 )," Proc. Nat. Acad. of Sci., U.S.A., Vol. 73, 1976, pp. 1416-1417.

[NSWB]

A. Nagel, E. Stein, and S. Wainger, "Balls and metrics defined by vector fields I; Basic Properties," to appear in Acta Mathematics.

[NSW]

A. Nagel, E. Stein, and S. Wainger, "Boundary behavior of functions holomorphic in domains of finite type," Proc. Nat. Acad. Sci. U.S.A., Vol. 78, 1981, pp. 6595-6599.

[NSWV]

A. Nagel, E. Stein, S. Wainger, "Hilbert transforms and maximal functions related to variable curves," Proc. of Symposia in Pure Math., Vol. XXXV, part I, 1979, pp. 95-98.

[NVWW1] A. Nagel, J. Vance, S. Wainger, and D. Weinberg, "Hilbert transforms for convex curves," Duke Math. J., Vol. 50, 1983, pp. 735-7 44.

420

STEPHEN WAINGER

[NVWW2] A. Nagel, J. Vance, S. Wainger, and D. Weinberg, "The Hilbert transform for convex curves in Rn,, to appear in Amer. ). of Math. [NVWW3]

• - - - , "Maximal functions for convex curves," Pre print.

[NW]

A. Nagel and S. Wainger, "Hilbert transforms associated with plane curves," Trans. Amer. Math. Soc., Vol. 223, 1976, pp. 235-252.

[NW2]

- - - · · · "L 2 boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multi-parameter group," Amer. J. of Math., Vol. 99, 1977, pp. 761·785.

[NE]

W. Nestlerode, "Singular integrals and maximal functions associated with highly monotone curves," Trans. Amer. Math. Soc., Vol. 267, 1981, pp. 435-444.

[P]

R. Paley, "A proof of a theorem on averages," Proc. Lond. Math. Soc., Vol. 31, 1930, pp. 289-300.

[R]

N. Riviere, "Singular integrals and multiplier operators," Ark. Mat., Vol. 9, 1971, pp. 243-278.

[PS]

D. Phong and E. Stein, "Singular integrals related to the Radon transform and boundary value problems," Proc. Nat. Acad. Sci., U.S.A., Vol. 80, 1983, pp. 7697-7701.

[S]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

[SBC]

, Boundary Behaviour of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972.

[SH]

----,"Maximal functions: Homogeneous curves," Proc. Nat. Acad. Sci., U.S.A., Vol. 73, 1976, pp. 2176-2177.

[Ssp]

----·"Maximal functions: Spherical means," Proc. Nat. Acad. Sci., U.S.A., Vol. 73, 1976, pp. 2174·2175.

[SPL]

____ , Topics in Harmonic Analysis related to the LittlewoodPaley Theory, Princeton University Press, Princeton, 1970.

[SI]

- - - - · "Interpolation of linear operators," Trans. Amer. Math. Soc., Vol. 88, 1958, pp. 359-376.

[SWA]

E. Stein and S. Wainger, "The estimation of an integral arising in multiplier transformations," Studia Math., Vol. 35, 1970, pp. 101-104.

[SW]

____ , "Problems in harmonic analysis related to curvature," Bulletin of the A.M.S., Vol. 84, 1978, pp. 1239·1295.

AVERAGES AND SINGULAR INTEGRALS

421

[SWE]

E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton.

[STR1]

R. Strichartz, "Singular integrals supported on submanifolds," Studia Math., Vol. 74, 1982, pp. 137·151.

[STR]

J. Stromberg, "Weak estimates on maximal functions with rectangles in certain directions," Ark. Mat., Vol.15, 1977, pp. 229-240.

[STRO]

J. Stromberg, "Maximal functions associated to rectangles with uniformly distributed directions," Ann. of Math., Vol. 107, 1978, pp. 399-402.

[V]

J. Vance, "LP boundedness of the multiple Hilbert transform along a surface," Pacific J. of Math., Vol. 108, 1983, pp. 221-241.

[WE]

D. Weinberg, "The Hilbert transform and maximal function for approximately homogeneous curves," Trans. Amer. Math. Soc., Vol. 267, 1981, pp. 295-306.

[Z]

A. Zygmund, Trigonometric Series, Vols. I & II, Cambridge University Press, London, 1959.

INDEX approach regions admissible, 245 non-isotropic, 261 non-tangential, 244

convex curves, 411 curvature, 321, 361 and Fourier transform, 321, 325, 268, 375

Ap classes, 73, 353 area integral, 92 atomic decomposition, 114, 156, 159, 340

DeGiori-Nash regularity theory, 144, 158 Dirichlet problem, 132, 243 domain of holomorphy, 211

Bergman kernel, 230

duality of H1 and BMO, 114, 340

B.M.O. (Bounded mean oscillation), 9, 94, 331, 340 BMO(R~ x R~), 101

electrostatics, 163

Bochner-Martinelli formula, 196 Bochner-Riesz summability, 344 Calderon-Zygmund decomposition, 48 Campbell-Hausdorff formula, 267

exponential mapping, 267, 275 Fatou's theorem, 245 finite type, 269, 283, 324 Fornaess imbedding theorem, 226 functional calculus, 27

canonical coordinates, 275 Carleson measure, 16, 94 Cauchy-Fantappie formula, 195 Cauchy integral, 5, 8, 186 on a Lipschitz curve, 143 Cauchy-Riemann ~quat ion, 201

g-functions, 393 Hardy space, 144 HP spaces, 89 HP(R! x R;) , 101 harmonic measure, 141

convergenge of averages over spheres, 358, 361 along curves, 361 along vector fields, 361, 406

Hartogs extension phenomenon, 207

covering lemmas, 60

Henkin integral formula 219, 336

heat operator, 254

423

424

INDEX

Heisenoerg group, 257, 335

non-commuting vector fields, 267

Hilbert transforms along curves, 361, 372 along vector fields, 362,406

oscillatory integrals (first kind), 308 (second kind), 335, 344, 349

hydrostatics, 177 hypoelliptic differential operators, 281

Poisson integral, 187, 243 hi-Poisson integral, 102

Kohn (canonical) solution, 209 Laplacian, 266

Rellich-type formulas, 150, 166 restriction theorems, 325, 344

Korn-type inequalities, 145 Korteweg-de Vries equation, 25 Laplace equation, 132, 242 Leray form, 194 Levi-pseudoconvex, 214, 257 polynomial, 227 Lipschitz domain, 133, 145 Littlewood-Paley-Stein theory, 48, 53

Sobolev estimates, 145 space of holomorphy, 42 space of homogeneous type, 251 stationary phase and quantum mechanics, 415 Stein-Weiss spaces, 96, 160 steepest descents, 412 Stokes theorem, 189

local singular function, 210

surfaces (non-zero curvature), 321

Lu Qi-Keng conjecture, 235

systems of elliptic equations, 133, 163

maximal functions, 48, 49, 245 strong, 60 spherical, 359, 393 on curves, 391, 400 on vector fields, 406

Szego kernel, 193, 265, 296, 297 Transference theorem, 39

method of layer potentials, 133, 143

Vander Corput lemmas, 309, 370

Mobius transformation, 186 multilinear Fourier analysis, 18 multiparameter differentiation theory, 57 multipliers, 72

weight norm inequalities, 72 Zygmund conjecture, 67

Library of Congress Cataloging-in-Publication Data Beijing lectures in hannonic analysis. (Annals of mathematics studies ; no. 112) Bibliography: p. Includes index. 1. Harmonic analysis. I. Stein, Elias M .• 1931- . II. Series. QA403.B34 1986 515'.2433 86-91452 ISBN 0-691-08418-1 ISBN 0-691-08419-X (pbk.)

Elias M. Stein is Professor of Mathematics at Princeton University