Beijing lectures in harmonic analysis

Annals of Mathematics Studies Number 112

BEIJING LECTURES IN HARMONIC ANALYSIS

EDITED BY

E. M. STEIN

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY 1986

Copyright C 1986 by Princeton University Press ALL RIGHTS RESERVED

The Annals 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. Pa. perbacks, 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

vii

PREFACE NON-LINEAR HARMONIC ANALYSIS, OPERATOR THEORY AND P.D.D. by R.R. Coif man and Yves Meyer

3

MULTIPARAMETER FOURIER ANALYSIS

by Robert Fefferman

47

ELLIPTIC BOUNDARY VALUE PROBLEMS ON LIPSCHITZ DOMA INS

by Carlos E. Kenig

131

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

by Steven G. Krantz

185

VECTOR FIELDS AND NONISOTROPIC METRICS 241

by Alexander Nagel

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS by E. M. Stein

307

AVERAGES AND SINGULAR INTEGRALS OVER LOWER DIMENSIONAL SETS 357

by Stephen Wainger

INDEX

423

v

PREFACE In September 1984 a summer school in analysis was held at Peking University. The subjects dealt 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

Beijing Lectures in Harmonic Analysis

NON-LINEAR HARMONIC ANALYSIS,

OPERATOR THEORY AND P.D.E.

R. R. Coifman and Yves Meyer

Our purpose is to describe a certain number of results involving the study of 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:B1 -1 B2 is said to be real analytic around 0 in

B1 if we can expand it in a power series around 0 i.e. 00

F(f) = I Ak(f ) k=0

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

Ak(fI...fk):B1 xB1...xB1 B2 (linear in each argument) such that Ak(f) = Ak(f, f, f) and k

(1)

IIAk(fI...fk)IIB2 < Ck

11

IIfjIIB1

j=1

for some constant C . (This last estimate guarantees the convergence of the series in the ball IIf1I < C .)

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 Bi (the complexification of B1 ) and the extension is holomorphic from B c to B2 i.e., F(f +zg) is a holomorphic (vector valued) function of z e C, JzJ < 1 , Yf, g sufficiently small. The converse is also true. Any such holomorphic function can be exx the kth Frechet differenpanded in a power series, (where Ak is k!

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 holomorphic" and prove some boundedness estimates. Let L denote a differential operator like a(x) dx

xfR,

a(z)az

zEC

= div A(x) grad, A _ (aij)

aij(x) ai j(x)

x f Rn

x e Rn

the coefficients a(x) (or aij(x) ) will be assumed to belong to some Banach space B1 of functions (for example L°° ). It is natural to ask when such objects as: L-1

,

fL sgnL, e-tL , e-t\/L ,

or more generally, c(L) (where (k: C C ), can be defined as a bounded operator (say on L2 or some Soboleff space), and a functional calculus developed i.e., (k1(L)02(L) _ 0102(L).

NON-LINEAR HARMONIC ANALYSIS

5

Many questions arise:

a) Does F(a) = c(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 (A(L). b) What is the largest domain of coefficients a for which we have estimates for c(L) ? This question is the same as asking what is the largest BI 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°°, B.M.O. and other "exotic spaces." We now start with a fundamental example related to the Cauchy integral. We let La

l+a -dx

with hall < 1 a(x), real valued.

If we define h(x) = x+A(x), A'(x) = a. We then have

Laf=\_d f°h-110h=i

UhdxUhlf

where

Uhf =f°h. Of course, in this case, if we use the Fourier transform we can define

(I_ d)f idx

r eix&O(e)f(e)de

R. R. COIF MAN AND YVES MEYER

This gives, for example 00

sgn

d

elx6sgn 6 (e)de = n dx) f = J

f

P. V.

f t dt = H(f

x-t

-00

Thus we can define

n sgn(La )f = nUhsgn

r li dx a Ulf h =pv J

00

f(t)(l+a(t) x-t+A(x)-A(t)

dt

-00

(where we used the observation that 95(ULU-1) = U¢(L)U-1 ). We view

F(a) = sgn L. as an operator on L2(R) and wish to know whether it is analytic on L°° or if we can replace a by complex a and still have a bounded operator. If we do this, writing a = a + ip jfaIl0 < 1 , we find

F(a) f =

f

f(t) (1+ia+i$)

x-t+A(x) + iB(x) - A(t) - iB(t)

f t)[(l+a)/(1+a)](1+a)

x+A(x)-t-A(t)+i(B x)-B(t))

dt

dt

= UhCUhif1

where

h = x+A(x) Cf =

('

J

f(t) (1+Bi(t)) x-t+iB1(x)-iB1(t)dt

-00

f1(t) = f(t) i+a Bi(t)

B1 = B°h-1

NON-LINEAR HARMONIC ANALYSIS

7

Since Uh is bounded on L2 it would suffice to prove that C is bounded on L2 for all B such that B' is small. We could also try to prove this by expanding -irr sgn(La)f = J (xf t)+A(x)-A(t) dt = C' (-1)k

r

(A(x)_A(t)\kf(1+a)dtJ

-

x-t

/

x-t

Observe that the operators are of the form

T(f) = ft

(Ax_ A t) f(t) dt = fk(xt) f(t)dt

We will prove Theorem I: Let T a Cc(C) and A(x) such that

IA xx_A t

I

< M and T(f) = p.v. J

(`A(xx-y (Y) f(y) dy

Then the operator T is bounded on L2(R) (and LP 1 < p < oe ). 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

as we can see its boundedness in L2 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. COWMAN AND YVES MEYER

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

Let I' be a rectifiable curve through 0, s be the arc length parameter

s

z'(s) = ea(s) i.e., z(s) = r eia(t)dt 0

The Cauchy integral on r is given as:

f

cr(f) = p.v.

00

f t z' t dt

z(s)- z(t)

-00

r

J

00

00

1

z s- Z (t s-t

f t z' t dt s-t

rJ 0rz` ss-t-z t/I fs-tt)

dt

-00

if we assume 0
*

s-t

we can take c e C o (C)t/i(z) =

on 8 < lz l < 1 and obtain the bounded-

z ness on L2 of C11 (from Theorem I). Condition * is the so-called chord arc condition and * for S small is equivalent to a e BMO with lIaIIBMO small (see [3]). If we think of C as an operator valued functional of a, 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

NON-LINEAR HARMONIC ANALYSIS

T(f) = p.v.

where IK(x,y)I <

C

Ix-YI

9

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

C moreover they were also IaxKI + Ia KI < Ix-y12 Y

antisymmetric i.e., K(x,y) = -K(y,x) .

(For example, K(x,y)

(A(x) -Y (Y))

i1

--y

-0 c C4, A' c L°° .) Recently

G. David and J.-L. Journe found a necessary and sufficient condition for

such operators to be bounded on L2 (or LP ). This condition is simply that T(1) 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 c BMO(R) if 1 /2 IIbII* =

uIII fI

where mi(b) =III I b(x)dx

and I is an interval (or a cube in Rn ), and that this norm is equivalent

to the following "Carleson" norm

rr

JJ I

where '/it =

1 /2

111

*b12 dxtdt

0

0 (t) 0 C o C. fcidx =0 (ci a'0) (see [51).

A basic treason for the frequent occurrence of functions in B.M.O. is the following simple fact.

PROPOSITION. If T is as above and T is bounded on L2 then T maps L°° into B.M.O.

R. R. COWMAN AND YVES MEYER

10

Proof. Let b c L°° and let I be given. Consider I = 21 and write Tb = T(bXI) + T(b(1 - XI)) = -51 + b 2

Clearly

f

1 /2

1 /2

Ib-m1(b)I 2

lill

+

r III f ID2-ml(b2)I

1/2 2

I

The first term is dominated by 1 /2

2(-L II

JI


r1I2dx

f

1 /2


I

For the second we observe that

T(b2)(X) - T(b2)(u) =

<

I

f[K(x,y) - K(u,Y))b2(y)dy

f

III dylIbll,. < CIIbII . IX-YI2

Ix-y1>111

Integrating in y we get IT(b2)(X)-mI(T'(b2))I < CIIbII,,.

which shows that second term is bounded by CIIbII. We have thus shown the necessity of the condition T(1) c BMO. Before stating the theorem precisely we would like to reformulate it somewhat.

NON-LINEAR HARMONIC ANALYSIS

Let k eCU(RI) with f6dx = 1 and

11

=f(x). Let ft(u) _

(U-xand similarly for Vit (u) . We claim that under the preceding assumptions on T we have lI
1

1+

(xY)2

In fact, assume for simplicity, that S6 is supported in (-1,1). Since f c&du = 0, if we assume Ix-yj > 3t

fr(z){j'K(z ,u)

II =

<

J'Ic!'(z)I

dz

t 21ot(u)ldudz
t

ly-xl2

IY-xl

(where we used the fact that ly-z I < t , Ix-ut < t, Ix-y l > 3t and the ). hypothesis layK(x,y)I < If Ix-yj < 3t we use the antisymmetry of k(x,y) to write Ix-yi-2


I2 ffK(z,u)Otu)Ot(z)-.At(z)ot(u))dzdul

but lp (u) ¢t (z) -fir (u)

t

(z) I < Iu

t Iu-zl < t, Ix-yj < 3t and IK(z,u)l <3

1

and the fact that Iu-x I < t , 1

imply

lz-u I

,.Ot >I

Combining these estimates proves our claim.

12

R. R. COIFMAN AND YVES MEYER

We can now state

THEOREM (G. David, J. L. Journe) [7]. Let T:`D - D' such that for some

E>0 1 <- t

1

1+IuyI

I+E

= pt(u-v)

t

and I!

*

S t

vll+E = pt(u-v)

1 + Iu t

Then the necessary and sufficient condition for T to extend to a bounded

operator on L2 into L2 is that T(1) 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. J k(x,y)f(y)dy where

IxyI 1

1°

Ik(x,y)I <

2°

Ik(x,yl)-k(x,y)I < Iy-yI IE

Ix-yI'+E

for Ix-yI > 2Iy-y' I

and 1 E

k(x1,y)-k(x,y)I < Ix x I

x-yI

for Ix-yI > 2Ix-xI I

I

E

for some E>0. 3°

II + II < t

NON-LINEAR HARMONIC ANALYSIS

13

(This last condition followed from k(x,y) = -k(x,y) and 1°, 2°) then the conditions * are verified. It can be shown that if * is valid then T can be represented as a limit of integral operators whose kernels satisfy the conditions 1°, 2°, 3° We will refer to 1°, 2° as standard estimates, and to 3° as the weak cancellation (or boundedness) property. This last condition is independent of 1°, 2°, and can be proved in a variety of ways, as we shall see. To see how the theorem can be applied to reduce the degree of nonlinearity a of a "polynomial" and to obtain estimates consider

f(y) C(aIf) n = ft!J,A(xX_A(Y)ln Y J Y !

dY

We check by a simple integration by parts that Cn(a,l)(x) = Cn-I(a,a)(x)

If we make the induction hypothesis that Cn_I(a,f) is bounded on L2 it would follow by the preceding remarks that Cn_I(a,f) maps L°° to B.M.O. from which we deduce that Cn(a,l) a B.M.O. and that 3C > 0 such that IIC5(a,f )IIL2 <_ CnllaIILjIfIIL2

We will see later that this reduction of Cn 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

Ptf=95t*f

Qtf=qit*f.

We start with the fundamental theorem of calculus writing for f e Co(R)

14

R. R. COIFMAN AND YVES MEYER 1 /E

Tf = li

PtTPtf

o

Eir

I

(PtTPtf) dtt

t

E

=

We'll study only the first term (since the second is its adjoint). We start by observing that t _2

pt 2g

(

) = t 2 2(te)g(e) = 20(t6)01(te)g(e) = (Q1 tQtg)

where ip1(x) = xq5(x) and Q 1,tg the first term as

=

iA1,t *g. This permits us to rewrite

00

-2

f Q1 tLt'tf

dt

t

0

where

Ltg = QtTPtg = J t(x,Y)g(Y)dY

and

et(x,Y) = < T*Ot , Ot >

Also observe that Lt(l) = f et(x,y)dy = = At *T(1) = Pt*b (here all integrals converge absolutely since let(x,y)I < pt(x-y) ). Before proceeding we remark that since Iet(x,y)l < pt(x-y) we can

think of Lt as an averaging operator on the scale t, and of Ptf as being essentially constant on that scale i.e., Lt(Ptf) would look like Lt(1) Pt(f) . (This would be exact if Pt is a conditional expectation and

NON-LINEAR HARMONIC ANALYSIS

15

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

r

QI,tLtPtf dt =

0

r QL,t1Lt(1) Pt(f )t Lt 0 00

+

J0

QL,tlLtPtf-Lt(1)Pt(f )Idt

r 0

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

J

<e J Ifl2dx

(this is valid using only *). In fact, let g e L2

j1 =

fJQit()(LtPt)f-Lt(1)Pt(f)}dx tdtl 1

<

rTQ1,t(g)I2

4xtdt)

°(f ftxYxPt(f )(y)

- Pt(f )(x))dy

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

R. R. COIFMAN AND YVES MEYER

16

1 /2

(j//'Pt(x_Y)IPt(f)(Y)_Ptf )(x)12

dxdydt

1 /2

fff Pt(u)IPt(f )(Y)-Pt(f )(Y+u)12

dudydt t

-

dudedt1

1I2f(e)2

')I2leit1

`

(fffpt(u) 1u S I$(te)12 Itel 8

t

=

J

(6)1

2 du ddt t

1/2

00

f I0(t)12ts dt

P(u) us

t

1/2 1^

I

1/2

/2

(j(2de)1/2

o

As for the first term, we proceed as above and are led to estimate 1 /2

(f*bX)I2t*f2 This is dominated by

llf 112

d tdt

if and only if (fit * b(x))2 d tdt 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 interva 1 I

v(I) < C111

I =1(x,t) cR2:(x-t,x+t)
I

NON-LINEAR HARMONIC ANALYSIS

17

Carleson's lemma states that

fFP(xt)d(xt)
,

where F*(x) = sup IF(x-y,t)j . This is proved by observing that if OA = {x:F*(x)>A1 then outside OA F(x,t) A1) < v(OA)
and the LP result follows by integration. We now would like to make a few comments concerning the principal term: 00

f Qt((,At * b) (0t * f )) dt

= 77 (b; f)

0

This is-essentially equivalent to a simpler looking version 00

J

(art*b)(0t*f)dt

0

which is the basic bilinear operation in (b, f) commuting with translations and dilations i.e.: If bT(x) = b(x-r) and bx(x) = b(Ax) 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 n.

18

R. R. COIFMAN AND YVES MEYER

One cannot conclude this section without mentioning that IT is also the principal term in the Bony linearization formula and is called paraproduct by him i.e., given F f C°°, f Aa, 0 < a < 1/2 one has the remarkable formula (Bony)

F(f)=rr(f;F'(f))+e where

e e .12a (f EAa iff If(x)-f( y)1 < cIx-yIa)

.

The proof of this linearization formula is the "same" as for the T(1) theorem and is achieved as follows.

F(f) = lim F(f * ct) = -

do

fr

t -2 F(f * qt) at

0

J

00

F'(f * fit) f * Vt !Lt (here ct = t 2- ct)

0 00

f

F'(f)*(btf * btdt+e =rr(f;F'(f))+e

0

and it is quite easy to verify the e e A2a if f e Aa. (This method 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

is multilinear in the ti We will assume that Ak commutes with simultaneous translation of ti i.e.

Ak(th,tz,...,tk)(e)

*

=Ak(tI...tk)(e-h)

where th(IA )

= ti(V, - h) .

ikie,thi

Clearly if we take ti(9) = e linearity in each argument that

(® =

e-ihki

ti(O) we find using the

k A h(l ki)

Ak(e

e

ik19

, ...,e

ikke

)(e) = Ak(e

ik10

...e

ikkO

)(e-h)

taking h = 0 we get Ak(e

(e) _ A(kl ... kk) e

e

More generally if ti(0) = F ti(j)e' 0 we find k

A(tlt2...tk)(e) =I...LA(jl,...,jk)tl(j1)t2(j2)tk(jk)e jk

ie(! j i) 1

j1

where

A(jl ... jk) = A(ellle, ...,ellke)(O)

Before continuing let us observe that for k = I , eike form a basis diagonalizing all linear operators commuting with translation, in fact, all such operators are norma I. For k > 2 it isn't as simple. Consider for a moment a finite dimensional vector space V and a bilinear transformation AN xV - V. We can ask when is it true that

20

R. R. COWMAN AND YVES MEYER

there exists a basis in V el

eN a linear operator A defined on

V®V diagonalized by ei®ej A (ei®ej) = Ai,j ei®ej such that

A(ei,ej) = Ai j e,(i,j) for some map n(i,j) of

(1,..,N)2 -(1,...,N) i.e. we want the diagram to commute

ti A V®V

where

v(ei, ej) = ei®ej n(ei®ej) = en(i,j)

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

Ak(fI...fk)(x)

ffeix(e1+...ek)VeI...ek)(eI)f(e2)...f((k)deidek =

Rk and

21

NON-LINEAR HARMONIC ANALYSIS

nk(eieix, ...,

elekx) (x) =

eix(e1+...ek)Ak(eI

... Xk) .

(Note that this realization is only valid for multilinear operations verifying some mild continuity conditions.) This realization permits us for example to show that the study of fo lAt * b (kt * fat can easily be converted to the study of

J to * (iAt*b 95t* fdtt 0

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

0

(' t/it*b(kt*f 0

if we take we have

dt

_

('eix(eI

J

(f(tei) 3(te2)dt

2)

2

(ow

with support in 1 <

2 and

supported in lei < lU

5l'1(t(e1

where 0 I(() is any function equal to 1 on 1.9 < lel < 2.1 . In this 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.

R. R. COIFMAN AND YVES MEYER

22

PROPOSITION 1. Let I'k(61

6k)1 <

then

1

(max

k2

k

A(fI...fk) =

ei

re

I

4e1... ek)f(e1) ... f(ek)del ... d fk

maps L2 x x L2 L2 continuously. This is an elementary result which we leave as an exercise. It would be desirable to obtain more subtle results for L2. THEOREM [5].

Id

Let A(e,...4k) be such that

al, Ca ,

dial

jai _S [.n] +

a=(a1"'ak)IaIai

2J

then

A(fI...fk): LPIxLp2...xLpk,L4 k

>pi>1

>Pi

More generally one can take operators of the form

o(f I" f) k

=f J

I"

k

1

dd d4k 1

with o(x, d) such that 1a

dX o(x,d)I < Ca1R

+IeF

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 e LP , f 2 = a e L°° and write

A(f, a) (x) = IJ

eix(e1+e2)x(41,

e2)a((1)f(e2)del de2

[fk(x_tx_)a(t)] where k(x 1,x2) = A(x1x2) satisfies

f(y)dy

Ik(x 1, x2)I <

IVkI <

x12 +Cx22 ,

3

Ixi

thus k(x, y) = f k(x-t, x-y)a(t)dt verifies conditions 1°, 2°, and 3° of the T(1) theorem: thus to verify the boundedness in L2 (for a fixed) it suffices to calculate T(1) and T*(1). These can easily be calculated, in fact T(1) = J eixel X(e1, 0)9(S 1)dS

l

which is a linear Calderon Zygmund operator applied to the bounded func-

tion a. Thus it is in BMO. Since T*, (in f, for a fixed) is given by the symbol 'k*((1, e2) _ X(el, -el -'), verifying the same hypothesis the result follows for T*(1). Property 3° can be verified once we observe that the same induction shows that

IIT(e'xe)IIBMO < C .

and that this condition implies

f IT(-Ot)ldx < c on any interval of length t III

.

24

R. R. COIFMAN AND YVES MEYER

In fact Ot(x) = feiX6

t

(et)de4 lIT(ot)(x)IIBMO

,

now assume cbt(y) is supported in I. lox-y(y)l

fIT(9t)(x)I <_

dy

-t <

if lx I > 2 It l thus

JTt(x)Idx < c I

if dist (I,I) > t

.

Since T(4t) is in B.M.O. it follows that

t > III

fIT(t)(x)-miT(t))

ftTdx < c ,

1

thus we obtain

J

IT(Ot) ,At 15 t

.

provided trtOt 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

NON-LINEAR HARMONIC ANALYSIS

25

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) 0 "U

+a36u"u xER,t>0. &A3

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

u(x,t) = 1: Ak(v) = I uk(x,t) where k

f ix(16i) uk(x,t) = J e 1

ak(e,t)v(e1)... v(ek) del ... d6k

(here v is some unspecified functional parameter) plugging into the equation we obtain r3uk

at

n-1

,auk

3

ax 3

j=1

k

it

we find first that ak(e, t) = e

/k

ak(6)1

1

k

6i3

r)x Lr uiuk-l

e3

1 ak(e), and that

k ( l ; J )=_3(e1)

13

k-1

Y aj(61 ... ej)

6k)

j=1

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

26

R. R. COIFMAN AND YVES MEYER

under permutations of ek, so that the equality should be true only after symmetrization.)

It can be shown (not so simply) that e1+...+ek ak

(e1+Q (e2+N ". (ek-1+ek)

solves the recurrence above. Thus

uk(x,t) =13 i

J

vx,t

If we let

;k-1

1

1

=

k

k i=l vx,t(ei)de1 ...d,k

eixe+te 3 v(S)

p. v. f --L- v,

uk(x,t) = iax J

and

fV 0 d71

u(x,t) = _ J ,U-Cv)-1(1)dri _

where

satisfies the integral equation

f

I I- ' T4,

d

NON-LINEAR HARMONIC ANALYSIS

27

This set of remarkable formulas constitute the so-called method of "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 -0 ItbIL,. <E0.

ij

1

The bij(x) are complex valued functions. We would like to prove L2 estimates for such functions as e-tL, eit L, eitV'L, e-tVL

These provide us with estimates for solutions of the initial value or Dirichlet problem for 2

-1u=Lu, au= 2 -Lu

u= -Lu,

at

a2 -Llu=0

/

&2

or more generally for F(L) where F is a bounded holomorphic function in a sector jarg zj < S containing the spectrum of L (this can be shown to be true if e0 is sufficiently small). We define formally

F(L)

2ni

J

r

where r is the boundary of the sector.

d

R. R. COWMAN AND YVES MEYER

28

We let 92

R1=

R =

ax

"

-L

(C-L)-I

=

i

1

2

2

+A-Lr bil

=

I

fix) =

(I-BR) (C +A)

(C+A)-1(I-BR)-t

,

=

(C+A)-t(BR)k

We thus find 00

00

F(L) = I J (C+A)-t(BRekF(C)dC = IAk(B) . 0

r

0

To better understand such expressions we consider the first two terms

(A0(B)f) =

J r

Q F(C)dI

1

2ni

1

r

C_ICI2

F(C)dCf(C)

=F(IKI2)f(C) = m(C)f(C) .

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

c gIk

and

thus A0 is a Calderon-Zygmund kernel operator mapping L2 - L2, L°° - BMO. We now consider A1(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 00

AI(B) =

t B a2 1 F(it) dt t it+A axiaxl itt+

f

J

00

and consider separately t > 0, t < 0. Change variables t = ± , to S2 obtain 2 terms of the form 2

J

1

i+s2t

0

B s2D;D

1

µ(s) as

µ(s) = F(1/s2) .

D.D.

D.D.

i+s2A

Now write s2DiD) _ s2A i+S2O i+s 2O

This gives f

D.D.

J

I

A

i+s2O

D.D.

00

1

B

j µ(s) ds s which we recombine with the other

1

0 1+s2O

term corresponding to t < 0 to get

A0(B) B

D_J .D f

The second term is of the form

1

If'o 0'

1+s2O

B

1

1+s 2O

µG)

s ds

DiDI

The operator in curly brackets has a kernel satisfying *. Thus, to check boundedness in L2 we need to calculate T(1) , T*(1). 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 r0 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 N, z cc, z =x1+ix2 since e 161x12x2 =

(61 -'62) el x

and the spectrum is the whole complex plane. We are thus led to consider for 0 e Co(C1) expressions of the form

=-2ai

fx)1(idxAd. C

C

These formal expressions need to be given sense and one should prove that such an expression is (oq,)(L). In the case L0 = 2

easily justified using the Fourier transform. Since d

21 1(f

1

dX A d f (e) = 21

4(X)

1

ax

X-6

aX- X-L C

dX

= O(Of(e)

(here we used the fact that

at 0 ).

tai az

=

S(z), where S is the Dirac mass

z

We see that

O(L0) = fe'x6OC6)i(6)d6

If we now consider L = -L ITO

like to understand the nature of

=

f

cv(z-Y)f(y)dy .

where a E L°°, Ila11 < S0 we would

31

NON-LINEAR HARMONIC ANALYSIS

(L) = 2ni f c(A)

aX L-A as A as

for functions / homogeneous of d°0. For example, we would like L (this will give us L = L L which will turn out to be very important). We can, in this case, calculate explicitly

(L-A)-1

,

in fact, L-1f = J

"(Y-W )

f(l+a)fdv(w)

.

We now observe that if h(z) is such 21

h=l+a and h(z)-zeL°°

then =

L(ei(h(z)C+i )) =

LXC

CXC

from which we find

(L-C) f = XL if

and

XL-1 X f

X

or more precisely

n

f

(L-C)-1f and

=

e'{(h(z)-h(w))C+(z-w

1

#

f(w)dw

e"(h(z)-h(w)X+(z-w)f# dh f(w)dw

32

R. R. COIFMAN AND YVES MEYER

c(L)f

(

)ei1(h(z)-h(w))C+(z-w)Z1dr

f(w)dw

f1fo

fk(h(z)-h(w),z-w) , f(w)dw where

f)eidC

k(u,v) = C

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

Lf=-1 L

fw (z-w)2

dw-!!f. 0z

Since Lh = 1+a, for this to be bounded we must have !Lh c L° i.e., a * 2 c L°°. In other words, in order to have a functional calculus in z

it T (consisting of bounded operators in LZ(CI)) it is necessary to assume that a c L°° and the 2-dimensional Hilbert transform (or Ahlfors Beurling transform) of a is also bounded. This version of H°° 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

33

We proved earlier that the commutators

-yy

A(x) = a(x)

Ck(aIf) _ J(A(x)_A(y))kf(y)d

sati sfy IICk(a,f )II1 < Ckllallk llf 1

2

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

f

f(y)

x-y+iA(x) - iA(y)

dy

is a bounded operator on L2 if IIA'II, < c and that A(x)-A(y)

I

i

x-y x-y

< ec lei IIA'Ilo

f(y)dy

IIf II2

L2

It was proved in [4] by a careful analysis of the functional calculus interpretation of the Cauchy integral that in fact, < c(1+k4) IIaIIkIICk(a,f ) I I

llf 112

L2

This more precise result implied a much stronger statement. THEOREM 1 [4].

Let 0 E C11(R). A real valued IIA'II. < - then

x-y


X-y y

(A(x)_A(y)'f(y)

L2

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 A(x)-A(y) x-y IITe(f)112 =

fe

(y) dy

< C(IIeajl-+1)NIIf II2

x-y

2

for some N>0, C>0. In fact, this will imply that for ¢(x) such that

+IeIN)d5 <

we have A( x ) - A( y )

P

fc){fe

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

(

xydy

d 2

2

<

fiei

IlTfII2de < cf 112

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

i

TA(f)= re

x-y

dy

grows like 11A'1111 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).

NON-LINEAR HARMONIC ANALYSIS

35

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 e 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) (depending continuously on I) for which Ix eI: Ib(x)-C(I)I < aI > N III Then b e BMO and IIbIIBMO < cNa.

The main idea to estimate the BMO norm of T(1) is to replace inside each interval I , TA by an operator TA where AI = A on a large I

and Ai has a smaller Lipschitz norm, and then compare (1) to TA(1). This is achieved via the following lemma, the first of

fraction of TA

I

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

IEI >

III

4 and

CI-3M M.

36

R. R. COIFMAN AND YVES MEYER

In that case consider the smallest function AI > A with A'(y) > 23 .

Ik

then have AI(y) = A(y) except for disjoint intervals Ik on which AI(y) = M . But

< 2M II-UlkI +

+

fiA(Y)dY = "_.UIk

'fUlk

=2MIII-3MIUlkI 3MIUIkI < (2M

- m1(Ai)) II I < (2M - M) III =III

i.e.,

IuIkI < 4 III

.

Since I< Ai(y) < M2, we have 3M-3M
M

We

NON-LINEAR HARMONIC ANALYSIS

37

We consider the function

2M-A'(x) = Ai

IA; I < M2 .

Then we have ml(Ai) > M and we construct A 11 as above. A11 = 2M(x-a)-A(x) except on a set of meas <

III

4

A(x) = 2M(x-a) - A11 = Al

A'(x)=2M-AI=A1

3M-3M
The main result is the following.

THEOREM (G. David). Assuming that there exist S > 0, c > 0 such that for each I there exists KI(x,y) satisfying standard estimates uniformly in

I

with T1(f) = J K1(x,y) f(y) dy

satisfying

IITI11L2(L),L2(I) < Co

and there is a subset E C I with IEI > 6111 such that Hx rE , Vy eE ,

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

.

38

R. R. COWMAN AND YVES MEYER

Then T maps L°° to BMO with IITIIL°°,BMO

C77 C0 . <-

iMA(x)-A(y)

If we let a(M) =

x-y

sup

x-y

II f e

dylisMo a direct application

IIA-liar 1

of this theorem choosing for each interval I A1(x)-A1(y) im

K1(x,y) =

e

x-y x-y

shows that

a(M) < Ca( M) i.e., a(M) < c1(1+M)N for some N or . A(x)-A(y) II x-y e f(y)dy L2 < C(1+IIA'II00IIflIL2 x-y t

IITf11 2=

Proof. We consider f e L°°, f = fXl + f(1-X1) = f1+f2, checks that

llf1l00 < 1 .

One

f IT(f2)(x)-T(f2) (x01 < clll I

from which it follows Ix el:ITf2(x)-c1I > cI < 770111 if c is large enough. We now claim T(f 1) is well approximated by T1(f 1) , which we con-

trol, so that we want now to estimate

f E

IT(f1)-TI(f1)Idx

NON-LINEAR HARMONIC ANALYSIS

39

where E = I - UIi , E' = I - UIi where Ti = (1+s) Ii

J

fIKxy)_Ki(xY)ldY

xEE' yd

=

J

IK(x,y)-K(x,Yi)+KI(x,Yi)-KI(x,Y)IdY

xEE' I.i

where we have used the fact that K(x,y) = KI(x,y) outside Ii and yi are endpoints of Ii, (K(x,yi) = KI(x,yi)). This integrand is dominated by the Marcenkiewitz 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 operators. 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 1 -parameter group of measure preserving transformations on a measure space (X,dx) and A(a I . . .a k, f )

a k+1 multilinear operation on R given by

A(a; f) = ff k(x-t, x-t2 ... x-tk, x-y)JJai(ti) f(y)dtdy Rn

=J

satisfying

40

R. R. COWMAN AND YVES MEYER k

I IA(a; f)II, p<_ [I IIaiII,oIlfIIL 1=1

Then the multilinear operator A on L°'(X)k x LP(X) -> LP(X) defined by

A(A,F) = J

frk(t1,t2. tk,s)[JAi(Ut .x)F(Usx)dt ds 1

satisfies

I}A(A,F)IILP(X)` 11

IIA'IIL°°(X)IIFIILP(X)

(note that the constants are the same).

The examples we have in mind are the following. X = Rn, dx Lebesque measure, e a unit vector Utx = x-te

If we take Hf

fR

.

f x t dt, A f = f x -te dt . Or consider

f

Ut(ei9) = ei(e-t)

X = T1

4f - c rf(0-t)

J

1t

dt

ctg t

If we take k

Ak(a,f) =

f(A(x)_A(y)dy

x-y

f

SE--y

k

f(A(x)_A(x_t))

A? t dt

NON-LINEAR HARMONIC ANALYSIS

41

then Nk,e(A' f) =

ftf

/

k

t'

A(x-te),j f(x-te) dt t

Multiplying by i2(e) and integrating on IeI = 1 in Rn we get

S2(e)

A(x)-A(x-te)

k

t

f(x-te) dt t

f

(A(x)-A(y)lk x-y I K(x-y)f(y)dy where

k(Y) _

WYIYI) 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 L0 for example, we took L0 = -A and L = -A+Y- blJ

ax-E

e0 and

then proceeded to expand functions of L, F(L) as a power series in terms of the coefficients of L-L0 i.e., we wrote 00

F(L) = F(L0) + I Ak(b) k=1

where the Ak(b) was an operator valued homogeneous polynomial of dAk 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 holomorphy 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 flow to our first example where F(a) = sgn(1i dx)

where a is a function on R, and F(a) is an operator on L2(R). The series expansion was f(y) (7 A (X

F(a)f=1: J

(1-a(y))dy,

y

0

A'=a

Let us consider for simplicity the commutator series Fo(a) f > ` I (A(xX y (Y))k X_ y fly

J

=

(f )

0

We wish to find the smallest norm

III

III

(i.e., the largest Banach space)

for which *

IIAk(a)IIL2 L2 <_ Ck IIIaIIIk

in particular for k = 1 we must have IIA1(a)IIL2 L2 <_

Ca

If we decide to define IIIaIII = IIAI(a)II(L2

.

L2 it certainly defines a

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

NON-LINEAR HARMONIC ANALYSIS

AI(a)f =

We have already shown that

f

A(x)-A(y ) f(y ) dy . x-y xx-y

1IA I(a)11

L2L2 < c Ila II, and it can easily be

shown that IIalI, is dominated by cIIAI(a)II

to consider the L2 norm of

2

L L 2.

(In fact it suffices

(x-x 0)2 A I (a) ()(I) - 2(x -x 0) A I(a) ((y-y 0) XI) + A I (a) ((y-Y o)2 X I)

on the interval I whose center is x0. By letting I shrink to x0 and

using the L2 estimate on AI(a) one obtains an estimate for a , for simplicity one can assume that a is smooth since the estimate does not is equivalent to the L°° depend on this condition.) Thus our norm III

III

norm. The estimates * have already been proved in this norm. We may also find the domain of holomorphy of F0(a), in fact

F0(a)f

f(y)dy = fT-_ A (x1X -A 1 _y -f x-y (A(x)-A(Y))

f(y)dy

x-y

We have seen by the method of boosting for the Lipschitz constant

that this will be bounded on L2 as long as ,/A(x)-A(y))

1

1 - A(x) -A(y) x-y

1`

clearly the case whenever infI x,y

x-Y

for some q5 c C°°

A X_- - 1I > 0 and we conjecture

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(l1 dx) . [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) = fo er(a(t))dt and consider the L2 operator valued functional

aa)f

. 217i

p.v.

f

00

z(s)-z(t) dt =

Ak(a) f

_ ,p

here again one can define the norm IIIaIII = IIAI(a)IIL2,L2

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 IIaIIBMO < So then 1 -So <

Iz s -z t

I<1

-

S -t

,

and this is precisely the condition permitting us to write

C(a) = f 4(z ss_z t) fLt) dt giving rise to a bounded operator on \ S-t Thus BMOis the natural space of holomorphy. As you recall from S. Krantz's lecture one can express the Szego projection S(a), projecting L2(1', ds) L2(R, ds) onto H+(I', ds) (H+ is the space of functions in L2 admitting a holomorphic H2 extension to the "left" of T ) 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 L2.

Riemann mapping function.)

Another remarkable example leading to an "exotic" space of holomorphy and to interesting geometry involves the functional calculus in L = 1L - . As already seen it was necessary to assume that a E L°° z

and a *

12

E L°° with small norm. (This was obtained by considering the

z

example T(a) = L .) It turns out that T(a) is analytic in a , relative to the norm IIIaIII = IIail0 + Ila *

a

II.., and the condition really means that

Z2

there exists a bilipschitz map h(z):C -C such that ai 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 AI(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] [2]

[3]

[4]

A. P. Calderon, Cauchy Integrals on Lipschitz curves and related operators. Proc. Nat. Acad. Sci., U.S.A. 75 (1977, 1324-1327. R.R. Coif man, D. G. Deng and Y. Meyer. Domaine de la racine caree de certains operateurs differentiels accretifs. Ann. Inst. Fourier 33, 2 (1983), 123-134. R.R. Coifman, Y. Meyer, Lavrentiev's curves and conformal mappings, Rep 5. 1983, Mittag Leffler Inst., Sweden. R.R. Coifman, A. McIntosh and Y. Meyer. L'integrale de Cauchy definit un operateur borne sur L2 pour les courbes lipschitziennes. Annals of Math. 116 (1982), 361-387.

[5]

R. R. Coifman and Y. Meyer, Au deli 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.

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. [10] E. M. Stein, Singular Integrals and Differentiability Properties of Function, Princeton University Press (1970). [71

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 classical 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

1.

The maximal function, Calder6n-Zygmund decomposition, and Litt lew ood-Pa ley Stein theory We hope here to review briefly some aspects of the classical

1-parameter theory of these topics. The three are inseparable and we hope to stress this. We begin with the fundamental

CALDER ON-ZYGMUND LEMMA. Let f(x) > 0, f t L1(Rn), and a > 0.

Then there exist disjoint cubes Qk such that

(1) a <

IQ

'

I

f(x)dx < 2na

k

Qk

(2) f(x)
f If(x)Idx

(3) IU QkI <<

a

-

Rn

Proof. Let Rn be subdivided into a grid of congruent cubes so large that IQ fQ f < a for all of them. Subdivide each cube in this collection II

into 2n congruent subcubes. Select from these the cubes Q' such that 1 f, Qf > a . For these Q' we stop the bisection process. For the

10

rest, we continue until we first arrive at a cube Q' such that 1 f ,f > a at which point we stop. 1

I

Q

The cubes Q' at which we stop are then our Qk. By construction f > a. Let Qk be the cube containing Qk which was bisected

IQkI fQk

Qk. Then

1IkIf1k f < a.

It follows that

and since we did not stop at

k,

MULTIPARAMETER FOURIER ANALYSIS

f 1Q

k

1

k1 1 f

49

f< 2na ,

f < 1Q k

Qk

proving (1). Notice that (3) follows from (1) because 1Qk' < a fQk f so summing on k, we have

('

IUQk1
f

f

Qk

Rn

Finally, (2) follows, since for each x I UQk, x belongs to a sequence of cubes Ck whose diameters converge to zero and such that Ck

It follows from Lebesques theorem on differentiation of fc k f < a.

integrals that 1(x) < a 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) = sup

If(t)(dt

1

r>o IB(x;r)1

B(x,r)

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

sup

1

xEQ dyadic cube IQI

JI If(t)I dt Q

(Recall that a dyadic interval of RI 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 Q1,Q2 are dyadic either Q1 f1Q2 = 0, Q1 C Q2 or Q2 C QI .) Then the following simple lemma sheds some light on the relationship of the Calderon-Zygmund lemma to the Maximal Operator.

ROBERT FEFFERMAN

50

LEMMA. Let f > 0 c L1(Rn) and a > 0. Let Qk be Calderon-Zygmund cubes as above, and let Qk denote the double of Qk. Then there exist positive numbers c and C such that (1) UQkC 1Mf > ca(

(2) UQk ) (Mf > Cal

(3) Furthermpre, if Qk are dyadic, then UQk )1MSf > Cal.

Proof. (1) Let x c Qk. Then there exists a ball B(x;r) such that x c Qk C B(x;r) and B(xx;rl < Cn. Then k Mf(x) >_

Ifl >

1

B(x;r)

,/

B(x;r)

IQkl

1

(IX)l) IV

IfI > 1 a Cn

Qk

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

ff

I

B(x;r)

f+ I

f

B(x,r)fld[UQk]

Qi

< ajB(x;r)I }

Qi

< alB(x;r)I ± 2na

Key point : if Qj fl B(x ;r)

0 then Q) C B(x;l Or) so that

I IQiI
Y IQjI

f < Cna IB(x;r)l

.

MULTIPARAMETER FOURIER ANALYSIS

51

The use of the Ca1deron-Zygmund lemma is apparent in the Calder6n-

Zygmund Theorem on Singular Integrals: Let X and N be Banach spaces and let B(X,N) denote the bounded operators from X to N K: Rn x Rn/lx=yl -, B(X,N) satisfy (1)

B(x

<

Ihl1

Ix-

n++8

.

Let

for IhI < Ix-yI 2

and for some S > 0. and

(2) if Tf(x) = fRn K(x,y)f(y)dy for f c LP(X), and suppose for some

p0> 1

IITfII

<< CIIf1I

LPO(X)

LPO(N)

Then, for T we have I#XI ITf(x)IN > all <

IIf IILI Q

(X)

and

IITf1ILP(N) < CPIIfIILP(X) for 1 < p < po

.

Proof. Let a > 0 and f c L1(X). Set 1

f dt

if x cQk,

IQ kI

Qk

g(x) = f(x)

if x / Qk .

and b(x) = f (x) - g(x) . Then Il ITg(x)IN > a1I <

PO

IlgllL ° L

As for Tb(x), suppose x I UQk.

< a: IIgiILI SI: IIfIILI(X) (X) < (X) - a

ROBERT FEFFERMAN

52

Let bk(x) = XQk(x) b(x) ;

f

bk(x) dx = 0. Then k

Tbk(x) = J

K(x,y)bk(y)dy

.

Qk

Let Vk be the center of Qk; then

J

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

bk(y) dy = 0

J

Qk

Qk

so

Tbk(x) =

J

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

Qk

and ITbk(x)IN <_

diam(Q )s +3 Ilbkll Lt (x) Ix-yk1

summing over k we have diam(Qk)s

f IT(b)(x)IN
X/Uijk

x'!k

dist(x,Qk)n ,

Ilb k II

L

t dx < CII f II t

L (X)

Thus

I{Tb(x)IN > all S Okl +1 llflll

<_ SL IIflll

.

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 If = f * K where K(x) is a complex-valued function satisfying

MULTIPARAMETER FOURIER ANA LYSIS

(a)

(0)

53

IK(x)I < C/ Ixln

I

K(x)dx = 0 for all 0 < P1 < p2

P1<_lxIp2

and

S

(y)

IK(x+h)-K(x)I < C IIh°

IhI <

IxI

2

The Riesz transforms Ref = f * xJ/Ixln+l are especially important since

they are related to HP spaces and analytic functions. For a Calder6n-Zygmund singular integral T , it is easily seen to be bounded on L2(Rn), since K(e) e L°°. Also, since T* is also a Calderon-Zygmund singular integral, T is bounded on the full range of LP(Rn), 1 < p < _. 2. Littlewood-Paley-Stein Functions. The most basic, simplest of these are the g-function and S function defined as follows: Let A e Cc`(Rn), ek = 0. Let t/rt(x) = t-n to \t/ for t > 0. Then

Rn

=J

g2(f )(x)

If *c1it(x)12 dt

t

0

S2(f)(x)

=ff

if *

t(y)I2 anal

r(x)

where r(x) _ {(y,01 Iy-xI < ti. Then it is a basic fact that 'IIS(f)IILp < when 1 < p < oo. If, say, Vi is Lpf suitably non-trivial, (radial, non-zero is good enough) then the reverse

CPIIfIILP and Ilg(f)IILp
P

Ilf 11

inequalities hold: IIS(f)IILp _ cplifliLP and IIg(f)IILp>-cPIIfIILp

ROBERT FEFFERMAN

54

Now take S(f) . We want to point out here that S is a singular integral. In fact define K:Rn F. L2(F(O); dydt) by K(x)(y,t) = &t(x-y). Then

S(f )(x) = If * K(x)IL2(r,dtdy/tn+1)

and K satisfies IK(x+h)-K(x)I < C

IhI <2 IxI

IxIh' 1

Also by a Fourier transform argument

IISfII

.

so the

< cilfiI LZ(R n)

I- 2(R n)

Calder6n-Zygmund theorem applies. In fact, the adjoint operator also maps Lp(L2(P)) - 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 < w. 3. The Hardy -Litt lewood Maximal Operator as a Singular Integral. Let

O(x) E C(Rn) and suppose for IxI < 1 , O(x) = 1 and for IxI > 2, O(x) = 0. Then define K:Rn+, L°°((0, 00);dt) by K(x)(t) = -ot(x) _

t-no(x/t). Then IoxK(x)(t)I = It (n+t)0k x

< CV

Ixln1

00

fl

+i

(since if t > IxI/2 , V (x/t) = 0 ). Again L
InI1 IxI

and we also have

IIf *

K11L°°(L<_ CIIf II

L°

if IhI <

IxI

2

since If * ot(x)I < II-kII1

IIf II°

Then Mf(x) -If* K(x)I ,° so M is bounded on Lp(Rn) , p'> 1 and weak 1-1. The Estimates for Pointwise Convergence of Singular Integrals on L1(Rn) . Suppose that K(x) is a classical Calder 6n-Zygmund kernel and let KE(x) = K(x) )(IXI>E(x), for e > 0. We are interested in the 4.

existence a.e. of lim f * KE(x) for f e Ll(Rn). In order to know this, it E-40

55

MULTIPARAMETER FOURIER ANALYSIS

is enough to show that T*f(x) = sup If * KE(x)I satisfies the weak type E>0

estimate IIT*f(x) > all < a f

If I .

R

It turns out that by using the Hardy

Littlewood maximal operator it is not difficult to prove T*f(x) < CIM(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

LI(Rn). This inequality follows easily from the observation that T* is a singular integral. Let 1

if IxI < 1

O(x) c C (Rn), O(x) =

0 if lxI >2 and set K,(x) = K(x)[1 -0 QL)]

Then

IK,(x) -K(x)I

I In XIxI<2e (x) so that T*f(x) < sup if * KE(x)I + Mf(x), and so we need only show that E>0

sup If *kI is weak type (1,1). In order to do this let H:Rn , L((0,-);de) E>0

be given by H(x)(E) = II

(x). Then IH(x)-H(x+h)IL <

bounded from L2 - L2(L), so H is weak 1-1 .

C InI 1

and H is

Ix

5. The Maximal Function as a Lit tlewood-Pa ley-S tein Function. Let

f c L2(Rn), f(x) > 0 for all x. Use the Calderon-Zygmund decomposition with a = Ci , j c Z for some C > 0 sufficiently large, to get (dyadic)

cubes Qk where 1 f f - C). Define f .l decomposition,

as in the Calder6n-Zygmund

IQ)kI Q)k

1

f

if x c Q

IQ,

k

Qjk f(x)

and A f =fj+1-fj, then observe that: i

if x

U Qk

56

ROBERT FEFFERMAN

(1) jf lives on k Qk and has mean value 0 on each Q. (2) Aif is constant on every Qk for i < j.

(3) fJ - 0 as j,--o and f.J

+Do

f

as j -.+00 so f =

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

i

f

are orthogonal so that

1/2

If IIL2(Rn)

F Af. j=-

= (Y IlojfIIL2)= II (Y lojf(x)12)

1/2

I1L2

Finally, observe that the square function (I IA f(x)12)1 /2 is j

essentially just the dyadic maximal function. In fact, if CJ << MS(f)(x) then x e Qk for some k and Aj(f Xx) 'ti CJ. It follows that 1 /2

C i

Iojf(x)12)

> cMSf(x)

11

Before finishing this section, we shall need estimates near L1 for the maximal function.

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

J Mf(log+ Mf )k-1dx < 00 if and only if

r If l(log+ If I)kdx < 00 .

Q

Q

The proof runs as follows': If f e L(log+ L)k then

IlxIMf(x) > all < C

J lf(x)l>a/2

and so

f(x)dx

57

MULTIPARAMETER FOURIER ANALYSIS IIMf II L(log

f(log a) k-1 a

J

L)k-1 <

lf(x)Idxda <

If(x)l>a/2

1

If(x)I

J

Qp

llf II

f

Q (log a)k-ldadx

1

L(Iog L) k

Conversely, (Stein) Calder6n-Zygmund decompose Rn at height a > 0. We have

J

f(x)dx < J f(x)dx< Y J

M&f(x)>Ca

k

UQk

f < Ca j IQkl < CaI{Mf > call .

Qk

k

This yields 00

ra

5 If(x)I(log+Msf(x))kdx <

J

If(x)ldx(log

a)k-lda

Qk I

M6f(x)>Cna

00

<

r J

a a

If(x)Idx (log

a)k-1da

MSf(x)>Cna

1

<

I

f

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-

ROBERT FEFFERMAN

58

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 - fix, 8 > 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 of 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

(xl,x2, ,xn) - (Six1,82x2, ,Snxn), where Si > 0 is arbitrary. This is the "strong maximal operator," M(n), defined by

M(n)f(x) = sup 1 xER IRI

JR

If(t)I dt

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 IIxIM(n)(f )(x) > al I < a Ilf II I(Rn L

)

For instance when n = 2 and when f8 = 8-2X(Ixl I<8/2)x (Ix2I<8/2) then for

1x

1

1

,

Ix2I > 28,

M(2)(f8)(xl,x2) =M(1)(XIxII<8/2)(xl)M(1)(XIx2I<8/2)(x2) ti

I lI I

I

and

I#x EQ0IM(2)(f8)>aI I ? I#xI Ix1I Ix2I < a , and 8< 1x11<1 II ti a log !

if a=1/8.

MULTIPARAMETER FOURIER ANALYSIS

59

If we have a weak type inequality, we must have i1M(2)(fg) > all

Ilfsll

a so that llfsll > c log . and the smallest Orlicz norm for which this holds is the L(log L) norm. A similar computation in Rn reveals that for M(n) to map LD boundedly to Weak LI we must have Lip C L(log L)n-I L)n-t The next theorem shows that indeed M(n) does indeed map L(log boundedly into Weak LI . THEOREM OF JESSEN-MARCINKIEWICZ-ZYGMUND (1935) [1]. For

functions f(x) in the unit cube of Rn we have lixeQo,M(n)(f)(x)>ail a llfll

L(Iog

L)n_t

(Q0)

The proof is strikingly simple. Define Mi to be the 1-dimensional 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 t , 3E2)1 say R = I x J . Then

ffif(xt.x2)idxidx2 = III f(th ff(xi.x2)dx2)dxi

R

I

R

(2.1)

J

lf(xl,x2)ldx2 < Mx2f(xt,-2)

111

J so (2.1) is

< lIl

M2f(x1,x2)dxl <Mx1(Mx2f)(x1 ,x2)

60

ROBERT FEFFERMAN

Thus, for all (x ,x2) 'r QO' M(2)f(x1,x2)

<- Mx ° Mx2(f )(xl,x2) 1

We have seen that Mx2 maps L(log L)(Qo) boundedly into L1(Q0) so that IIMx2fIIL1(Q0)

-

CIIfIIL(log L)(Q0)

and finally I1xEQoIMx1Mx2f(x)>aII < a'IIMX2fIIL1 (Q


I1fIIL(log L)(QD)

Now, this method of iteration in the proof above gives sharp estimates Now,

for M(n), and it may be suspected that the whole story of 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 understood. The model for our method is the following, where the operator. M(n) is M(n-1 ) controlled by COVERING LEMMA FOR RECTANGLES OF THE STONG MAXIMAL

OPERATOR [3]. Let 1Rk1 be a given sequence of rectangles in Rn C B(0,1) whose sides are parallel to the axes. Then there is a subse-

quence 1Rkl of 1RkI so that

MULTIPARAMETER FOURIER ANALYSIS

61

(1) IUkkI > cnIURkI 1

n-1

/

(2)

IleXp(I X'lk)

II'LI(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 e IM(n)f(x) > at there is a rectangle Rx containing x with

LIfI>a.

(2.2)

I

Without 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 IUkkI <_ a Ilfll L(log

L)n-I(Qo)

By (2.2), IRkI < a fW IfI and summing we have

IURkI

r

IfI 1" X,k < a

IIf IIL(Iog L)n-1

xRR

k

exp(LI /(n-1))

Qo

Now, let us prove the covering theorem. We shall proceed by induction

on n. Assume the case n-1 . Let R1,R2, ,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 xn side length is multiplied by 5. Then we describe the procedure for selecting the Rk from the Rk: Let K1 = R1 Suppose R12, ')Rk have already been chosen. We continue along the list, and each time we consider the rectangle R we ask whether or not

62

ROBERT FEFFERMAN

IR n [U(Rj)d]I < z [RI

where the above union is taken over all the Rj , j
U

Rn

(Rj)d

>21R1.

kinR=c kj L before R Let us slice all rectangles with a hyperplane perpendicular to the xn

axis. Then if slices are indicated by using S's instead of R's

,

Isn[U(Sj)d]I > 2 ISI so that M(n-1)()(U

(W

on UR j , where M(n-1) is acting in the

1) >

jd

2

x1,x2, , xn_1 coordinates. By the boundedness of

M(n-1 )

on, say, L2

(by induction) we have

IURjI < CIU(Rk)dl < C' I IRkI < C'IURkl

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

U

(Rj)d

Wjnkk j
If we again slice with a hyperplane perpendicular to the xn axis,

kn Ujl <2 (kl j
MULTIPARAMETER FOURIER ANALYSIS

63

and if

so that, if Ek =-9k - lUk^`Sj we have IEkI > 2 1. 4) t L(log L)n-1(So) we shall show that

<

(2.3)

X k (k

C II(bII

L(log

L)n-1

so

which will give

\1 /(n-1)

f exp S

X

1

k/

0

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

fI

Xgk0dxl..dxn-1 < I k

S0


J 0
f

'

IEkI-L

k

fO

<

M(n-1)(4))dxldx2...dxn-1
L(log L) n-1

S0

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.

ROBERT FEFFERMAN

64

To illustrate the method of the machine, we consider a maximal operator whose relation to multiplier theory will be studied below. Suppose in R2 we consider the class 91 of all rectangles of arbitrary side lengths which are oriented in one of the directions ek = 2-k, k = 1,2, , measured from some fixed direction, say the positive

x-direction. Define the maximal operator m by mf(x) =

sup

1 f If(t)I dt

xEREQ IRI 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 IImf(x)>afl <

a IIfIIL2fR21

The proof consists of showing that, given a sequence of rectangles 3Rkl belonging to S?j , there exists a subfamily {RkI such that (1) IURkI > cIURkI

and (2)

k11L2(R2) < IURkI1/2

ti

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

k

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 thai their longer side lengths are decreasing. Then consider the rectangle R ti following Rk_1- Consider in particular IRI Y IRURjI j
=

IRI J

I XW1dx .

R j
MULTIPARAMETER FOURIER ANALYSIS

65

If this is less than 1/2 , select R as Rk. If not go to the next rectangle on the list, and apply the same test to it. In this way we obtain

the desired Rk. Notice that

II y XWkI2 =

f

X j,kXk,Xkk =

j

2j

xAk +

j
IRkI k

1

(2.4) 2

k

L.r fXWi +

I IRkI < 2 Y IRkI < CIURkI

1r j
k

k

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

IRI

I

before R

IRfRjI > 1

.

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

ROBERT FEFFERMAN

66

Then by dilating S if necessary, we can assume that Rj is centered at the same center as S. Now the point is that (2.5)

iSI

-

c

JRI

In fact,

h

hH/B'

IRjnSI

hL

IS 1

- LO'

and

IRjnRI ti h h O'-9

_

h e(e'-0)

h

IRI

and our inequality (2.5), taking into account that f ti L, is

I > I or 9'- 6 > c O' 0' - (0'-0) and this in our case of Bk = 2-k is valid with c = 1/2 . (If ek = (1 _ )k, c = e.) Then summing over j in (2.5) we have

I$11

fL & be

S

re

R

.

I

I

JR

before R

in other words

M(2)(I X, } > 2 c on URj and by the boundedness of M(2) we see that

IUR;I _CIIYX&112
2

c

67

MULTIPARAMETER FOURIER ANALYSIS

by (2.5). We have controlled m by M(2) here in just exactly the way M(n) was previously controlled by M(n-I). 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 !Rkl there exists lkk} a subsequence so that (1) IURkj > cIURkl

(2) IILXR IIp.
k

Then IIfIIp1P

;mf> all < CC

.

In fact we have, by definition IRkl so that tmf> of C URk and 1 fR If I > a for all k. By the covering lemma, select the class 12k

I

I

kl.

k

Then Ikki < a f&

IfI

and so

Iv kl < a ff 1 X" k

IIf IIpII Y

`<

)
a IIfIIp}uRk)I/P.

a IIflIpiURkII"p

<_ C

and the estimate on Ilmf > all follows by a division of both sides by I U'k I I /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 (A I1 '021 ,'On of the positive real variable t , with each Oi(t) increasing and the family of rectangles

68

ROBERT PEFFERMAN

1Rt{t>o given by Rt = iIIt

t2(t)

,

t2 t)J .

Form a maximal operator M

defined by

M(f )(x) = u JR._

f lf(x+y)I dy Rt

Then M is of weak type 1-1 , just as in the special case of the HardyLittlewood operator where fi(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 ¶ is the class of all translates of the Rt and if R,S r 3 and R R S 10 and if R corresponds to a bigger value of t than does S , then S C k, the 5-fold dilation of R. Next, he considered the collection of rectangles Rs,t , s,t > 0 where Rs,t

r t t >' L c(s,t) i(s,t) E flx l -2'2J =[-2'2] -2 L2

where 0 is a function increasing in each variable separately, fixing the other variable. In other words, Zygmund next conjectured that since is a 2-parameter family of rectangles in R3, the corresponding maximal operator, which we shall call MZ should behave like the model 2-parameter 1

operator M(2) in R2: 11Mzf(x) > a, JxJ < 1 #{ < a 1{"L(1og L)(Ixl<1)

Not long ago, using the methods we have just discussed, Cordoba was able to prove this [7]. Let us give the proof. Suppose IRkl is a

sequence of rectangles with side lengths s,t, and 0(s,t) in R3. We must show there exists a subcollection 1Rk4 such that (1) IURkI > c}URkf

(2) 111

kllexp(L) < C

69

MULTIPARAMETER 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 IRkf1 ['Uk Rj11 < 2 (Rkl

,

that there are finitely many Rk and that the Rk are dyadic. (In fact, we may assume this because if IRI fR IfI > a for some R e 91 containing x ,

then there exists a dyadic R1 whose R1 (double) contains x such that IR' 1R1 +fj> C .) Now let R1 =R and, given R1, ,Rk, select Rk+l

as follows: Let Rk+1 be the first R on the list of Rk so that

1 fexP(x)dx
(R

j
ti

We claim that the Rk satisfy

Rk be R1,

,

`x

RN and let

X

Ukj

j = Rj

) < C'. To see this let the

exp(I X

k

- U Rk. Then k>j

N

I exp

i

j

=

r exp

J

N

Xk )dx

N-1

('

+J

ex

X+...+ Jexxk=1 1

il N

'kN-1

t1

and

j

fexp


X k

fex(kk=1 k<j

}

so we have

Now let us show that 1URjI > clURjl . Let R be an unselected

rectangle. Then

ROBERT FEFFERMAN

70

1 fex(x)dx>C 1R

I

R

ti where the sum extends only over those chosen R k which precede R.

Let us slice R with a hyperplane in the xl,x2 direction. Call S,Si the

slices of R and R Then

1 fexP())dxidx2>C. I S1

S

ti (Again we sum only over those Sj which appear before S.) Now, each Rj appearing before R has the property that its x3 (or z ) side length ti exceeds that of R. It follows that each corresponding Si . has either its xl or x2 side length longer than that of S. Call those j havyng longer x t side length than x 1 length of S of type 1, and the other of type II. Put S = I x j . Then

Jbexp(,.+jX C < II1 f f exp( IxJ

,,dxldx2

X.9i)dxidx2 = I1111J1 f I

exp

III

I

J

II

C X

dxt

-Lfexp IJI J

so it follows that

MxI[exp(YXWAMx2Cexp(lXk A >C on URA; hence

URAC MxI

[ex()]>

U

Mx[exp(X)]>/ LL

\\

I

X-,

;}

dx2

MULTIPARAMETER FOURIER ANALYSIS

71

so

JURjI < C' Ilexp(y' Xj )IIL' < C_IUkjI So far what has been done suggests the following general conjecture

of Zygmund which says: Let 0, i = be functions which are increasing in each of the variables ti > 0 separately. Define a k-parameter family of rectangles

2,

0(tl,...,tk) +S6(t1,...,tk) Rt1 ,t2,...,tk i=1

and a maximal operator on Rn by

MJ)(f)(x) = t1 t 2

supsk >o

Rt1,...,tk 1

f R

lf(x+y)ldy t1,

,tk

Then lUxJlxI < 1,M,(D(f)(x) > all
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 R3, consider all rectangles 6 of the

form s x tc1(s) x tc2(s) where Ebi 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 < 1 in R3 , there exists a rectangle S E 6 such that R C S and ISI < CIRI . Clearly, then % cannot be any better than the 3-parameter operator M(3). To do this let (k1 and 02 be increasing, 0, and continuous on [0,11 so that on each interval 2-(k+1) < s < 2-k ,

2(s) () takes every value between 1 and 1s

C.2(k+1) . Then given three

side lengths p1, p21 p3 < 1 of R , simply choose s according to the

ROBERT FEFFERMAN

72

following: say 2-k-1 < p1 < 2-k. Choose s e (2-k-1, 2-k) satisfying

'Ys

1(s)

P3 (since 1 < P3 <

- P2 - P1

P2

< 2k+1 we can do this). Then choose t

so that t(Ai(s) = p2. This guarantees t-02(s) = p3, and we are finished. We can make [2-k-1 2-k]

L2

.

assume every value between 1 and C2k on

bt letting 01(s) = e-1 /s and 42(s) = ¢1(s) on

2-k-1, 2 -k] and 952(s) -

3.

01(2(ss))

'k1(2 , 2-k-1 for all s e L2-k-1, 2 2 kl

Multiparameter weight-norm 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 Rn 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 R1 by Muckenhoupt [9] and Hunt, Muckenhoupt and Wheeden [10], and in Rn by Coifmart 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 [ii]. It is no coincidence that the class of weights w for which the HardyLittlewood maximal operator is bounded on LP(w) is exactly the same as the class of w for which all Calderon-Zygmund operators are bounded on LP(w). This is the so-called AP class of Muckenhoupt.

73

MULTIPARAMETER FOURIER ANALYSIS

A nonnegative locally integrable function w(x) on Rn is said to belong to AP if and only if for each cube Q C Rn p-1 IQI

fwdx) Q

IQI

w-1/(p-1)dx

('


Q

The smallest such C is called the AP norm. We say that w e A°° if and only if, whenever Q is a cube and E C Q, if IEI/IQI > 1/2 then w(E)/w(Q) > -q for some rl > 0. Let us list some properties of AP classes: (a) If p > 0 and w e AP then pw a AP with the same norm as w. (/3) If w e AP and S > 0 then w(3x) a AP with the same norm as w. AP, where 1 1, = 1. (y) If w e AP then w-1 /(P-1) a P

P

(8) If w e AP then w e A°°. In fact, if w e AP by (a) and (13) it is

enough to show that if IQI = 1 , and fQ w = 1 then IEI > 1/2 implies w(E) > r).

(For, in general if Q is arbitrary of side 8 and I E I > 1/2IQI , consider w(8x) on Q/8 and multiply w(8x) by the right constant p to have fl pw(Sx) = 1 . Then pw(Sx) on E/8 would have /8

[pw(Sx)](E/8) > [pw(bx)](Q/8)

w (E) >

rl

w(Q)

But by the AP condition, p-1

r)_1

< (f W-1/(P-1) JE

rw-1/(P-1)

<

JQ

E

and so

JE

w>C.


ROBERT FEFFERMAN

74

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.: If w c AP then w satisfies a Reverse Holder Inequality: (E)

IQI

f

1/(1+8) w1+s

C8 IQI

<

fw

Q

Q

for all cubes Q with 8 > 0. The constant CS may be taken arbitrarily close to 1 as 8>0. (C) From (E) it is immediate (see also (y)) that w c AP implies w c Aq for some q < p. (r)) If f is a locally integrable function in some LP space and 0 < a < 1 then (Mf )a c A 1 , i.e., M((Mf )a)(x) < C(Mf )a(x) (for w c A implies w c AP for all p > 1 ). To prove this let f e Lp(Rn) be given and a c (0,1). Let Q be a

cube centered at z, and Q its double. Then write f = XQf + XcQf = f1 + f0. We must show that

I M(f 1)a dx < CM(f )a(z )

IQI

Q

and

M(fo )adx <- CM(f)a(x) .

1

IQI . Q

As for the first inequality,

)1/a M(f1)adx

J

Q

If J

Z1

iIdx If

II ff1.

MULTIPARAMETER FOURIER ANALYSIS

75

(This is an immediate consequence of the weak type estimate for M on L1 .) This shows that

iQl

f M(f1)adx

<M(f)a(x)

<

ICI J

Q

if I

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

ICI

f If of dx > 2 M(f d(i) C

Now, if x f'Q then any cube C' centered at x which intersects

c

is

contained inside a cube of comparable volume centered at i so that

Ilfol < A Id flfoldx. IC

II

C

C

We have proven that for all x EQ, M(foxx)
I1

G

a

lf

fM(fo)adx

.

C

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 e AP, 1 < p < - [121. In the first place if f = w-1 /(P-1) XQ and if M is bounded on LP(w) one sees right away that

w(Q) IQI

f Q

or

p w-1/(P-1)dx


f Q

w-P/(P-1)wdx

ROBERT FEFFERMAN

76

i

w(()

fw-l /(P-1)

1


IQI

1QI

Conversely, assume w c AP. Calder6n-Zygmund decompose f c LP(w) at

heights Ck where k c Z , and C is large (to be described later) and get Calderon-Zygrpund cubes {Q } 1 so that

Ck and f fIQ 1

k Qj

(y is a large constant dependent only on n ). Then P 1 f (Mf)Pwdx
k,j

Rn

k,j

J

j

w c A°° therefore w(Qk) < C"w(Qk) and so the above

Now w c AP

J

J

expression is P


k

W(Q)

°(Ql)

IQk

k, j

fo odx < by the AP condition on w

1

1

Qk 1

p

o(Qk)

(*) < C "" kj

1

o(Qk)

f (f o 1) odx

where o = w-i /(p-1)

k

Qi

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

If Ek = Qk - U Q then choosing C large enough insures that Q>k

IEkI > J

and so

IQkI 2 j

, and since a c AP

u .E A°° we have o(Ek) > rlo(Qk) J

J

77

MULTIPARAMETER FOURIER ANALYSIS

(*)

I < C I o(Ek) (0,(Qjk)

where Mo(f )(x) =scupp

JQig

(fo I)do < C

Mp(for) do

R

n

a

g(Q) fQ IfIda. Now the same proof that works to

show that the Hardy-Litt lewood maximal operator is bounded on LP(Rn)

proves that if or satisfies a doubling condition, then for p > 1 , Mo is bounded on LP(do). It follows that the second inequality is


r

fPol-Pdx = j fPw dx .

Note that the operator M11 f(x) = sup 1 f IfIdµ and its boundedness x(Q µ(Q)

Q

on LP(dµ) enter in a crucial way the proof of the weight norm inequalities for the Hardy-Littlewood operator. This operator Mµ is interesting in its own right, since it is natural to ask what happens if we replace the Godgiven Lebesgue measure by another measure dµ. In fact, if µ is any measure finite on compact sets and if, in the

definition of Mµ we insist that the balls be centered at x then Mµ is bounded on LP(µ), 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 Mµ as well. In fact, if w c AP it is not hard to see that M(f )(x) < CMw(fP)I /p(x)

ROBERT FEFFERMAN

78

(Indeed, 1 /p /p11

I f f dx = I1 Q

II

ffw'/Pw4/Pdx<_L Q

1Q1

(

(JrW_P'IPdXV'

Q

Q

(p-1)/p

p

1/p I

w

(ffPwdx)

1Q1

fw_h/(P_1)dx)

JfPw) Q

I

< CMw(fP)1/p(x) . )

Q

The proof would be complete if Mw(fP)1/P were bounded on LP(w). Unfortunately, Mw(fP)1/P is bounded on Lq(w) only for q > p. But if we

notice that w e App w e Ap_f then the proof is complete. Anyway, it is clear that the operator Mµ arises naturally. Next we shall study its n-parameter variant M(n) just the strong maximal operator, only taken relative to the measure µ

M()(f)(x) =sup µR)

r IfI dµ R

R a rectangle with sides parallel to the axes. Unlike the case where n = 1 , restrictions must be placed on µ whether or not R is centered at x . The following result gives conditions on µ which are rather unrestrictive, and which guarantee the boundedness on LP(IA) on Mn) [13]: THEOREM. Suppose µ is absolutely continuous and non-negative on Rn, and that the Radon-Nikodym derivative of µ, w(x) satisfies an A°° condition in each variable separately, uniformly in the other variables. Then Mµ) is bounded on LP(dµ) for all p > 1 . We require the following lemma.

MULTIPARAMETER FOURIER ANALYSIS

79

is uniformly in A°° 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 ELI > L, then LEMMA. If

IRI

fE w

>q, for some q>0.

IR w

Proof. The proof is by induction on n. Assume this for n -1 . Consider a rectangle R as above, R= I x J where I is n-1 dimensional and J one-dimensional, and a subset E of R such that

For each x e I let Jx = i(x,y)IyEJI be a vertical segment. Since Ei R

it is easy to see that for x in a set I' of measure > 100 III

> 2

have IE n JX I > 100 IJX1 Now.since w is uniformly A°° in the xn variable, (3.1)

J

wdxn>77

wdxn ,

x e I' .

JX,nE

But also if we fix any xnEJ then

(3.2)

fwdx'? il' J wdx' I

I

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

J wdx>r) E

f w dx It'JxJ

we

ROBERT FEFFERMAN

80

and integrating (3.2) in xn e J gives,

fw dx

f wdx>71' I'xJ

IxJ

which shows that- fE w>rfr?'fRw and proves the lemma. Proof of the Theorem. We prove a covering lemma, namely, if 1Rk1 are

rectangles with sides parallel to the axes, there exists 1Rk1 so that w(URkL < Cw(URk) and

III XkIILp(W) <

Cw(URk)1/p'

To prove this order Rk by decreasing xn side length, and then select a rectangle R when IR n [U(Rk)d]I < 2 IR I

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

RknR=0. Now if we slice R , an unselected rectangle, with a hyperplane perpendicular to the xn direction we have Is n [U(k)d]I > 2 I,SI -= w(S n [U('k)d]) >

so that Mwn-1

71

(S)

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

k Now the Rk have disjoint parts property with respect to dx and so they have this property with respect to w dx also. It follows that w(U(Rk)d) < 5' wOk)d < C

L w(Rk) < C'w(URk)

Therefore w(URk) < Cw(URk) .

.

MULTIPARAMETER FOURIER ANALYSIS

81

let us slice the Rk with a hyperplane

Now to estimate III

perpendicular to xn , calling the slices 9k* Then

so since w fA` in

2

we have w('Sk - U Sj) > r7w(Sk) (if j
and we test

w(E) = JE

fRn-1I Xkow

Ir

Owdx < C y w(tk)

1

w (gk)

f

ow dx

§k

< C J M(n-1)(O)wdx. w k

is bounded on LP(w) so this

U SjBy induction k - -9k - j
This shows that

IILP(w)<

IIX1j

III X3rk1I

LP

(w) < Cw(USk)l

/P'.

Raising this to the

pth

power and integrating in xn we have 11 1 XWk1ILP(w) < Cw(URk) .

REMARK. Given this covering lemma, cover the set IMwn)(f) > al by Rk fw >a. Then we need only estimate w(URk) of such that

w(k) fRk

the covering lemma. But

w(URk) <

w(Rk) < a ff Y Xkrkw < IIf !I

2

LP(w)

a

I I f IILP(w)

w(URk) 1 Jp

.

II IX kIILP (w)

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. Suppose % denotes the family of rectangles with side lengths of the s and t > 0 are arbitrary. (Suppose form s, t , and the sides are also parallel to the axes.) Define the corresponding maximal operator M by

Mf(x) = sup 1 f IfIdt. xERER IRI

R

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

fMfPw
rw

P-1

lR

II

IwJJ


R

for all R E %. To prove this result we need the following.

LEMMA. If w c AP(R) then w satisfies a reverse Holder inequality.

Proof. Since w is uniformly AP in the x1 variable, w will satisfy a reverse Holder inequality uniformly in that variable:

1/(1+)

fw(xlfrP)i4dxI) I


III

fw(xiP)dxi). I

MULTIPARAMETER FOURIER ANALYSIS

83

(C independent of p ). Fix I, an interval of the x1-axis, and define a measure in the x2, x3 plane whose Radon-Nikodym derivative W(p) is defined by

1/(1+8)

fw1+&(xi:P)dxi)

III

W(P) =

I

We claim that W satisfies an A°° condition uniformly (in I) relative to the class of rectangles whose side lengths are t , II It in the x2,x3 plane. Then let S be such a rectangle in the x2,x3 plane and E C S such that IEI/ISI < 1/2. Then

JW(P)dP

C.

E

fIII

E

I.

fw(xiP)dxi)dP

III f f w(xi,P)dxldp

=C

I

.

ExI

Since w c A°°(t), ffExl w < (1- 71) AR w and so fE W(p)dp < C(1-rl) fsW(P)dp ,

and by taking S small enough, C will be so close to 1 that C(1-71) < 1 and W is uniformly A°° on the collection of all such S. Therefore since S is just 1-parameter (just a linear change of variable in one of the x2 or x3 variable away from squares) we have that W satisfies a reverse Holder inequality: (For 8' some value < S ) 1/(1+8') ISI

f W14dp


w

rW. S

This shows that

I111 fwixi) 1

1

S

and so

I

(1+8) 1(1+8')

(1+S') dP

1

< Cf1 - ISIf(Ti wdx 1

S

I

dp

ROBERT FEFFERMAN

84

I1 rW1+ R


Now on to the theorem. Because w e AA(JR) and w-1 /(P-1) a AP'(91) satisfies a reverse Holder inequality, w EAp-E(at) for some E> 0. It follows that Mf(x) <

and it just remains to show that Mw is bounded on Lp(w). A quick review of the proof that Mwn), n = 3 , is bounded on LP(w) reveals that all we really used was that w satisfy an A°° condition in the x1 and x2 variables as well as a doubling condition in the x3 variable: w((R)d) < Cw(R). All of these are satisfied by our w here, and this concludes the proof since Mw(f) < M(3)(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 R2, and consider the following basic question: For which sets S C R2 is XS(e) a multiplier on LP(R2) for some p 2 ? For XS to be a multiplier of course means that, if for f e C R2) we set Tf (6) = XS(e)f(6) then we have the a priori estimate IITf1ILp(R2) <- CIIfliLp(R2)

In his celebrated theorem, Charles Fefferman showed that if S is a nice open set in R2 whose boundary has some curvature then XS(e) will only be an L2 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 < -0, just because of the boundedness of the Hilbert transform on the LP spaces in R' . The case that remains is the one we consider here. Let 01 > 02 > 03

> On > 0n-1 ~ 0 be a given sequence of angles 0 and let

85

MULTIPARAMETER FOURIER ANALYSIS

N N

I

I

0

1

I

I

2

4

Figure 2

R0 be the polygonal region pictured above. Then we shall define TO by

T0f (f) = XRe(04) Consider as well the maximal operator M0 defined by

Mef(x) =

sup

1

r if I dt

XEREB0 IRI ,J

R

where B0 is the family of all rectangles in R2 which are oriented in one of the directions 0k, but whose side lengths are arbitrary. We claim

ROBERT PEFFERMAN

86

that the behavior of T9 on LP(R2) for p > 2 is linked with the behavior of Me on L(P/2)((p/2)' is the exponent dual to p/2 ). More precisely, suppose that To is bounded on LP(R2) and we assume the weakest possible estimate on M0, namely J{MOXE > 2 I1 < CIEI. Then

Me is of weak type on L(P12) . Conversely, if Me is bounded on L(P12) (R2) then TB is bounded on Lq(R2), for p'< q < p [16].

To prove this assume first that To is bounded on LP(R2). Then the first step is to notice that this implies that

I(YITkfkl2) /2

(fkl2)1/2IILp(R2)

1ILP(R2)
This is proven by observing that if we dilate the region Re by a huge RP"

p to get RB and translate R0 properly (by Tk) then R8" kk looks like a half plane with boundary line making an angle of ek with the positive x axis. Then if rk(t) are the Rademacher functions, and T

XRp,T k f

T8 ' k f

then

e

T0,Tkf

_

and

(lII

li (

`

Lp

`Te(eTk xf k)I2/

II

' LP

1

r f J R2


iT x

p

i Irk(t)T0(e k fk)(x) dtdx =

irk *x

J

0

0

('

fJ

R2

('

J 0

1

P k(x)

dtdx
TT (Irk(t) e R2

(IfkxI2)' lam

/2:1 ll

\IIP

fk)I dxdt

LP

87

MULTIPARAMETER FOURIER ANALYSIS

Taking the limit as p -> oc, we have 1/2 12

)

II (YW JTkfk

11LP(R2) < CII (lfkl2)'

/211LP(R2)

where Tk is the Hilbert transform in the direction 9k. The next step is to use the above inequality to prove a covering lemma for rectangles in 30. Let 1Rk1 be a sequence of such rectangles. Select R given that R1, have been chosen provided IR n [ U R ] I < 2 IR I

.

Then if R is unselected M0(XU ) >

j
on R

2

so that IURkl

Let Ek = Rk - U

j
6KXUi'1k > 2}I << CIURkl

and let fk = Xt

.

.

Then looking at the picture

k

below, since IEkl > 2 IRkI on at least a set of measure >

Figure 3

100

I-rkI

the

ROBERT FEFFERMAN

88

segments pictured contain at least 1 /100 of their measure in EEk. If we duplicate the rectangle Rk as shown, on these segments Tkfk > 1/100. Applying Tk = Hilbert transform in the direction perpendicular to 0k to

Tkfk we see that Tk(Tkfk) > 1/100 on all of Rk. Repeating twice more we get

1/2

/2I1LP


1

I+

Il

IfI

(1

Ia(tk12)


LP

IIxUiz k11 LP

This shows that M0 is of weak type (p/2)'. Conversely, assume that M0 is bounded on L(P12) . Define Sk < 61 < 2k+1L Then if Sk also stands for the = 16 _ (61 e2)I2k multiplier operator corresponding to Sk , we see that

IITOfIIq

I1 /2IIq Y I{ (skTofI2)

)1/2IIq

(IskTkf 12

To estimate II(11 ISkTkfl2)1/2IIq , let II¢II(q/2)' =1 and let us estimate

fITkskfI20

f IT k(Skf

)12.b

But in R1 we have the classical weight norm inequality for the Hilbert transform:

fjHfj20
IfI2M(01+e)1/(1+e)

forall e> O.

It follows that (3.3) is

(3.4)

< if ISk(f)I2Me(kl+e)t /(l+s)

M0 is bounded on L(q/2)/(1+e) if that (3.4) is

a

is sufficiently small. It follows

MULTIPARAMETER FOURIER ANALYSIS

<

CN (sf2\fl110/2

89

C'IIfIIL9

proving that TO is bounded on L9 . 4.

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+ _ 1z=x+iyIxcR1,ycR1 ,y>01 which satisfies the size restriction

(1+00

I /p IF(x +iy)Ipdx

< C for all y c 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 +00

f _ increases as t

IF(x+it)Idx

0, we have +00

IIFIIHI defsu

N

J -00

IF(x+it)Idx ti IIuHI1

1

+ IIvIII(RI)

So we may view the space HI through its boundary values as the space of all real valued functions f of LI(RI) whose Hilbert transforms are LI as well.

ROBERT FEFFERMAN

90

If we want a theory of HP(Rn) then, following Stein and Weiss we may consider the functions F(x,t) in R++1 = 1(x,t)IxeR11,t>01 whose values lie in Rn+1 : F(x,t) = where the ui(x,t) satisfy the "Generalized Cauchy-Riemann equations," n

Cal

i=0

i (x, t)

0 (t = x 0)

1

and

dui

dtrj

J

for all i,j

.

1

These Stein-Weiss analytic functions are then said to be Hp(R++1) if and only if 1 /n

sup J t>O

iF(x,t)ipdx

= JIF11

HP(R°)

[171.

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 Rn+1 , then the boundary values u1(x) satisfy ui(x)

Ri[u0](x) where Ri is the ith Riesz transform given by R1(f )(x) _ f *Ixln+i

In particular we may consider an H1(Rn+1) function (by

identifying functions in Rn+1 with their boundary values) as a function f with real values in L1(Rn) each of whose Riesz transforms Rif also belong to L1(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 Rn+1 and bounded there, u is given as an average of its boundary values according to the Poisson integral:

MULTIPARAMETER FOURIER ANALYSIS

u(x,t) = f * Pt(x); f(x) = u(x,0) and Pt(x) =

91

cnt (Ix I 2

+t2)(n+1)12

Let F(x) _ ((y,t)l lx-yl < t(. 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) =

lu(y,t)l , then u*(x) < cMf(x) sup (y,t)cI'(X)

Unfortunately, if u(x,t) is harmonic, for p < 1 , u = PLO, and 1Rn lu(x,t)lpdx < 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) < 00 for a.e. x c Rn. On the other hand, suppose F is Stein-Weiss analytic, say F c H1(R++1) Then a beautiful computation shows that if 1 > a > 0 is close enough to 1 (a > nn1) then A(lF la) > 0 so that lF la is subharmonic. If

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 = [F la (which has JG1 'a(x,t)dx < C for all t > 0 ) we see that G* < M(h) for some h c L1 'a. Now M is bounded on L1 Ia so that M(h) c L1 La and so G* c L1 Ia. It follows that F* a L1 . Just as for a random f c L1(Rn) we do not necessarily have Rif c L1(Rn) (singular integrals do not preserve L1 ) it is also not true that for an arbitrary L1 function f that for u = P[f], u* c L1 . But if f c H1(Rn+1) then u* c L1(Rn). Thus the nontangential maximal function F*(x)

=

sup

lF(y,t)l c L1(Rn)

(y, t) (F(X)

if and only if the analytic function f c

H1(Ri+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,

S2(u)(x)

= ff

JVu12(y,t)dtdy

r(x) which we already considered in the first lecture. As we shall see later, for a harmonic function u(x,t), IIS(u)IILp ti IIu*IILP 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 Ipvl = IVul

.

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

IIu*IILP<00<1IF*IILP<00 It is interesting to note that the first proof of

FcHP(R+) IIS(u)IILP

11u* 11

LP ' 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++1 and requiring that u* or S(u) belong to Lp(Rn).

93

MULTIPARAMETER FOURIER ANALYSIS

It turns 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 e Hp we could

take u = P[f] which of course satisfies Au = 0. This is not necessary. If f is a function and 0 E C- (Rn) with IR 4 n=1 , then we may form f* (x) =

sup If *Ot(y)I (t,y)EF(x)

,

(kt(x)

C n O(x/t) and if lr a C0 (Rn) is suitably non-trivial (say radial, non-zero) and fci = 0 we may form S2 (f Xx)

= ff If * At(y)I2 dndt t

Then C. Fefferman and E. M. Stein have shown that IIf ll

Hp(Rn)

IIf *II

Lp(Rn)

IISq'(f )II

Lp(R )

for 0 < p < -. .

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 HI with BMO, which we shall now discuss. A function O(x) , locally integrable on Rn is said to belong to the class BMO of functions of bounded mean oscillation provided I fI0(x)-cQldx <

M

for all cubes Q in Rn

IQI

Q

where 0Q = IQI fQ 0. The BMO functions are really functions defined modulo constants and

11

IIBMO is defined to be sup

IQI

fQ 10 -IQI

ROBERT FEFFERMAN

94

According to a celebrated theorem of C. Fefferman and Stein, BMO is the dual of H1 (181. 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 > 0 is a measure in Rn+1 and Q C Rn is a cube. Let S(Q) = 1y,t)Iy Q, 0 < t < side length (Q)1. Then we say that A is a Carleson measure-on R++1 iff g(S(Q)) < CIQI . The basic property that characterizes Carleson measures is

ff Iu(x,t)Ipdi (x,t) < Cp

p> 0

J u*(x)pdx Rn

Rn+1

for all functions u on R++1 In connection with this type of measure there is the characterization of functions in BMO(Rn) in terms of their

Poisson integrals. A function O(x) on Rn with Poisson integral u(x,t) is in BMO if and only if the associated measure dµ(x,t) = IVuI2(x,t)t dxdt

is a Carleson measure. C. Fefferman proved this and used it to prove that every function in BMO acts continuously on H1 :

J f(x)O(x)dx

< CIIfIIH1(Rn)10IIBMO(Rn)

.

Rn

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) in R++1 IIS(u)IILp

IIu*IILP for p> 0

.

95

MULTIPARAMETER FOURIER ANALYSIS

We begin with the estimate flu*llp < CpiIS(u)llp , and to do this we shall show that

ltu*>Call < c

12

S2(u)(x)dx +

J

a

S(uYa

From this our claim follows. This is because for )Lg(a) = If Igi > all we have

Ifu*IILp J ap

a

00

00

r P'g(u)W)dO +Xg(u)(a

IXu*(a)da < Jr ap-1 0

0

0

0

<

f

#xS(u)(,6)

0

f aP3dadfl + 00

f aP_1As(aa 00

0

00

Assuming p < 2 as we clearly may, this is

f

OP-1's(u)(R)df3

NS(u)IIP

.

0

To prove the estimate on Itu* > Calf , we set the notation that M(XE) > 2 , and then claim

1.

da

a

5 S2(u)(x)dx > c fJ lVu(y,t)l2t dtdy where R = Ur(x) R

s(u)
In fact, if (y,t) c R then IB(y,t) flf S(u)>ajI < 2 lB(y,t)I

x4iS(u)>at

ROBERT FEFFERMAN

96

Then

f

S2(u)(x)dx

xJS(u)aI

S(u)
J ivul2(y,t)tl-ndydt

J

=

dx

T(x)

(4.1)

Ioul2(y,t)tl-nIIxl(y,t)tr(i),x/IS(u)>a#ildydt

= ff R n+l

But for (y,t) e R ,

IIxk(y,t)cI(x),x/IS(u)>aiij = IB(y,t)ncIs(u)>alI >2 IB(y,t)I and (4.1) is

(4.1) > c JJ IVu12(y,t)t dydt

as claimed.

II. The next step is to write

IVu12 =

A(u2), and apply Green's theorem

to R :

ffA(U2)(y,t)t dydt = R

Now

f !) t - u2 Alt da aR

> c > 0 for some c so the above gives

f u2da < C aR

J S(u)
S2(u)(x)dx +

J

u(pu) tda

.

aR

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

u(Vu)t do, = J

f

aR

u(Vu) t da

aR

where OR is the part of aR above IS(u) > al. It is not hard to see that

lpult < a on aR so that 1 /2

(+

J

all' /2 (f u2da

Jul lpult da<

aR

aR

Putting all of our estimates together, we see that

J

f u2da < C aR

S2(u)(x)dx + a2 IS(u) > all

.

S(u)
ti

III. Next we wish to define a function f by f( x) = u(x, r(x)) where (x,r(x)) c aR defines the function r. We claim that in the region R

lul
(4.2)

This is done by harmonic majorization. It is enough to show this on OR and this in turn is just saying that for any point p c aR , JU(p)l is

dominated by the average over a relative ball on aR of u + Ca. This follows from the estimate JVult < Ca on aR . Anyway, from (4.2) we have, for x I IS(u) > al, u*(x) < CP[f ]*(x) + Ca , so that finally l1u*>Call <4M('(x)>all

r S2(u)(x)dx +CI1S(u) > all

J S (u)
This completes the proof.

ROBERT FEFFERMAN

98

The proof that IIu*IIp < CPIIS(u)IIp which we just 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 IIu*IIp ? CpIIS(u)IIp

LEMMA. Let f(x) and g(x) c L2(Rn), and suppose 0 c C° (Rn) radial and u = P[f ] . Then

ff IouI2(x,t)Ig*ot(x)12t dtdx R n+1 +

< fn'

Rn+1

R

for some 1i c C'(Rn) with f;& = 0

real-valued).

Proof.

ff

A(u2)(x,t)Ig*ot(x)I2t dtdx

Rn+1

=

f Rn+1

rfV(u2)(x,t) V[(g * ot(x,t)2)t]dt dx = -2 fJu(xt)Vu(xt). V[(g * ot)2](x,t) t dt dx Rn+1

Rn++1

+

+

f

-2 ,

u(x,t)

R n+1

_ -2

ffuVu 2(g*q5t)(x)p[g*q5t(x)]tdtdx -2 n+t R+

ffu i (g*q5t)2dtdx n+1

R2

= 1+ 11

,

99

MULTIPARAMETER FOURIER ANALYSIS

where

ff

I

u(tV(g*.kt))t-1/2. Vu(g*0t)t1/2dtdx

Rn+l +

ff

<

1/2

1/2

U2 1 g*otl 2 dt dx

I

Rn++l

rIV,I21g*ot(x)I2 t dtdx

RJn++I

and

II

=fu

(g*ot)2dtdx = -

Rn+l

ffu

+ J u2(x,0)g2(x)dx

R n+l

u !AL (g*ot)2dtdx

ff

-JJ

Rn

u22(g*(kt)

Rn+l

A

f2g2dx

fn R

We see that

2 ff u +

(g*q5t)2dtdx <

55u2(g*) (g*)dtdx + fn f2g2dx Rn++l +

but

(g*-Ot)dtdx = ff u2(g*ot)t I (g*ot) dt dx Rn+1

Rn+1

.

ROBERT FEFFERMAN

100

But

so

ff_2u -5211 (g*Ot)t(xi(k)t * gldt dx +

S=

I (fg/

(g*(kt)(g

1

(ffIvu2(g*ct)2 t dtdx)\

1)/2

fu2(g*(xjc)t)2 d tdt

+

r u2t

2

Ydx

1/2,

* (xio))

dt dx

1 /2

1/2 *(xio))2 dtdxi)t

t

Putt ing this together gives

fflvuI2(g*t)2tdxdt C [Y

f 1

/

(

1u2 g*

2

14)

dt dx

)rru2(g * ( X

(ai4)

2

d tdx

+f

f2g2

dt dx

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 5.

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 Rn, x -Sx, 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 R2 ) (x1,x2) (S1x1,82x2), For convenience we shall work in R2 but all of this could just as well be carried out in R n 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(R2) for the one-parameter space of functions on R2. Let (x 1,x2) c R2 and denote by 1'(x) the set 61,82>0.

i(yi,tl,y2,t2)I Iyi -xiI < ti, ti > 01 c R+ X R+

Let u(x,t) be a function in R+ x R+, x c R2 , t c R x R+, which is biharmonic, i.e., harmonic in each half plane separately. Then the nontangential maximal function and area integral of u are defined by sup Iu(yl,tl,y2,t2)I (y,t)EF(x1,x2)

u*(xi,x2) = and

S2(u)(x) = ff IV1V2u(y,t)I2dy1dy2dtidt2 F(x)

More generally, if S6 c Cc (R2) and if

I 0 = 1, otI,t2(xi,x2) = tl1t21

x

then for a function f on R2 f*(x) =

sup if *0t t (x) 2 (y,t)cr'(x) 1

and if tA c Cc '(R2) and

x' tz

.

ROBERT FEFFERMAN

102

fb(x1x)dx1 = 0

f(J(xix2)dx2

=0

for all x2 ER1

for all x1 a R1 ,

then

S ,(f )(x) =

ffIf*&tt(yi,y2)I2 dyl d Y2 dt I 't

t

dt 2 2

t

2

F(x)

Given f(x1, x2), we define its bi-Poisson integral by u(xl,tl,x2,t2) = P[f](x,t) = f * Ptl't2(x1,x2), where the bi-Poisson (or just Poisson for short) kernel is defined by Pt1) t (xl,x2) = 2

P x1 P tl

2

and

(T2

where P is the 1-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* E LP(R2) where u = P[f 1. It is not hard to see, just as in the single parameter case that f or any O .E C- (R2), f.0 = 1 IIf*IILP - IIu*IILP for p > 0

,

so that we may use any approximate identity which is sufficiently nice to define product Hp spaces. In terms of area integrals, we also have, for O(x1,x2) suitably nontrivial, say i/i even in x1,x2 and not - 0, I1SV'fIILP ti I!S(u)IILP

p > 0, u = P[f] .

To complete the chain of equivalences , we would like to know that IIS(u)IILP

IIu*IILp

MULTIPARAMETER FOURIER ANALYSIS

103

In fact, this is true, but is not obvious, and so we intend 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 of looking at the one-parameter case than one is used to. Proof that of S(u)e LP , then, u* a LP (Gundy-Stein) [211. We can assume that u = PR]. Then since the 2-parameter area integral is invariant under taking Hilbert transforms in each variable separately, we see that we may write

f =f+++f+_+f_++f__ where f++ is supported in eI, e2 > 0 and f+_ is supported in e > 0, `'2 < 0, etc., and we have S(f++) a LP. By reflection we may assume f is analytic (i.e., P[f] is bi-analytic) and then show that f* a LP. But for u = [f] a bi-analytic function, we know that for a > 0, Iu(xl,tl,x2,t2)Ia is subharmonic in each half plane (xi,ti) a R+ separately; this implies that

lu(x,t)la < P[If(x)la] .

If a
So what we must show is that IIf1ILp <_ Cplls(f )IILp,

P>0.

To show this let us define some notation. In R I , if f( x) has Poisson integral u, we let Qt be the operator (or kernel) which takes

to tpu(x,t) , so that

f

ROBERT FEFFERMAN

104

S2(f)(x)=

fflf*Qt(Y)I 2 dy2t t

F(x)

Then going back to our present situation where f is given on R2, we define 00

-Qtf(x1,x2) _

f(xl-y,x2)Qt(y)dy

and Q t similarly. Then let us define a Hilbert space valued function F(x1,x2) a L2

(0);

dydt t

by

F(x1,x2)(y2,t2) =Qt f(x1,x2+y2) 2 Now we know that in the one-parameter case iiS1fIIp > cpIlflip and a

glance at the proof of this fact reveals that it remains valid for Hilbert

space valued functions. Fix x2. Then 00

00

S1(F)(xl,x2)pdx1 > cp -00

f IF(xl,x2)Ip 2

L (r)

dxl

-00

and integrating this in x2 ,

J1S1(F)(xl,x2)Pdxldx2 > cpJjIF(xl,x2)Ip 2

(5.1) R2

But fixing x1, since IF(x1,x 2)1 integral of f(x 1, - ) at x2 , we have

dxldx2

R2

)

L2(iis the value of the one-parameter

MULTIPARAMETER FOURIER ANALYSIS

J

IF(x1,x2)Ip 2

L J,)

dx2 > cp

f R

R

105

If(xl,x2)IPdx2

1

and so (5.1) is greater than or equal to

cp

ff

Iflpdxldx2

R2

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

SI(F)(xl,x2) = S(f )(xl,x2) the two-parameter area integral of f. Next, let us prove that IIS(u)IILP < cpllu*IILp [21]. The proof is a simple iteration of the one-parameter case given previously. To begin with, we recall Merryfield's lemma: Let q c Cc (R I) be supported in [-1, +1] and have f q5 = 1 . Then there exists Vi c Cc (RI) whose support is also contained in [-1, +1] with f i/' = 0 and such that if u = P[f],

fJIvul2(x,t)(g*t(x))2t dtdx < C J f2(x)g2(x)dx + R2

RI

jJu2(x,t)(g*cfrt(x))2 dt dx

+

R2

Introduce the notation tVu(x,t) = Qtf(x) , u(x,t) = Ptf(x), g*qt(x) _ Pt(g)(x) and -Qt(g)(x), Qt, i = 1,2, will denote the operator

acting in the ith variable. Then we estimate

106

ROBERT FEFFERMAN

jf

(5.2)

[Qt1Qt2f(xl,x2)]2[tt Pt2g(xt,x2)]2

dt1dt2 dxldx2 t it2

R2xR2

Fix x2, t2. Then (5.2) is

<

ff ir

X

dtdx

x2,t2 xl,t2

f{Q?f(x)]2{g(x)]2 dx1 dt2 xl =1+11.

f

+

(PtlQt2f(x))2(Q11P2 g(x))2

t2

t

2 2

x1

Now

I=

d

ffPt2Pt1f(x))2(?g(x))2 dtdx t

X1t1 x2 t2

fjfI

f(Pf(x))2(tg(x))2dx2!1 dxI

x1,t1

x2

+

Then

II =

fff(Pf(x))2(Qtg(x))2dx2dt2/t2dxi + fff2(x) g2(x)dx1dx2 X1

So the inequality we seek in the product case is

MULTIPARAMETER FOURIER ANALYSIS

(5.3)

J

107

J

(Qtf )2(x)(Ptg)2(x)dxdt/t < c

(R+)2

(R+)2

(Pt2f(x))2(Qt2)2(x)dx2dt2/t2

+

J

J

x1ER1 (x2,t2)ER+

f

J f2(x)g2(x)dx

J

Rz

x2ER1\(x1,t1)ER+

It is now easy to see that

IIS(u)IIL

P

< CpIIu*IILP, 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)E 1

Iu(y,t)I 0(x)

10

where r1010(x) = {(y.t)I ly;-xiI < 1010ti, i =1,21 .

This is an irrelevant change, since a trivial computation shows that IIu*Iip for iIu*IIp is, for a larger aperture, no more than a constant times a smaller aperture, the constant depending only on the apertures involved. In (5.3), take c(x) = 1 for all Ixl < 1/3, and g(x) = Xu*(x)
108

ROBERT FEFFERMAN

I

S2(u)(x)dx when u - P[f]

M()( u*>a )<1 /200

<J J IV1V2uI2(Y,t)IR(Y,t)flIM(Xu*> )<1/20011dydt

< ff 1V1V2ul2(y,t)t1t2dydt R*

where R* = t(y,t)I IR(y,t)fllu*>aII <

200

IR(y,t)il and where R(y;t) is

the rectangle in R2 with sides parallel to the axes and with side lengths 2t 1,2t2 centered at y. Notice that if IR(y; t) fl lu*>all < 100 IR(y; t)I then g * (kt(y) = Ptg(y) > c for some c > 0. It follows that

f

S2(uXx)dx < ff IV,V2u12(Y,t)Pt(g)2(Y)dyt1t2dt (R+)2

M(Xl *>a #)<1 /2 00 u


5J

u2(Y,t)ot(g)2(Y)dy tat2 +

(R+)2

+

ff P2 f (Y)2 Qt 2(g)2(Y)dy

J Y1

f If f J

Y2 (Yi,tl)fR+2

Pt1)f(Y)2. 1

(Y2,t2)eR+

t

(g)2(Y)dydt

t 1

l+

1

ff2g2dy

J R2

= i + ii + iii + iv.

If Qt(gXy)' 0 then u*(x) < a for some x f R(y; t). But then lu(y,t)l < a so i is less than or equal to C onsider

i:

a22

109

MULTIPARAMETER FOURIER ANALYSIS

a2

fft(g)2(y)dy tdt

=

12

a2

fft(1_g)2(y1y dt < a2II1-gIIL2 < a2lju*>a}I (R+)2

(R+)2

Consider ii: If Qt2)(g)(y1,y2) 1 0, then there exists x2 such that

Ix2 -Y2I < t2 and u * (yl,x2) < a. This implies that IPt2)f(y1,y2)1
ii is less than or equal to

ff(2)()2(Y)th2dY)dYl J

a2

t2

2

Yl

Y2't2)

Ag ain

ff

dt 2 Qt2)(g)2(Y)

dY2 dyl 2

Yl

Yl

Y2.t2

<

J Y1

g)2(Y)dY2 t2 dYl

J Jv

2

2,t2

f(1-g)2(Y1,Y2)dY2 dYl < Ilu*>atI . Y2

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

J u*(x)
f 2(x) dx <

J

(u*)2(x) dx

.

u(x)
So we have

J {u*
S2(u)(x)dx atl+

J

u*(x)
u*2(x)dx

ROBERT FEFFERMAN

110

and we have seen before that this implies that 1IS(u)II

LP

< CpIIu*Il

LP'

2>p>0.

The next topic that we shall consider is that of duality of H1 and BMO in the product setting. In the classical case there were four results which expressed this duality. 1) The characterization of Carleson measures p for which the Poisson transform f -- P[f] is bounded from LP(dx) to LP(dp), p > 1 . 2) The characterization of functions in BMO(R1) by a condition on their Poisson integrals in terms of Carleson measures. 3) The characterization of functions in the dual of H1 by the BMO condition.

4) The atomic decomposition of H1 . Let us try to guess what the analogous theory should look like in product spaces. For simplicity we consider the dual of H1(R+xR2). Then what should an element of BMO(R2 xR+) look like? We might look at tensor products of functions in BMO(R1) to get a feel for the answer. So, for example if 951 and 952 are in BMO(R1) then S1(x1)S2(x2) might be our model. Of course, this function S(x 1,x2) satisfies

(5.4) J1

d

195(x1,x2)-c1(xl)-c2(x2)j2dx2dx2
R

for the appropriate choice of functions c1 and c2 of the x1,x2 variable. A Carleson measure in R+ x R2 would be a non-negative measure p for which (5.5)

ffP[fIPdii < Cp ffIf(x)I1'dx, p>1 (R+) 2

R2

where P is the bi-Poisson integral. The obvious guess is that p

MULTIPARAMETER FOURIER ANALYSIS

111

satisfies (5.5) if and only if p(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 x J) = S(I) x S(J) for R = I x j . In terms of these Carleson measures, it is not hard to show that 95 satisfies (5.4) if and only if its

bi-Poisson integral u satisfies dp = IO172uI2(y,t)tlt2dt is a Carleson measure. And finally, all of this in some sense is equivalent to asserting that every f e HI(R2 xR2) can be written as I Akak where Y' IAkj < CIIf1I 1 and H

ak(x1,x2) are "atoms," i.e., ak is supported in a rectangle Rk = IkxJk such that

r

ak(x1,x2)dx1 = 0

for all x2

ak(x1,x2)dx2 = 0

for all x1

Ik

J Jk

and 1

IIak112<_

IRkI

1 /2

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

ffIP[f]Pdl < Cp J Iflpdx (R2)2

R2

From here it is not difficult to produce examples of functions O(x1,x2) which satisfy

112

ROBERT FEFFERMAN

1J I

Ic(xi,X2)-C1(x1)-C2(x2)12dxldx2 < C

R

where C 1, C2 depend on R, yet I LP(R2) for any p > 2. Therefore, this condition is not strong enough to force 0 to belong to the dual of H1 In other words the simple picture of the structure of H1(R2 xR+) and BMO(R2xR2) 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 classical 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 e H1(R2 xR2)*, if u = P[qS] we have a Carleson condition with respect to open sets satisfied by the appropriate measure. To describe this result, we make the following definition ([23], [24], and [251).

Let 9 C R2 be an arbitrary open set, and let R(y; t) be the rectangle in R2 centered at (y1,y2) = y and with side lengths 2t1 and 2t2. Then S(i2) the Carleson region above SZ is defined as S(Q) = U S(R) = f(y,t) E (R+)21 R(y; t) C SZ# . RCSZ

Then we say that µ > 0 in (R4)2 is a Carleson measure if and only if ,u(S(SI)) < C101 for every open set SZ C R2 f c H1(R2 xR+)* if and only if for u = P[f ] ,

dµ = 1V1V2uI2tlt2dt1dt2dy1dy2 is a Carleson measure.

In fact, this follows immediately from the inequality (5.3). To see this, notice that IV1V2uI is invariant under the Hilbert transform HXi(i = 1,2) so that if we prove this when f e L°°(R2), we will have proven it also when f is of the form

113

MULTIPARAMETER FOURIER ANALYSIS

gl + Hx 92 + Hx2 93 + Hx Hx 2g4 1

gi c L°° .

1

A function a(x) on R2 will be in H1(R2xR+) if and only if a c L1(R2), H a,Hx2 a, and H xl H a cL1. xl x2 In fact, if a c H1(R2xR+) then S(a) e L1(R2) hence so are S(Hx a), 1 S(HX a) and S(HX Hx2 a); therefore Hxia, Hx1 Hx a e L1 . Conversely, 2

2

1

if a, Hxia , and Hx Hx2 a e L1(R2) then we can form F

, F+_, F_+

1

and F__ c L1(R2) such that a = c F++ and reflections of the F++ are boundary values of bi-analytic functions. A bianalytic function F with (distinguished) boundary values in L1(R2) has F* c L1 by a subharmonicity argument applied to IF I', a < 1 . So a* a L1 and a e H1. Let

® L1(R2)i by

$ e H1(R+xR+)*. Define a map from H1(R+xR+)

i=1

0(f) = (f, Hx l f, H x2 L H xl Hx 2f) . Then

Ij6f jI

l

lif 11

HI .

0 is obviously one to one, so 0-1 = s exists

and is bounded on Im(6). The map $ ° extends, by Hahn-Banach to an element of the dual ®L1 = ®L°°. Then

$(f)=P° s(f,Hx l f,Hx2 f,Hxl Hx2 f)

=

rfgl

rJ

=J f

+ Hx f g2 + Hx f g3 + Hx Hx2f g4 1

2

1

(g1Hx 1g2+Hx 2g3+Hx Hx2g4) 1

Thus every element of (Hl)* is of the form

fcH1.

114

ROBERT FEFFERMAN

So it suffices to show that if f c L°°(R2) with u = P[f] then dp

=

Ip1V2I2(y,t)tlt2dydt is a Carleson measure.

But in (5.3), take g = XU(x1,x2), and notice that if f 0 = 1 , supp fi(x) C

[-1,1] then Ptg(x) = 1 if (x,t) c S(Sl). This is because for such (x,t), R(x; t) C SZ and g * 95t(x) = fR2 ot(x -u)du = 1 . It follows from (5.3) that

ffviv2u2tit2dxdt < CIIfIIoo f IQtgi2I at dx

X1 \(x2t2)

/

X2 \(X1,t1)

< CIIf11211gII2 < C

11f 112

ICI

6. Duality of H1 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 1-parameter case. In what follows we shall be working with functions in HP(R+xR+) or BMO(R2 xR2) only. The theory for R2 xR2 x x R2 or for Rn f1 x Rm+1 is quite similar and only requires minor changes. Now let I1 be the family of all rectangles with sides parallel to the axes and sR4d be the subfamily of 91 whose sides are dyadic intervals. If f( x1,x2) is a sufficiently nice function on R2, and q' c C°°(R1), iii is even, ib' 4 0 real valued and supp(qi) C [-1,1], and i/i has a large

115

MULTIPARAMETER FOURIER ANALYSIS

number of moments vanishing, then for 00

ft

1

1

2

(x1,x2) _

1)0(x2)

tl

X

t2

t11t21,

f

=1

0

we have dtldt2

f *tPtl,t2(Yl,y2)otl,t2(xl -y1,x2-y2)dyldy2

f(x1,x2) =

tit t2

R+x R+

In fact, taking Fourier transforms of both sides, for the right-hand side we have 00

ff f(St

dtia22 = f(e)

r r J

0

RZxR2

00

I

(t1C1,t2C2)12 dtldt2 =j(e)

J

0

We can use this representation to decompose the function f as follows: R c 8Rd . Set ?1(R) = 1(y,t) c R+xR+I y c R , P1 < t1 < 2 Pi where Pi , i = 1,2 is the side length of R in the xi direction 1. Since R2xR2 = U W(R), if we define + + Rc91d

fR(x1,x2) _

f(y,t)

ff f=

1 fR, and each fR is supported in Rc9? d

R the double of R and has the property that

ROBERT FEFFERMAN

116

ffRx2x2 dx1 = 0

for all x2

ffR(XlIX2 )dX2 = 0

for all x1

I

J

where 14 =TxT. It will be convenient to define a norm

I

I

R on functions supported on

a rectangle R , as follows. N

IfIR

_

oaf IIIa111ia2

IaI=O axa1ax22

where N is a large integer. With these preliminaries we can pass to a theorem characterizing (H1)* in a number of useful ways. THEOREM [2$]. For a function on R2 the following are equivalent: (1)

e H 1(R+ x R+)*.

(2) 95 = g, +Hx1(g2) + Hx2(g3) + Hx1Hx2(g4) for some 91,92,93 and

g4 in L°°(R2). (3) If u = P[¢] in R+ x R+, then ffIviv2uI2(y.t)tit2dt < CIt I, for all open sets Q C R2 S(Q)

(4)

If ¢(y,t) _

* lt(y), then

ffIy,o2dy.
fore!! open sets [i C R2

.

117

MULTIPARAMETER FOURIER ANALYSIS

(5) 0 can be written in the form

cRbR where bR(xl,x2) are

7-

ReRd

supported in k in R2.

IbRIR < 1 and

I cR < CIcI for all 9 open RCSZ

Proof. To begin with, we proved in section 5 that (1) --> (2). It is also trivial that (2) - (1), since if f e H(R2xR+

f J

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

and since f eH1, Hx

I

2

Hx 2

fHx1 Hx (f Xx) g(x) dx 2

(f) rLl.

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.do this via the atomic decomposition of H1 which we shall describe here only enough to derive our implication. We shall present the decomposition of H1 in greater detail later. Let f e H1(R+xR+). Then SO(f) e L1(R2) (this follows by vector iteration, just as in the argument that S(f) e L1 implies f e L1 ). Consider the sets SZk = ISO(f) > 2k1, k e Z. Set ak(x l ,x2) =

I

f R(x)

Re91d

IRflUk1 >1/21RI IRf1Slk+1I < 1 /2I RI

Then, as we shall elaborate later on ak(xlPx2) = H1(R+xR+) atom where

ak(x I x

l(xl,x2)IM(2)(Xci )(xl,x2) > 1/101 k

) 2

is an

118

ROBERT FEFFERMAN

Then f = 1 Akak where Ak =

and by the strong maximal theorem

IS'EkI < CISZkI so that I Ak < C IIS f,(f )IIL1 < C,IIf 11

Hi

Now consider c(x1,x2) satisfying (4). Then it will be enough to show that

J ak(xl,x2)'O(xlx2)dx
and then simply sum over k. But

(6.1)

5ak(xl.x2)(xlx2)dxldx2 =2ki R

I

fIf(x)c(x)dx

k

2

where the sum is taken over k

={Rdtdl IRrH kI>2 IRI but IRfQ2k+1I <2 IRI}.

Then (6.1) becomes

56(x)

RY k

fffYttx_Y) d y dt dx 2ki 91(R)

1

R9k

ff 2k1

kI

RU c%k

MR)

kl

t i tit Sff(y,ty,t'y__ 2

(R)

/2

1 /2

If

ate (y,t)I2dY

ff I56(Y.t)12dy d t2c2 RU -1 91k 91(R)

MULTIPARAMETER FOURIER ANALYSIS

(6.2)

ff

IS,,(f)2(x)dx >_ U

J

119

If(Y,t))2dy td 1t2

RE9

k 91(R)

fl k/Qk+1

To show this

S, (f Xx)dx

ff

=

J I)k/uk+l

xE?k/SZk+1

(Y,t)12dy

If

t2

F(x)

> J J jf(y't)12Ifx (? k/Ik+1 I(y,t) Er(x)1I dy 2t2

12 Suppose that (y,t) c %(R), R c R k . Then if the aperture of

I' is large

enough, then (y,t) E I'(x) for all x c R. Since IN x E(?k/Ik+1) 'R' I>

IRI

, R 9k

2

we see that

t1t2 (observe that t1t2 ' IRI )

{x

2

for (y,t) c

U %(R), and this proves (6.1).

RcRk

But then

SV(f)2(x)dx < (2k+1)21 kI

il kAlk+1 and combining this with (6.1) yields

U ff If(y,t)I2dy t2t2 < C2k IS kI

RED

k W(R)

12

ROBERT FEFFERMAN

120

As for

R(R. 91(R)

if we observe that for any R E 91k I 91(R) is contained in S(?lk), we get

U

ff

I0(y,t)I2dy

RE9lk

tat2 < if

195(y,t)I2dy tat- < C ?1

S(? k)

91(R)

by (4). This shows that I f a k Odx I < C and completes the proof that (4) implies (1).

We shall show next that (2) implies (4). Let g e L(R2). We claim that if 1 C R2, then

f Ig(y,t)12dy

dt CIIgII2lgl 00

tit-00

where g(y,t) = g *Vit t (y). To show this, observe that since 1

2

C R(y,t). Hence, if (y,t) f S((1),

supp(i4') C [-1,1],

g *0t(y) _ (gX 1) * qt(y) and so

ff S((1)

ff (g)(y,t)dy dt

Ig(y,t)12dy tit 2

(R+)2

12

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

IIgX 1112 f 0

MULTIPARAMETER FOURIER ANALYSIS

121

Since, f +1 1 q, = 01 i(0) = 0 and fr E Co°

()

=

0(161 -N)

as

I

CI

- 00,

for each N > 0.

so 00

f

de

<

0

It follows that

f

Ig(Y,t)I2dy

dt < C IIgII,2oIjjl

tlt2

S(U)

as claimed. If we wish to prove that the same Carleson condition holds for g replaced by Hx1 Hx2 (g), then we proceed as follows. Observe that

t

1/2

(.rf EHxHx()l *t(Y)I2dY t S(f)

=

1/2

12

JJJ

S(SZ)

where T = Hx1Hx2(0). The function T splits into a product of gJ(1)(xl).T(2)(x2) where 'y(1) is odd, Co° and decreasing at oo like Ixil-N (depending on how many moments of rr vanish). Now suppose we choose r?(x) on R1 so that supp(q) C (1/4,4), 71 E C°°(R1), 77 even and E

-q1/ x

00 k=-o \2k/

rl/ 2kl

Set

= 1. Let i 0(x) = E ,Ix 1l and for k > 0, let k<0 \2k/

_ 77(x) k

Then

122

ROBERT FEFFERMAN

(a) supp(Wk,j) C 4R(0; 2k,2i) (b) 'Yk,j

is odd in each variable separately

(c) Tkj is Cc (R2) and

(.9)aq,j

(d)

=

0(2-kN-jN) as k,j

o if

Ial <2

If

By Minkowski's inequality, we have 1 /2

1 /2

(ffig * 'Ftl2dy dt

(6.3)

("k,j)tl2dy tdt

<

tlt2

S(fl)

k,j

Jf19*

12

s(1)

Now, to estimate

kj

I9 *(Tk,j)t12dy tat 12

we use the same argument as that given above, except that now supp(q1k j) < 4R(0; 2k,2)) and not the unit square. If (y,t) a S(cl), then R(y;t) C B and the support of (Wk,j )t(. -y) will be contained in R(y,2kt1,2jt2) C IM(2)(X[2) > 2-(k+j)I =

jj

k,j

Thus

I 5 Ig*('Fk,j)t(y)I2dY

dt2

=

S(Q)

S(B)

11gx

ff

112 ff

kj 2

0

f 0

1(9,,,,

kj) *'Ykjl2dy tat2

1

2 <1i Ill

1

2

kit

I'kj(6,,C2)12

JJ 0

1

2 2

0 00

MULTIPARAMETER FOURIER ANALYSIS

123

By the strong maximal theorem ISkj I < C(k+j)2k+1-, and it is easy to de see that fo 00 fo kj2 decreases like a large power of 2-(k+l) as

k,j

oo.

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 Km with multipliers showed that for multiplier operators I, KI, 1, ei(e) that f, Kif E L1(Rn) implies f E H1(Rn) if and only if the ei

separate antipodal points of Sn-I , i.e., if and only if for every e t Sn-I there exists i such that ei(e) I ei(-e). The way Uchiyama proves this is to show that the dual statement is

,

true, namely, every (k E BMO(Rn) can be written as (6.4)

go + I Kigi for some 90,91,-,gm c

This depends on a simple lemma.

LEMMA. If ei are as above, then given f c L2, and a vector v c Cm+1 , there exist functions go,---, gm E L2 so that

go + I Kigi = f and g(x) = (go(x),

g1(x),..., gm(x))

v for all x E Rn

To prove the formula (6.4), Uchiyama decomposes 0 = I CIO1 as in our (5), and applies the lemma to get functions g1(x) such that Kg1(x) = C1-O1(x) for which g(x) is perpendicular to the correct v, and the result, when modified only slightly to gI, has the property that gI E 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

ROBERT FEFFERMAN

124

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(f) c L1(R2) then f* or L'(R2) [27). Suppose S,&(f) E L'(R2). Then in our discussion of duality we defined atoms ak(x) =

fR(xl,x2) 2kls kl RERk

To simplify this notation we define w = itk and A(x) = 2klSkrak(x) . Let q51 and q52 c Cc (R1) with 0'(x) > 0, f q5' = 1 , and supp(O') C [-1, +11. Set 1t2101(t1).02(x)

-011't2(x1,x2) = ti

Define w = Mt21(X(a ) >

to .

We need to estimate A * q5t

(x) for

1t

2

10

x / CO. To do this let us make the following definitions. If R is a rectangle then R1, R2 will be its sides, so that R = R1 xR2 . Let

AI(x)

= I fR(x),

A?(x)

= L f R(x) . 91k. IR2l=2J

JR11=23

Then to estimate A * 6t1,t2(x) , since supp(q5t( -x)) C R(x; t) = S, in the definition of A we need only consider those fR for which R f1S For any such rectangle R c Rk , since R C (0, minimumr (sill 1`

where w denotes again

,

R il <

IS

Slsi q11 /2

2

Mt21(Xw) >

1010

Then

=P

0.

125

MULTIPARAMETER FOURIER ANALYSIS

IA

A * ctlt2x) =

I A * q5t1t2(x) 2i/Is2I
2)/Is1I
Y fR * Otlt2(x) RE8

where '23 C 91k consists of rectangles R so that iRi < p,

iR2II

I

SII

S2

< p and

kf1S 0. Thus R C''`9 for all R E 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' * q5t1t2x) we use the following trivial lemma.

LEMMA. On R1 suppose that fi(x) E C°° and is supported in an interval

0. Suppose a(x) is supported on disjoint subintervals of $, Ik whose lengths are all < yI$I. Assume that a(x) has N vanishing moments over each Ik. Then

f a(x)-O(x)dx aq

We estimate Al * (btt(x) using the fact that for each fixed x2, A1(- ,x2) has N vanishing moments over disjoint x1 intervals over l

length 2 .2) . (Actually, we sould have to break up A into 3 pieces to insure this, but we spare the reader this trivial complication.) It follows from the lemma that IA

*1

0t

1

(x)I <

)N+I(IA'

2i t1

1 I

I

* 1 X[-t 1 t 1 ] (x)) '

Convolving in the x2 variable, we have IAA *

(' IA'

N+1

-0t1,t2(x)I < C

11

IS

-gI

I dx'

ROBERT FEFFERMAN

126

g I

For this we get


Aj

1

IiI

2'/IS1 I
< CpN /2

1

II

f (IA2) J -9

)N1 /2

1/2

23/IS1 I<

Y 1 /2 r(IAhl2)

By symmetry 2

L.

Aj *

.0tlt2(x)

<

CpN/2 1 IAi

1 /2 121 .

2'/IS2I
Now let RCS , with

IR 1I

R 2I < IS2I

.

Then

S11

and also
IfR * .0t1,t2(x)I

IS

I

1

IfRI

Thus

r

f§I

<- C --L J L IfR *1t t (X)I 2

? ReAap

1

IS I

fR(x)

Y IfRI2 I

IlN /21 /2

R

(

/

RERk

CpN/4 1 I

1 /2

RE%k

(Is

I

1 /2

R E991k

To sum up our findings, we have seen that if x / Ei then

RN/4

127

MULTIPARAMETER FOURIER ANALYSIS

(6.5)

I)N/4

JA * otlt2(x),

C

Snl

fJ(Y)dY

SI

where 1/2

1/2 +((A2)2)

`1/2

-++ (I fR) RE%k

To finish the proof, we need another lemma:

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

LEMMA. Let g(x) _

REI

rectangles. Then 1IgU12 < U J

RE Proof. Let

f(x)h(x)dx

11h11

R8

<

tlt2

91(R)

= 1. Then

22

=f I ff _

If(Y,t)12dy dt

f(Y

,tt(x-y)dy

2

h(x)dx

R2 I

fffthYt)dYTTdt

W(R)

rf f &'t)12dy dt2)

(Pz JJ

1

9I(R)

/2

f

/2 Ih(y,t)12dy dt

tIt2)

R+)2

1/2

c

(51If(Y,t)12dY tat tit 2

1/2 IlS .(h)II2 <

(ffif

(Y't)Izdy dt

tIt2

ROBERT FEFFERMAN

128

Now, notice that, by the lemma,

IIJI12 < ff If(y,t)I2dy

at

f

<_

12

R E(R ) k

S ,(f)(x)dx < C 22k.

Ia 1

/0 kk+1

The same estimate holds for

IIA I12 .

Then

1 /2

(fA*2

A* < 111 /2

J

ti Also away from

< c1,011/2 IIA112 < c2k 1.1

.

('0

A*(x) <

M(2)(.1)(x) >

so 1 /2

fA*dx < (fM(2)(x2o(xdx)'(fM2a)2(x)dx R

< C I,,11121(,11 /22k = C2kIw1

/

.

It follows that IIA*111 < C2kkwi and also IIakIII < C. ROBERT FEFFERMAN DEPARTMENT OF MATHEMATICS UNIVERSITY OF CHICAGO CHICAGO, ILLINOIS 60637

BIBLIOGR APHY

Ill [2]

B. Jessen, J. Marcinkiewicz and A. Zygmund, Notes on the Differentiability of Multiple Integrals, Fund. Math. 24, 1935. E. M. Stein and S. Wainger, Problems in Harmonic Analysis Related to Curvature, Bull. AMS. 84, 1978.

MULTIPARAMETER FOURIER ANALYSIS

[3]

[4]

129

A. Cordoba and R. Fefferman, A Geometric Proof of the Strong Maximal Theorem, Annals of Math., 102, 1975. 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. A. Cordoba, Maximal Functions, Covering Lemmas, and Fourier Multipliers, Proc. Symp. in Pure Math., 35, Part I, 1979. F. Soria, Examples and Counterexamples to a Conjecture in the Theory of Differentiation of Integrals, to appear in Annals of Math.

[7] [8] [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.

ROBERT FEFFERMAN

130

[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, Carleson 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 H1 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 lack 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' ([191), 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 of Section 3 comes from the recent expository article "Recent progress on boundary value problems on Lipschitz domains" ([231). The results explained in Section 2, (b) and Section 3 are unpublished. Full details will appear elsewhere in several joint papers. Acknowledgements. 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 Au =

132u

i=I 9X?

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

7

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 oD , the boundary of D . The classical Dirichlet

problem is to find and describe a function u that is harmonic in D, continuous in 5, and equals f on 09D . This corresponds to the problem of finding the temperature inside a body D when one knows the temperature f on 3D. The classical Neumann problem is to find and describe a function u that is harmonic in D, belongs to C1(D), and satisfies represents the normal derivative of u on XD . N = f on r3D , where 09 This corresponds to the problem of finding the temperature inside D when one knows the heat flow f through the boundary surface X. 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

a

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 e Ck,a(aD), and the bound llullck,a(D) < Ck,allf llck,a(3

k = 0,1,2,...

0
134

CARLOS E. KENIG

study the so-called 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 B. In this case we have a lot of symmetry at our disposal and everything can be done explicitly. Let do denote surface measure of aB.

THEOREM. Suppose that 1 < p < oc and f c LP(3B,do). Then, there exists a unique harmonic function u in B such that lim u(rQ) = f(Q) r-,1

for almost every Q e aB , and

(*)

fu*(Q)Pdo(Q) <_ Cp ff(Q);Pdc7(Q) aB

aB

where u*(Q) = sup Iu(rQ)I. O
The theorem asserts that fr(Q) = u(rQ) converges to f(Q) not only in LP norm, but also in the sense of Lebesgue's dominated convergence. In the analogous estimates to (*) in the Neumann problem, u is replaced by the gradient of u. In that case the estimate fails for p = 00, even if aN 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 IX-QI < (l+a) dist(X, dB), for some fixed a > 0, then u(X) has the limit f(Q) for almost every Q . The theorem is most easily proved by writing down a formula for the 2

solution, u(X) = fas P(X,Q) f(Q) do(Q) , where P(X,Q) = n Iin IX-Q 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 preliminaries

(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 e aD there is a

ball B with center at Q, a coordinate system (isometric to the usual

coordinate system) x'= (xi, , xn_I), xn, with origin at Q and a function (h: Rn-I R such that 4(O)=O, IOW)--0(0I <Mlx'-y'I and Df1B = (X =(x,xn):xn>o(x'){(1B .

If for each Q the function 0 can be chosen in CI(Rn-I), then D is called a C1 domain. If in addition, 74 satisfies a Holder condition of order a, IV(*')-V (y')I <-Clx'-y'la ,

we call D a Cl,' domain. Notice that the cone re = l(x,xn):xn<-Mlx'1{ satisfies ren B CCD. Similarly, ri = 4(x ,xn) : xn>M1x'I ! satisfies ri n B CD. Thus, Lipschitz domains satisfy the interior and exterior cone condition.

The function 0 satisfying the Lipschitz condition 10(x')--*(y')I < Mlx'- y'l is differentiable almost everywhere and 170 e L00(Rn-I), 111741I,, M.

136

CARLOS E. KENIG

Surface measure a is defined for each Borel subset E C d D fl B by

f

a(E) =

(1 + IO95(x')12)I

/2 dx

E

where E* = ix': (x', O(x')) eE 1.

The unit outer normal to dD given in the coordinate system by /2

exists for almost every x'. The unit normal at Q will be denoted by NQ. It exists for almost every Q e dD , with respect to da. (NO(x'), -1)/(1 + lvq,(X,)I 2)1

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 : AF = 8, where

n>2

1 (n-2)wnjX1n-2

F(X) =

1-

TIT

log jXJ

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. with It provides a formula for a solution w to the equation Aa

( eCp(Rn),

w(X) = F *#(X) = fF(xYY)dv. Rn

It will be convenient to put F(X,Y) = F(X-Y). Notice that tYF(X,Y) _ 5(X-Y). The fundamental solution in a bounded domain is known as the Green function G(X,Y). It is the function on D xD continuous for X j Y and satisfying A G(X,Y) = 5(X-Y), X e D ; G(X,Y) = 0, X e D, Y e dD.

ELLIPTIC BOUNDARY VALUE PROBLEMS

137

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

5u(Y)(XY)dY

r u(Y)AYG(X,Y)dY = JD

D

=

=

5Eu(Y)AyG(xv) - Au(Y) G(X,Y)ldY = D

f[u(Q)

(X,Q) -

(Q)G(X,Q)J da(Q) =

Q

Q

aD

=

Ju(Q) aN G(X,Q)da(Q) 4

,

aD

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

(2)

5f(Q) aN (X,Q)da(Q)

u(X) =

Q

aD

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

CARLOS E. KENIG

138

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

5f(Q) aN (X,Q)da(Q) _

w(X) =

Q

0 1 «n

IX--Q InI<X_Q,NQ> da(Q)

aD

as a first approximation to the solution of the Dirichlet problem, Au = 0

in D, uI

D=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 e W, w(X) 2 f(Q) + Kf(Q), where K is the operator on aD given by

Kf(Q) = wn

J

<

f(P)do(P) .

p Q Inp

an Kf were zero, we would be done, and in some sense it is true that Kf is small compared to 2 f, when the domain D is smooth. In fact, aD If

has dimension n-1 , while it is easy to see that if aD is C°°,
C IP-QIn-2

Thus, the operator K: C(3D) C(aD) is compact. Therefore, by the general theory of Fredholm, the operator T = 2 I+K is invertible modulo

a finite dimensional subspace of C(3D). If D and cD are connected, T is actually invertible on C(3D). Therefore, the solution to the Dirichlet problem may be written

139

ELLIPTIC BOUNDARY VALUE PROBLEMS

<X-Q,N > u(X) = wn

g(Q)

IX-Ql °

do(Q),

dD

where g = T-1(f). The operator K is compact on C(O) even in C1'a domains because in this case

_Qln

I

I<

C n Q-

l+a) and so this

P procedure solves the classical Dirichlet problem in that case too. If D

is a C°° domain, K is compact from Ck'a(aD) to Ck,a(3D), k = 1, 2, , 0 < a < 1. Hence, if f e C°°(aD), u c C'(5). This approach can also be used to obtain results for f c Lp(aD) in C°° domains, and even on C1'a domains. We will now sketch the extension of the theorem for the ball stated in the introduction to C1'a domains. We first define non-tangential approach regions as follows: r (Q) _ IX eD : IX-Q I < (1+(3)dist (X, 3D)}

.

The non-tangential maximal function, with opening 16, of a function w

defined in D is Na(w) (Q) = sup I lw(x)j : X er0(Q)}

Because of the estimate

.

IP--P,QNp>

I
n

I

on

Cl,a

domains, it is easy to see that K is compact as a mapping on LP(dD). Also, standard arguments show that N16 (w) (Q)
where M is the Hardy-Littlewood maximal operator on X, and w is the double layer potential of f. Finally, from LP bounds for M, K and T-1 = \2

I+K)-1

,

one obtains:

140

CARLOS E. KENIG

THEOREM. Let D be a Cl,a domain, 1 < p < o. If f c LP(aD, da), <X-Q,N >

and u(x) =

(On

C p

1 1 f 11

w

LP(da)

f3D

IX-Qln

T-If(Q)da(Q), then JIN0uIILp(do) <

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

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


C <

IP-QIn-I

and so, even the LP boundedness, much less the compactness of K, is far from obvious.

Let us now turn to the Neumann problem. Let D be a smooth domain. We seek to solve Au = 0 in D, !AL I = f. By Green's formula, we aD

must have f3D fda = 0. When D and cD 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) = faD f(Q)F(X,Q)da(Q) = 1 n-2 IX-QlaNda(Q). Once again v is harmonic in D, and (Q) = 2 f(Q)-K*f(Q), where K* is the adjoint of K above, i.e. *Q 1 K f(P) = Wn 13D IP-Qln f(Q)da(Q). Thus, K is compact from

Cn faD

f(Q)

Ck,a(aD) to Ck,a(aD), and Fredholm's theory says that

=2

I-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) =

f(T1f)(Q)F(X,Q)da(Q), aD

and if f a C°°(3D), u e C°°(D). If D is CI'a, T is also invertible on

ELLIPTIC BOUNDARY VALUE PROBLEMS

141

the subspace of Lp((3D) of functions with mean value, 1 < p < w. Hence:

THEOREM. Let D be a C I'a domain, (D and cD connected). Let 1 < p < oo. Assume that f e Lp(3D, da), faD fda - O. Then,

u(X) = faD (T-If)(Q)F(X,Q)da(Q) is harmonic in D, Vu(X) has nontangential limits Vu(Q) for a.e. Q e aD, f(Q) = , and JIN,(Vu)JJLp(da) < CplifilLp(do).

What do we do when c9D is merely CI , 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 CI -domain, K: LP(OD) -, Lp(09D), 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 CI 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 of Zaremba and Lebesgue, we can solve the classical Dirichlet problem for A in D , for any f e C(dD). Given X e D , the maximum principle shows that the mapping f " u(X) defines a positive continuous linear functional on C(09D). Therefore, by the Riesz representation theorem, there is a unique positive Borel probability measure wX on 3D such that

u(X) =

ff(Q)da(Q) dD

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: wo = a/a(cB). (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 X principle). We fix X0 e 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 almost everywhere with respect to w. Conversely, given any set E C OD, with w(E) = 0, there is a positive harmonic function u in D with lim u(X) = + oo as X Q, for every Q e 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. If D is a CI,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 L. On CI 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 CI or even a Lipschitz domain, harmonic measure and surface 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 ( L2(aD, d o) . In fact, he showed that given a Lipschitz domain D , there exists s = (D) such that this can be done for f ( LP(dD, do), 2-e < p < o. 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 L. 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 C1 domains. (The best possible regularity result for harmonic measure on C1 domains is due to D. Jerison and C. Kenig (1981): if k =4, then log k ( VMO(aD).)

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 L2(aD, do) 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 L2 case, leaving the corresponding LP theory open. In 1981, R. Coifman, A. McIntosh and Y. 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 I 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 L2(3D, da) 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 L2 results of Dahlberg [5] 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 D C Rn , there exists E = E(D) > 0 such that one can solve the Neumann problem for Laplace's equation with data in Lp(r7D, do)

,

1 < p < 2 +E 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 HI(dD,da), which generalize the results for n = 2 in [20] and [211, and for C I domains in [12 ]. The key idea in this work is that one can estimate the regularity of the so-called 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 L2 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 CI 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. Invertibility is shown again by means of 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 estimates, 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. Laplade's equation on Lipschitz domains

(a) The L2 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 non-tangential region of opening J8 at a point Q c aD is I'18(Q) _ IX cD : IX-Qj < (1+,8)dist (X, 3))J. 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 even to the case n = 3 ), and to domains D C Rn , D = I(x,y): y>4i(x)), where : Rn-I R is a Lipschitz function with Lipschitz constant M, i.e. j0(x)-4(x')I < Mix-x'I . D- _ I(x,y):y M'Iz-xII C D. Points in D will usually be denoted by X, while points on aD by Q = (x, (k(x)) or simply by x, Nx or NQ will denote the unit normal to aD = A at Q = (x, 4(x)). If u is a function defined on Rn'A and Q c aD , u t(Q) will denote lim u(X) or X-3Q

XEIi(Q)

If u is a function defined on D, N(u)(Q) _

lim u(X), respectively. X-Q Xc e(Q)

sup

Iu(X)I.

XcF1(Q)

We wish to solve the problems

Au = 0 in D

Au = 0 in D (D)

(N)

,

u' aD = f c L2(aD,do)

a-N

f c L2(3D,do) aD

The results here are

THEOREM 2.1.1. There exists a unique u such that N(u) c L2((9D, do),

solving (D), where the boundary values are taken non-tangentially a.e. .

Moreover, the solution u has the form

u(X)

<X-Q'NQ>

1

(On

J aD

for some g c L2(3D, da).

IQ_XIn

g(Q) do(Q)

147

ELLIPTIC BOUNDARY VALUE PROBLEMS

THEOREM 2.1.2. There exists a unique u tending to 0 at -, such f(Q) that N(Vu) E L2(dD, do), solving (N) in the sense that as X -* Q non-tangentially a.e.. Moreover, the solution u has the form

u(X) = w n (n_2)

1 1X-Q1

n-2 g(Q)do(Q)

,

aD

for some g ( L20D, d u) .

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

g(x) =

(an

<X-QNQ>

J

IX-QIn

g(Q)do(Q)

aD and

f

Sg(X) _ - wn(

I n 2 g(Q)do(Q) IX-QI

aD

If Q = (x, 4(x))

,

X = (z ,y) , then

g(z.Y)=w f

[Ix-z I2 + [fi(x)-0(z)121n/2

g(x) dx

Rn-I

Sg(z ,Y) _ -

1 ca

n-2)

r

'J

1+IV _(X) I2

-n 2- g(x) dx

Rn-I [Ix-zI2+I4(x)-y]21

2

THEOREM 2.1.3. a) If g c LP(aD, do), 1 < p < -, then N(VSg), N(Xg) also belong to Lp(aD, do) and their norms are bounded by CilgjI p

L ((9D,du)

148

CARLOS E. KENIG

(b) lim

r

1

z)-q(x)-(z-x) VS6(x) g(x) dx = Kg(z)

Wn

Ix zI>E

a.e. and IIKgII

lim

[IX-zI2

<_CIIgII

LP(aD,do)

Lp(aD,do)

1
(z-x,(h(z)-S(x)) 1 +IW,(x)I2 g(x)dx

1

E- 06)n

exists

+[S6(x)-0(z)]2]n-2

[Iz-x 12 + [S6(z) -

S6(x)j2]./2

exists a.e. and in

1z-X I>E

Lp(aD, do), and its LP norm is bounded by CIIgII

LP(aD,do)

,

1 < p < oo.

(c) (xg) t(Q) _ ± 2 g(Q) + Kg(Q)

J

(VSg)t(z)=±1 g(z)Nz+- lim 2

n E -+o

(z-x,0(z)-0(x)) (1 + Q(k(x) g(x)dx [Iz-xI2+[(k(z)-(k(x)]2]n/2

Iz xI>E

g(z)-K*g(z), where K* is the

COROLLARY 2.1.4. (NzVSg)±(z)

L2(aD,do) adjoint of K.

2

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

Let 0: R - R be even, and C'. Let A,B: Rn-1 - R

be Lipschitz. Let K(z,x) = A(z) -A(x 916 z-x6 x 1 . Then, the maximal I

operator M*g(z) = sup

E-0

I

f

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

It XI>E

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

ELLIPTIC BOUNDARY VALUE PROBLEMS

149

IIM*gllp << CIlgllp

where C = C(M,0,n,p), and IIVAII00 < M, IIVBII. < M.

The proof of (b), (c) follow from the theorem above, together with the following simple lemmas. LEMMA. If f cCo(Rn-1), then

lim 1

E +0 Wn

c(z)-c(x)-(z-x) VOW f(x)dx iz-xI>E [Ix-zI2+[O(x)-95(z)]2]n/2

n--1

zk-xk

1

_-

n-I

J

1

X

[(z)_(x)1 x-z

axk

(x)dx

+t2)-n/2, where a(0) = 0, X'(t) = (1 and where the equality holds at every z at which 95 is differentiable, i.e. for a.e.z.

LEMMA. If a c Rn-1 previous lemma, then

1

c.°n

a

,

(a), f c Co(Rn-1) and A is as in the

I

a-O(a)- (a-x) V (x) f(x)dx = [Ia-xI2+[0(x)-a]2]n/2 n-1

=

2

sign (a --O(a)) f(a) - cI on n

fI

xk ak Ix-amn-I

fix) a

x--a)

c3f (x)dx c7xk

Moreover, the integral on the right-hand side of the equality is a continu-

ous function of (a,a) c R. 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 (2 I+K) and

150

CARLOS E. KENIG

I+K*

are invertible on L2(aD,do). This is the result of

G. Verchota ([33]). THEOREM 2.1.5.

(± 2

I+K)

,

(+

2 I+K*) are invertible on L2(3D,do).

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

2

L2(c3D,`do),

I+K*)J

are invertible. In order to do so, we show that if f e 1 I-K fll , where the constants 1 I+K* fIi II

2

II

L2(dD,do)

(2

L2(aD,do)

of equivalence depend only on the Lipschitz constant M. Let us take this for granted, and show, for example, that 2 I+K* is invertible. To

do this, note first that if T = 2 I+K*, IITf11L2 ? CIIfIIL2, where C depends only on the Lipschitz constant M. For 0 < t < 1 , consider the operator Tt = I + Kt , where Kt is the operator corresponding to the 2 by to. Then, To = 2 1, TI = T, and 03Tt : y domain defined LP(Rn-1)

cit

Lp(Rn-1) , 1 < p < - with bound independent of t , by the theorem of

Coifman-Mclntosh-Meyer. Moreover, for each t,

IITtf II

L2

> C Ilf II L2 ,

C

independent of t. The invertibility of T now follows from the continuity method:

LEMMA 2.1.6. Suppose that Tt : L2(Rn-1) -. L2(Rn-1) satisfy (a) IITtfIIL2 > C111flIL2 (b) IITtf-Tsf1IL2 < C2lt-sIIf!IL2. 0 < t,S < 1 .

(c) To: L2(Rn-1) , L2(Rn-1) is invertible. Then, T1 is invertible. The proof of 2.1.6 is very simple. We are thus reduced to proving (2.1.7) II(2 I+K*)fII

2

Od a` 11

JI

2

I-K*fII

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 e Lip (6), Du = 0 in D, and u and its

151

ELLIPTIC BOUNDARY VALUE PROBLEMS

derivatives are suitably small at -. Then, if en is the unit vector in the direction of the y-axis,

ay ado.

fiVui2da = 2 aD

aD

1 Vu 12 = 2

Proof. Observe that div(enIVu12) _

div a Vu = - Vu Vu + . div Vu = - Vu Vu .

Vu - Vu, while

Stokes' theorem now

gives the lemma.

We will now deduce a few consequences of the Rellich identity. Recall < < 1 . 1 that N xx (-V (x),1)/X/1 + IVO(x)12, so that (1+M2)I/2

COROLLARY 2.1.9. Let u be as in 2.1.8, and let T1(x), T2(x), Tn_I(x) be an orthogonal basis for the tangent plane to dD at (X, O(X)). n-1

Let Iptu(x)12 = I I 12. Then, j=1

J(

)2da
aD`

t3D

Proof. Let a = en-Nx, so that a is a linear combination of T1 (x), T2(x), ...,Tn-1(x). Then, IDu12

= ( /2 +

jVtu12, and so faD

= aN`+ . Also,

( )2do+

faD1VtuI2do=

( )2do+2

M faD TN ('u) do. Hence 2 faD faD IVttu12da - 2 f do. So, f C( faD IVtuj2da)1/2

(faD

N

faD(o)2do=

jVtuI2do+ C faD

2do)1/2, and the corollary follows.

152

CARLOS E. KENIG

COROLLARY 2.1.10. Let u be as in 2.1.8. Then, flVtul2da< c aD

f(

)2 do.

aD

Proof. faD IVuj2do < 2(f,3D IVu12d(,)1/2 GD I

I2do1/2, by 2.18,

and the corollary follows.

COROLLARY 2.1.11. Let u be as in 2.1.8. Then, faD IVtuI2da ti faD

olu I

In order to prove 2.1.7, let u = Sg. Because `of 2.1.3c, ptu` is con

tinuous across the boundary, while by 2.1.4,

C0N`±

now apply 2.1.11 in D and 5, to obtain 1.1.7.

=

This

(T. z I-K*) g . We

finishes the proof

of 2.1.1 and 2.1.2.

We now turn our attention to L2 regularity in the Dirichlet problem. DEFINITION 2.1.12. f E LP(A), 1 < p < ., if f(x, ¢(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 VxF(x,95(x)) is well defined, and belongs to LP(A). We call this Vtf.The norm in LP(A) will be IIVtf1ILP(A) THEOREM 2.1.13. The single layer potential S maps L2(A) into L2(A) boundedly, and has a bounded inverse.

Proof. The boundedness follows from 2.1.3a). Because of the L2-Neumann . The theory, and 2.1.11, IIDtS(f)I1 2(A) > CIIN(f)i 2 > CllfIi 2 L (A ) L (A) L argument used in the proof of 2.1.5 now proves 2.1.13. THEOREM 2.1.14. Given f c L2(A), there exists a harmonic function u , with IIN(Vu)IlL2(A) < CllVtfIl 2(A), and such that ptu = Vtf (a.e.) nonL

tangentially on A . u is unique (modulo constants), and we can chose u = S((), where g f L2(A).

I2 do

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 E C , and for 0<,8<2n, let D18= 1z EC:jargzI <J3/21.

We will consider the holomorphic function f(z) = zn/R, which maps DR

conformally onto Dn, the right-hand plane. We will also consider a bounded domain SZR CDR, with the property that an \0 is smooth, and

such that an

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

fl 1zI < 1 # = aDR n ilzI < 1 !.

v(x,y) = u(x,y)jn

v is harmonic in SZR, and v is identically zero near

R

the corner of aSZR, and is smooth everywhere else in

the arc length parametrization of an

11

.

Let s be

starting at 0. Then, it is clear

,

that as E L°°(aS2R). Let w(x,y) = Im f(z)l9

.

By the Cauchy-Riemann

equations,

I

I - ICI

EL°°(3113). I

However, N(Vv) (s) = N(Vw) (s) ti s- 1+771 16This function belongs to

Lp(ds) if and only if p v/R - p > -1. Fix now a p > 2, and choose so close to 277 that p n/R-p < -1 . Then, N(Vw) / Lp(ai2R). If

R

I - K*J were invertible/ in LP(a10) , then, since a-N E L°°(a11R), we \2 would have that S((2 I-K*) I (has a non-tangential maximal function in LP(a11 R). By the L2-uniqueness in the Neumann

problem, w-w is constant in 9R, but this is a contradiction. This shows that given p > 2 , we can find a Lipschitz domain so that I -- K*J is not invertible in Lp. The example can also be used to \2

154

CARLOS E. KENIG

I+K is not always invertible in Lq , when q < 2. In fact, fix q < 2 , and let p satisfy + 9 = 1 . Choose f3 so that show that

p'-''-p<-1. Let B =

and fix XeQ\IJzl<11.

X

Let o = w * be harmonic measure evaluated at X*, and k = d(''. We

rs

first claim that k / Lp(ds). In fact, let G(X) be the Green's function of

0 with pole at X*. Then, for s near 0, k(s) = = lim G s+EN > C lim v s+eN = C lim G(s+EN)-G(s) a

e-.0

E-e 0

E

E

O

E

(s) _

av (s)

s-1+v/I3,

(3N

where the first inequality follows from the fact that both G and v are positive, and harmonic on B, and 0 on ()iZP fl B (this is Lemma 5.10

in [19]). Assume now that 2 I+K were invertible on Lq(ds). Let g ? 0 e C(dl3 ), and h(X) be the solution of the Dirichlet problem with

data g. Then, h(X*) =

J

gda, = J gkds

aiiP

0-n

also, by the L2-theory, h(X*) = x[(2

I+K)-1(g]

(X*), where x is

the double layer potential. Let U be a ball centered at X*, contained in 9P. By the mean value property of harmonic functions and Harnack's principle, we have

h(X*) <

JJ

fhc(fNhds)

1 /q

1 /q

< C J ggds

because of the second formula for h(X*), and the assumed Lq bounded1 . But this implies that k ( LP(ds), a contradiction. ness of (2 1+K We now turn to the positive results. They are-

ELLIPTIC BOUNDARY VALUE PROBLEMS

155

THEOREM 2.2.1. There exists e = e(M) > 0 such that, given f c L'(aD, da) , 2- e < p < oo, there exists a unique u harmonic in D, with N(u) a Lp(3D, do) such that u converges non-tangentially almost everywhere to f. Moreover, the solution u has the form

<X-Q,NQ>

u(X)

n

I

-

X Ql

g(Q)da(Q) ,

aD

for some g c LP(iD,da) . THEOREM 2.2.2. There exists E = e(M) > 0, such that, given

f c Lp(3D, da), 2-E < p < ac, there exists a unique u harmonic in D, tending to 0 at oc, with N(Vu) c LP(3D, da) , such that NQ Vu(X) converges non-tangentially a. e. to f(Q). Moreover, u has the form

u(X) =

wn(n-2)

J IX-Q

In-2

g(Q)da(Q)

,

aD

for some g e Lp(3D, d a).

THEOREM 2.2.3. There exists E = E(M) > 0 such that given f c LP(A), < 1 < p < 2+E, there exists a harmonic function u, with IIN(Vu)lt

L'(A) CIIVtfIILp(A), and such that Vtu = Vtf (a.e.) non-tangentially on is unique (modulo constants). Moreover, u has the form

u(X) _ -

1

1

wn(n-2)

fIX-QIn-2

aD

for some g c Lp(3D, d a).

g(Q)da(Q) ,

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) ([51). 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) ([61). 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

invertible in Lp(dD, do), 1 < p < 2 + E,

\±

2 1+9 is

LP(aD, da), 2-E < p < co , and S : LP(dD, do)

r+ - I - K*) is i\`nvertible in

Lp(aD, do) is invertible

1
LP(dD,da)

This will be done by proving the following two theorems:

THEOREM 2.2.5. Let Au = 0 in D. Then JJN(Vu)JJ

1 < p < 2+E

C IIaNIILp(aD,da)'

.

THEOREM 2.2.6. Let Au = 0 in D. Then 1lN(Vu)JJ

Clivtull

LP( dD,do)

,

<

LP(aD,da)

1
<

Lp(dD,da)

We first turn our attention to the case I < p < 2 of Theorem 2.2.5. In order to do so, we introduce some definitions. A surface ball B in A

is a set of the form (x,4(x)), where x belongs to a ball in Rn-I . DEFINITION 2.2.7. An atom a on A is a function supported in a surface ball B, with hail L < 1/a(B), and with fA ada = 0. Notice that atoms are in particular L2 functions. The following interpolation theorem will be of importance to us.

ELLIPTIC BOUNDARY VALUE PROBLEMS

THEOREM 2.2.8. Let T be a linear operator such that C IIf11L2(A), and such that for all atoms a,

IITaII

157

llTffI

L2(A)

<

I,(A) < C. Then, for

1 < p <2, IITfIILp(A) «IIf1ILp(A). 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 < C . By dilation and show that if a = au is a atom, then IIN(Vu)II L1(A)

translation invariance we can assume that 0(0) = 0 , supp a C BI =

Let B* be a large ball centered at (0,0) in Rn, which contains (x, qS(x)), jxl < 2. The diameter of e depends only = C, by the L2-Neumann theory, on M. Since IIa II [(x, O(x)) : xl < 11.

o(Bjl

L2(A)

/2

I

faDnB* N(Vu) < C(faDnB* N(Vu)2da)1 /2 < C. Thus, we only have to

estimate

JCB*naD

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

larity theory for divergence form elliptic equations. Consider the biLipschitzian mapping (D: D - D- given by 4D(x,y) = (x, O(x) - [y-(x)1).

Define u* on D- by the formula u* = u at-1 , u* verifies (in the weak sense) the equation d iv (A(x,y) Vu*) = 0, where A(x,y) _ J X)

(X) ,

where X = 1-1(x,y) . It is easy to see that A c L°°(D-), and < A(x,y)e, t; > > CI; I2 . Notice also that supp dN C B 1 < B* fl aD. Define now I

for (x,y) e D

B(x,y) =

u(x,y) ,

A(x,y) for (x, y) c D

for (x,y) c D

and u (x, y)

u*(x,y) for (x,y) c D-

Because 0 = 0 in 3D\B* , 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, Id' =div B(x,y)ju = 0. In order to estimate u, (and hence Vu ) at -, we use the following theorem of J. Benin and H. Weinberger ([31]).

158

CARLOS E. KENIG

THEOREM 2.2.9. Let u solve Lu = 0 in Rn\B*, and suppose that

< -. Let g(X) solve Lg = 0 in

Ilall

g(X)

IXI > 1

,

with

IXI2-n. Then, u(X) = u. + ag(X) + v(X), where Lv = 0 in Rn\B*,

. IXI2-n -v, where v > 0, C > 0 depend L°°(Rn\B * ) only on the ellipticity constants of L. Moreover, a = C JB(X) Vu(X) . V &(X)

and Iv(X)I < C Ilull

= 0 for

where i/i c C°°(Rn),-

,

in 2B*, and 0 = 1 for large X.

IXI

Let us assume for the time being that u is bounded and let us show that if a is as in 2.2.9, then a = 0. Pick a tA as in 2.2.9. In D, B(x) = I , and so fD = f Vu V & = lim f Vu Vi , where

E-0 DP

D

D

DP = {(x,y): I(x,y)I < p, y > 4(x)+e 1, and p is large. The right-hand side equals lim e-0

faD1

aN

harmonicity of u, Jape

= lim J ayo

a-N

since, by the

aDp

= 0. Let ct! 1 = {(x,y) c aDp : y > fi(x)+e

P

and cDe

P12

\ (VP, I. P

= aDe

Then,

lim f

3Dp

[t/i-1 ]

au -3N

1] = f [0-11 a = JaD t/ia - JOD a -0 J3DP,2 [ i/i = 0 on supp a. Moreover, JD_ BVuo = fD Vu E

[0-11 au

= lim f

e-,0 aDP

J0

1

TR

a = 0, since

where 1* 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 < C, we know that assume that n > 4 for simplicity. Since Ila 11 L2(A)

u(X)

=cn JaD

f(Q)2 da(Q), with

IIfII

IX-QI

(x,y): y > 4(x)+1 {,

L2(A)

c L2(A), and so u c L°°(D1).

C

< 1 IX-QIn-2 1+


IQIn-2

Let now B be any ball in Rn so that 2B C Rn\B*, B is of unit size, and such that a fixed fraction of B is contained in D 1. Since N(Vu) c L2(A), with norm less than C, J2Bf1D IDuI2 < C, and moreover on B f1D 1 , lu(X)I < C. Therefore, by the Poincare inequality

+

159

ELLIPTIC BOUNDARY VALUE PROBLEMS

J

I2 < C. But, since

2B ([261).

solves Lu = 0, max [u1 < C(f

'

2B

B

rul21 /2 < C

Therefore, u E L(Rn\B*), IrII LOO(Rn\B*) < C. Hence, since

a = 0, Vu=Vv, and Iv(x,y)I VC/(IxI

+IyI)n-2+v'

diam B*, set b(R) = fA N(Vu)2, where, A

v > 0. For R > R0 = ( x,

I

2R

R

For each fixed R , let N1(Vu)(x) = sup I Ipu(x,y)I : (z,y) E I'i(x), dist((z,y), 9D) < SRl, N2(Vu)(x) = sup flVu(z,y)I : (z,y) E Fi(x), dist((z;y), (3D) > SR}. In the set where the sup in N2 is taken, u is harmonic, and the distance of any point X to the boundary is comparable to IXI. Thus, using our bound on v, we see that N2(Vu)(x) < C/IXIn-1+v ti C/Rn-l+v, and so JA N2(Vu)2 < CR1-n-2v Let now

/

R

R(x,y) : 4(x) < y < O(x) + CR, rR < Ix I < r--1R 1, r c \4 , 2) . By the

L2-Neumann theory in fZr,

f NI(Du)2du <

Ipul2dX < 3

I

R AR

r

R

in r from 1 /4 to 1 /2 gives

Ioul2da. Integrating

N1(Vu)2da < C J

JA,

,J

R

Q1/4\Q1 /2

r

u2

J

C1R
since Lu = 0 (see [26] for example). The right-hand side is bounded by 1 Rn = C CR1-n-2v, and hence b(R) < CR1-n-2v. Then, R3 R2(n-2)-2v

JA R N(Vu) < C(fAR

N(Vu)2)1

/2

n-1

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

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

DEFINITION 2.2.10. A function a is an H

(a) supp A C B, a surface ball, (b)

atom if A = Vta satisfies -1

IIAII

< 1/a(B), (c) f Ada= 0.

We will use the following interpolation result:

160

CARLOS E. KENIG

THEOREM 2.2.11. Let T be a linear operator such that

IITfII

I.2(A)

<

CIIfIIL2(A) and IITa1ILI(A)
1 < p <2, IITfIILP(A)
Hence, all we need to show is that if Au = 0, Qtu = Vta, and a is a unit size Hi atom, N(Vu) E L1(A). But note that if we let u(x,y)

(x,y) c D

u(x,y) _ -u*(x,y) (x,y) E D_

then u is a weak solution of Lu = 0 in Rn\B*, since uLaD\B 0. Then, u =

-i ag + v , but a = 0 since 'u -'U. must change sign at

°°.

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

results for CI domains in [12]. Some of the results one can obtain are the following: Let Hit(3D) _ Xiai : IIXiI < oo, ai is an atoml, Hi at(aD) = I Aiai: XIAiI < +°°, I

ai is an Hi atoms. THEOREM 2.2.12. a) Given f c Hat(cD) , there exists a unique harmonic

function u, which tends to 0 at -, such that N(Vu) ( LI(3D), and such that NQVu(X) -, f( Q) non-tangentially a.e. Moreover, u(X) = S(g)(X), g ( Hat . Also, ulaD E Hl,at(aD) b) Given f c H1,at, there

exists a unique (modulo constants) harmonic function u , such that N(Vu) F L1(cD), and such that VtulaD = Vtf a.e. Moreover, u = S(g),

g c Hat , and a E Hat(aD). c) If u is harmonic, and N(Vu) E L1(oD),

ELLIPTIC BOUNDARY VALUE PROBLEMS

161

d) f c Hat(aD) if and only if -N a E Hat(3D), ulaD E N(VSf) E L1(3D), if and only if \2 I-K*) f E Ha't(dD). then

Hl,at(c3D).

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 L2 results hold for all Lipschitz domains. We will now show that IIN(?u)IILp(n)
Let y=t(x,y)ER+:Ix(
IVu(z,y)I , and

sup (Z' Y) Ex+y

m

*(x)

sup

=

IVu(z,y)I . Our aim is to show that there is a small

(z, y) E x+y*

Eo > 0 such that fm2+Edx < c f clu .

f=

Ifl2+Edx

,

for all 0 < E < EQ, where

Let h = M(f2)1/2, where M denotes the Hardy-Littlewood

maximal operator. Let EX = Ix E Ra-1 m* (x) > At. We claim that

f

Im*>A,h
m2. Let us assume the claim, and

m 2 < CA2IEAI + Ca f Im*>X

prove the desired estimate. First, note that fm2< EA

f<X1 m2 +

J

t m *>I h>A

m2 < CX2IEXI + Ca

J m2 + J m2 I h>Jl l

t m *>A 1

1/2 . Then, by the claim. Choose now and fix a so that fEX f m2. For E>0, fm2+E=E f°°AE-1 f EA

0

Ih>At

E f0 AE-1 fEA m2dA < C E fO

Al+E IIm* >AIIdA+CE fQ c'o AE

,

m2dA<

Im>AI

1(fh>A m2)dA.

162

CARLOS E. KENIG

By a well-known inequality (see [141 for example), IEAI < Thus, fm2+E < CE fo AI+E Iim>a)lda+CE fo aE-1( fh>A m2)dA < Cc f M2+,+ C fm2 hE. If we now choose EO so that CEO < 1/2, for E < EO, fm2+E < C f m2 hE. If we now use Holder's inequality with exponents 2 2 E and 2+E

2

E

2+E , we see that fm2+E < C(f m2+E)2+E(fM(f2) 2 )2+E, and the E

desired inequality follows from the Hardy-Littlewood maximal theorem.

It remains to establish the claim. Let {Qkl be a Whitney decomposition of the set EA = Im*>AI, such that 3Qk C EA, and 13Qk1 has bounded overlap. Fix k , we can assume that there exists x c Qk such that h(x) < X, and hence, f2Q f2 < Ca2lQk l . For 1
rQk, and Qk,r = (x.y) : x c rQk , 0 < y < r length (Qk)}. Qk,r (and (D(Qk,r) ) is a Lipschitz domain, uniformly in k,r. Also, by construction

of Qk, there exists xk with dist(xk,Qk) ti length (Qk), and such that m*(xk)
Bkr = aQkrflR+\Akr so that aQk,r = Qk,r U Ak,r U Bk,r. Note that the height of Bk,r is dominated by Ca length (Qk), and that IVul <,k on Ak,r. Let ml be the maximal function of Vu, correspond ing to the domain Qk,r (i.e. where the cones are truncated at height ti P(Qk) ) Then, for x c Qk, m(x) < mI(x) + X. Also,

f m i < (using the L2-theory on

C J IVuI2da+ c Bk,r

flvuI2da+c f f2 < C J IVul2da+c)2IQkl Ak,r

2Qk

Bk,r

.

163

ELLIPTIC BOUNDARY VALUE PROBL EMS

Integrating in r between

and 2, we see that

1

a length(Qk)

5 m21 <- length(Qk) C

f 0

2Qk

2Qk

Thus, fQ m2 < Ca f2Q m2 + CA2IQkl. Adding in k, we see that k

f

im*>A,h<,1}

k

< Ca2IEAl + Ca f

im*>AI

m2

, which is the claim. Note also

that the same argument gives the estimate IIN(Vu)IIp < CIIVtullp

2 < p < 2+e, and the LP theory is thus completed. §3. Systems of equations on Lipschitz domains (a) The systems of elastostatics. In this part we will sketch the extension of the L2 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 ([81). 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

Lipschitz function 0: R2 y R. Let a > 0, µ > 0 be constants (Lame moduli). We will seek to solve the following boundary value problems, where u = (u1,u2,u3)

µ0u+(a+µ)Vdivu=0 in D (3.1.1)

ulaD =f eLZ(aD,da) µ0u + (k +µ) p div u = 0 in D (3.1.2)

A(div u)N+µl pu+(pu)tlNlaD =f EL2(aD,da).

164

CARLOS E. KENIG

(3-1 -1) corresponds to knowing the displacement vector u 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

..(X) = A S'1 17

4n IXI

'XI

41r- IXI3

r1+ 1 and A J2Lu 2µ

C= 1 1_ 2[µ

We will also introduce the stress operator T, where Tu = A (div u) N + ulVu + Vu tIN.

The double layer potential of a density g(Q) is then given by u(X) Xg(X) = fdD (T(Q)F(X-Q)}tg(Q)da(Q), where the operator T is applied to each column of the matrix F. The single layer potential of a density g(Q) is

u(X) = Sg(X) _

fr(xQ). g(Q)da(Q) dD

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

D, with N(u) c L2(aD,da). Moreover, the solution u has the form u(X) = ((X), g c L2(dD, do). (b) There exists a unique solution of (3.1.2) in D, which is 0 at infinity, with N(17q) c L2(r7D,do). Moreover the solution u has the form u(X) _ Sg(X), g c L2(dD,da). (c) if the data f in 3.1.1 belongs to Li(dD1 da), then we can solve (3.1.1), with N(V) c L2(iD,da). Moreover, we can take u(X) = Sg(x), g c L2(aD, 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', Sj be defined as above, so that they both solve

µ0u + (A+µ)p div u = 0 in R3\oD. Then: IIN(xg)IILP(aD,do,) < C11111

(a)

LP((3D,da)'

IIN(osg)IILP(aD,da) <- CII g IILp((3D,da)

(b)

for 1
g(P) + Kg(P)

(xi)±(P) = ± 2

: (Sg )j)± = ± jA 2 ni(P)g1(P) - ni(P) nj(P) < N(P), g(P)>} C

1

I

ll

+

+

)))

(.v.

f

aPi

F(P-Q)g(Q)da(Q)

aD

,

j

where K(P) = p.v. faD 1T(Q)F(P-Q)#t 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 L2(0,D, da) of ± 2 1+K, ± 2 I+K*, and the invertibility from

L2(aD, da) onto L2(aD, da) of S. Just as before, using the jump relations, it suffices to show that if u(X) = Sg(X), then IITulI ti LZ(aD,da)

Il7tullL20D,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 ars, 1 < r, s < m , 1 < i, j < n be constants satisfying the

ellipticity condition alb eiejifrls > clel21ii12

166

CARLOS E. KENIG

and the symmetry condition ars = asr. Consider vector valued functions

u = (u1, ,um) on Rn satisfying the divergence from system ars us = 0 in D. From variational considerations, the most TT ij TX- T natural boundary conditions are to Dirichlet condition (43D =f ) or the Neumann type conditions,

s

= n ars

av

1

ij

= f r . The interpretation of

problem (2) in this context is that we can find constants ail , 1 < i, j < 3 , 1 < r, s < 3, which satisfy the ellipticity condition and the symmetry condition, and such that p0u + (A+p) V div u = 0 in D if and only if

aXi a.. dXsj

=

0 in D, and with Tu = T u . In order to obtain the

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, NeCas identities). Suppose ars a us = 0 in D, ars = a, h is a constant vector in Rn , that

axi tj axj

>

and u and its derivatives are suitably small at oo. Then,

f h ene ij axi aj

rh our naarsej

aD

aD

ars our aus dv = 2

1 axi

aus da TX-j

Proof. Apply the divergence theorem to the formula

3

a [(h,ars - h rs -hj ars) ()ur ij ej if axi 1

am = 0 auaus

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.

167

ELLIPTIC BOUNDARY VALUE PROBLEMS

REMARK 2. Note that if we had the stronger ellipticity assumption that

ars i Sj >_ C tj JJv

12

< M1, that

Il

we would have, if X = (x,

Rn-1

- R,

e,t IiV uII

t

O0

L2(aD,da) ti Ildvll L2(aD,do)

In fact, if we take

h = en, then we would have

udo fhnEajjrs TXi 3T

J Ipurl2do < C aD

aD

I/2

fh,

2C

n E

s do, < 2C

ars

J pur12 da

i

Lj

aD

Thus,

030

fa D IVur12du < C I3 D

('

J

I/2 da

D (9D

2 do.

j

For the opposite inequality, observe that, for each r,s,j fixed, the vector hinLaes - hLnfars is perpendicular to N. Because of Lemma 3.1.5, J

I hen Pars

j

X do = 2 f(hneahinLar) W Nj do. aD

a`D

Hence, faD DuI2do< C(j

jptu12da)1/2('3D IVul2da)I/2 and so

Jii I2da
aD

ft

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

systems of elastostatics,

168

CARLOS E. KENIG

ars au

s

au r

JX-i ' Fxj

i ,

j

(div u)2 +

z

i au-

2 ,

i.i

which clearly does not satisfy aid e, es > C

ax;

t

i

dx.

f

i2, since the

quadratic form involves only the symmetric part of the matrix ( ;r). In

this case, of course

=

Tu = A (div UN + u1Vu+pa IN.

REMARK 4. The inequality 11vu 11

2

L (aD,da)

< C11ptu11

L2

(3D,da)

holds

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

our aus fft henears ,, ax; axj da =

algebraic argument. In fact, as in Remark 2,

2 faD (h n a ij - h. n ars) aue , aus da and for fixed r, s, j , e e a e, ax; axj (heeij n ars - h' n ee, ars) is a tangential vector. Thus, f

aD

C (j

2

IVtuI do)

I 12

-2 12 (f3D IVu1 do)'

h n ars our aus da <

N

Consider now the matrix drs

=

(ark n;nj)-1 . This is a strictly positive matrix, since ark e; ej77 t77 s > C 1e121'i12

ars our aus ij JX; aXj

Moreover, d rs UV ta" aut d rs nk art kgaXQ

nmameN, ave7v a

Q

r

(a'A - ars `us = d rs n. art aut , n sm i, Ri Ni ``av i, Ni ek 3Xk

0,U m

1

our - atr a°t our nmast my aXv vec)Xv aXe

= 1drsnkarkv n ma

that for t,r, f fixed, Idrsnkak1mast me

ml - avf

atT

ve

v

= d rs nk art `out kv3Xv

X Now, note

is perpendicular to N, by

our definition of drs, and the symmetry of ail = drsnkak8mamenv - avenv =

art kvaknvdrs ast me nm

n= atr atr n mem vlv nkdrs asr mem

amenm-amenm = 0. Therefore, JaD

--

atr n--Sis ast n -atrmenm mem mem

henedrsCdV (av da <

169

ELLIPTIC BOUNDARY VALUE PROBLEMS

C( faD IVtuI2da)1/2(faD IVul2da)1/2. Now, rs rs (3us niai] -- akinknjni

arsn

N-S

il r - akjnkn] aN rs

rs aus = niai]rs c3us akinkni ni aus ox ()X] -

=;niaij

A

I aus __ (mars - arsn ik k i]

ki knin J aX

I

I

J

au. 3Xi

But, for

-

i, r,s fixed, ars i]

arsn n is perpendicular to N, and so f h n d artn n ! aD e e rs kjk] aN

ikkl

a Qnineaur

Iptul2da)1/2(faD loul2da)1/2+faD

da
IVtul2dol.

We now choose h = en , so that hene > C , and recall that (drs) and

(artnkn) are strictly positive definite matrices. We then see that

I

J

Iotul2da

(ID

aD

-. 2

,12

Now, as IVul = Iptul

+ J jVtul2da

OD

.

OD

+ aN, 2 , the remark follows. 2

REMARK 5. In order to show that IaD Ivtul do < C fan

TuJ2da, it

suffices to show that faD Ipul2da< C faD IA(div u)I + pipu+Qut}I2da. In fact, if this inequality holds, we would clearly have that f3D Ipul2da<

C faD Ipu+putl2 da (Korn type inequality at the boundary). The RellichPayne-Weinberger-Necas identity is, in this case (with h = en ), faD nn I 2 Ipu+vutl2 +A(div But then,

431)

u)2da=2

Ipu12da< C(faD

1Vu1

faD

IX(div

da)1/2(faD Ik(div u)N+p{pu+putlNl2do)1/2

The rest of part (a) is devoted to sketching the proof of the above inequality.

THEOREM 3.1.6. Let u solve pAu + (A+µ)V div u = 0 in D, u = S(g), where g is nice. Then, there exists a constant C, which depends only on the Lipschitz constant of 95 so that

170

CARLOS E. KENIG

J IVu-12 da
aD

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

LEMMA 3.1.7. Let u be as' in Theorem 3.1.6. Then,

fN(v;)2 d a
aD

LEMMA 3.1.8. Let u be as in Theorem 3.1.6. Then,

fN(x(div u)I+µ)pu+Vut})2da
fijVu)I+µIpu+putIj da. aD

CID

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 ([91 ). Lemma 3.1.8 is proved by observing that if v

is any row of the matrix A(div u)I+l4pu+putj, then v is a solution of the Stokes system

= pp in D div v = 0 in D

Av (S)

vl3D =

f f L2(dD,da)

This is checked directly by using the system of equations pAu + (A+µ)p div u = 0. One then invokes 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 f L2(0,D, do), there exists a unique solution (v, p) to system (S) with p tending to 0 at oc, and N(v) e L2(0,D,da). Moreover {{N(v){{L2((?D,da)

I1

< CIIf L2(aD,do)

ELLIPTIC BOUNDARY VALUE PROBLEMS

171

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 3.1.10. Let F : RxRn -> R be a function of two variables t c R, x = (x1,. ,xn) E Rn. Assume that for each x , the function t H F(t,x) is Lipschitz, with Lipschitz constant less than or equal to M, and for each i , 1 < i < n , the function xi H F(t,x) is Lipschitz, with Lipschitz constant less than or equal to Mi, for any choice of the other variables. Given an interval I x J = I x J1 x x Jn , where the Ji'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) > F(t,x) on I x j . (b) If E = I(t,x) E I x J : F(t,x) = G(t,x)I, then IEI > 8 III IJ I is (c) For each i, the function Lipschitz, with Lipschitz constant less than or equal to Mi, and one of the following statements is true:

Either for each x , -M < 29 (t,x) < 45 49t for each x

- 45 <

(t,x) < M

or ,

.

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 u(X) solves the systems of elastostatics, so does u(X) + a + BX , where a is a constant vector, while B is any antisymmetric 3x3 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 pu(X*)-pu(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 Do C Rn be a fixed, C°° domain with {(x, 0) : IIIxIII = max Ixil < 1} C aDo, I(x,y):0 < y < 1 ,

IIIxIII<1ICDoCI(x,y):0R is Lipschitz, with IIIVoIII < M, we construct the mapping TO: Rn Rn by TT(x,y) _ (x,cy+rly*O(x)) where rJ E Co(Rn-I) is radial, f rl = 1 , and C = C(M) is chosen so that TOR+) C ((x,y) : y > o(x)I, and so that To

is a bi-Lipschitzian mapping. Also, it is clear that To is smooth for (x,y) with y > 0, and T0(x,0) _ (x, O(x)). We will denote by Ao the point T00,1). Lemma 3.1.7 is an easy consequence of LEMMA 3.1.11. Given M > 0 and 0 with IIIVOIII < M, there exists a constant C = C(M) such that for all functions u in Do, which are Lipschitz in Do, which satisfy pAu +(A+iz)V div u = 0 in Do and pu(Ao) = pu(Ao)t , we have IINo(V)IIL2(aD,do) < CIINo(A(div U _)I + y1vu+DutIIIL20D,da). Here No is the non-tangential maximal operator corresponding to the domain Do. 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 0 is such that IIIVOIJI < M, and there exists a constant vector a with III a III < M so that III7O -a III < e, then, for all

Lipschitz functions u on Do with pDu + (a+µ)V div u = 0 in Do, with Vu(Ao) = pu `(Ao) we have IINO(ou)IIL2(aD,da)

<

CIINo(a(div u)I+µ(Du+putIIIL2(dD,da) ,

where C = C (M, e) .

holds, then the corresponding estimates Note that if Proposition automatically hold for all translates, rotates or dilates of the domains Do, when 95 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 Do. The bottom Bo of aDo will be T95(dDOU (x,0): x ERn-I).

173

ELLIPTIC BOUNDARY VALUE PROBLEMS

PROPOSITION 3.1.12. Given M > 0, there exists a = 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 IINA(Vu)IIL2(aD,do)
holds. This is because in this case, the domain DS6 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

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. example). If

a

PROPOSITION 3.1.13. For all M > 0, e > 0, a e (0,0.1), if Proposition

(M,e) holds, then Proposition ((1-a)M,1.1e) 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,a) holds for any M,e. Fix M,a, and choose R so large that if e(10M) is as in Proposition R

3.1.12, then (1.1)Re(10M) > E. Pick now aj > 0 so that

II (1-aj)=1/10

j=1

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

suffices to show that IINO(Du)IIL2(3D,da)


where N,6 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 0 with IIIVO-a III < 1-1e, 111-a 1115 (1-a)M. We

will choose No as follows: Since a1) 0\Bo is smooth, it is easy to see that we can find a finite number of coordinate charts (i.e. rotates, translates and dilates of Do ), which are entirely contained in Do , such that their bottoms By are contained in c3Do, such that Tq((x,0): I IIx III < 1/2) cover dD(k, and such that the pi's involved satisfy IIIVIII < (1 - 2) M, and there exist u such that III ao III 1.11E

.

The non-tangential region defining N0l, on TV

is defined as follows: let F C

(x,0) : 11141 < 2

be a closed set. Con 2) sider the cone on R+, y = I(x,y) E R+: b;xI < y}, where b is a small constant. Consider now the domain DF on R+, given by DF = U ((x,0)+y). Then DF is the domain above the graph of a x,0): IIIxIII

x(F

Lipschitz function 0, for which IIIVOIII < 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) that T,(DF) is the and domain above the graph of a Lipschitz function 0, with 1.111E. The non which satisfies IIIVq JII < (1 - 10) M, IIIVO tangential region defining No, for Q e To ((x,0):

IIIxIII < 2

is then

the image under TV of (x,0) + y, with b chosen as above, suitably truncated, and where Q = TV,((x,0)). Let now, to lighten notation,

m=N,k(pu), m =NO(A(div)I+givu+V VI). For t > 0, consider the open-set Et = Im > ti. 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 TO ((x,0): IIIxIII < 2)

for a coordinate chart Do , each Uj contains T0(I.), where Ij is a cube in

IIIxIII < 2

and is contained in TV,(Ij), where Ij is a fixed

multiple of Ij . Finally, we can also assume that there exists a constant 770 such that if diam (Uj) < no, there exists a point Qj in 3D0, with dist (Qj,Uj) ti diam Up such that m(Qj) < t. Let now /3 > 1 be given. We claim that there exists 5 > 0 so small that if Ej = Uj fl Im > Pt, Fn < 80

175

ELLIPTIC BOUNDARY VALUE PROBLEMS

then a(E3) < (1-rJM)a(Uj) where 77M > 0. Assume the claim for the time being. Then

f

00

f m2dc=2 J

°° ta(E/3tlt

ta(Et)dt = 2162

0

dD0

r

2/32

0

00


=

f ta(Ui n E/3t)dt < 0

00

+2/32

0

r

00

ta(m>St)dt <

2/32(1-,1M) r ta(U )dt +

0

0

00

2

tof m>tIdt = (32

(1-77M) f m2da+

5 m2da. Thus, if

S

0

3D0

aDo

we choose /3 > 1 , but so that /32 . (1- riM) < 1 , the desired result follows. It remains to establish the claim. We argue by contradiction. Suppose not, then a(EJ) > (1- r)M) a(Uj). Let E) = T,I(E)). If 71M is chosen sufficiently small, we can guarantee that IEi -9911-1. Let now Fi _ Ei n Ij , and construct now the Lipschitz function Vi corresponding to it, as in the definition of NS6 . Thus,

>

,

III < (i_)vi

IIo - aIII < 1.111E . We now apply Lemma 3.1.10 to Vi, one variable at a time, to find a Lipschitz function f, with f > 0 on Ij , such that if F = Ix (Ii : f =01, then IFS nFjI > C a(Uwith IIIVfIII < (1 - 10) M,

and such that there exists af, with

III afIII < (1

- 10} M so that

IIIVf-afIII < 5 1.111E < e . 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 Dv,, which is contained in Do ), IA(div u)I + jA Vu+ Vut I I <5 t . 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 Ul into two types. Type I

176

CARLOS E. KENIG

are those with diam Ui > 710, and type 11 those for which diam Ui < ri0.

We first deal with the Uj of type I. In this case, Df has diameter of the order of

Because of the solvability of problem 3.1.2 for balls, and our normalization, we see that on a ball B C Do, diam B 1 , 1 .

A eB, we have fB Ipul2
Joining Af to A,0 by a finite number of balls, and using interior regularity results for the system EcAu + (A+µ) V div u = 0, we see that Ipu(A f)l < Cat, for some absolute constant C. Then

C o(Uj)162t2 <

I

m2do
(Fif1Fi)


f N2(pu)do,< aDf

C J Nf v

LVu(Af) - Vu t(Af) 2

-

a Df

1)

do <-

< C a(U)52t2 + C J Nf(A(div u )I+µ(pu+putF)2da aDf

by (M,e). The last quantity is also bounded by C a(Ui )a2t2, which is a contradiction for small S. Now, assume that Ul is of type II. Note that in this case there exists Q) c 3Do, with dist(Qj,Uj) diam U), and such that IV (X)I < 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 IV (Af)I in case 1, that for all X in a neighborhood of Af and also on the top part of Df, we have that IVu(X)I < t + Cat. Since for Q cTp(F,fIFj), m(Q) > 16t, and (3 > 1 , if a is small enough, we see that we must have Nf( Q) > m(Q). Hence, Nf Vu -r2u(Af) L

2

Dut(Af)

(Q) > (R-1-Ca)t >

177

ELLIPTIC BOUNDARY VALUE PROBLEMS 1

2

)t

if S is small and Q E TIA (F f1Fj). Thus, applying (M,e) to Df,

we see that CJ((,3-1)2t2o(U.) <

J Nf pu T (FJ.(1F.) J

LVu(Af) 20u t(Af1 )

faD Nf tpu -

C

[Vu(Af) 2 Vut(Af

2da<

1)

2da < Ca(U)82t2 , a contradiction if S

f

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 L2 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). We seek a vector valued function u = (u1,u2,u3) and a scalar valued

function p satisfying = Vp in D div u = 0 in D Au

(3.2.1)

ulaD = f E L2(c3D, do) in the non-tangential sense.

THEOREM 3.2.2 (also Theorem 3.1.9). Given f r

L2(0-D, do), there

exists a unique solution (u,p) to (3.2.1), with p tending to 0 at

oo,

and N(u) e L2(c7D, da) . Moreover, u(X) = Kg(X) , with g e L2(r7D, d a) .

In order to sketch the proof of 3.2.2 we introduce the matrix r(X) of fundamental solutions (see the book of Ladyzhenskaya [251), r(X) 1 aij (T'ij(X)), where rij(X) = 8n IXI + 1 XiXj , and its corresponding pressure 8n X 3

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

X1

Our solution of (3.2.2) will

4nIXI3

be given in the form of a double layer potential, u(X) = Kg(X) _ faD (H'(Q)F(X-Q){g(Q)do(Q), where (H'(Q) r(X-Q))if = Sijge(X-Q) nj(Q) +

178

CARLOS E. KENIG

arig

(X-Q)nj(Q). We will also use the single layer potential u(X) _

Sg(X) = faD r(X-Q)g(Q)do(Q).

In the same way as one establishes 3.1.4, one has: LEMMA 3.2.3. Let Kg, Sg be defined as above, with g c L2(cD, do,).

Then, they both solve Au = Vp in D, and D-, div u = 0 in D and

D-. Also (a)

IIN(Kg)IIL2((?D,da)

(b)

(Kg)±(P)

(c)

IIN(VS0IIL2((3D,da)

=

5

(P) == +-.)

CII9IIL2(OD,da)

nl(P)g3(P)

nl(P)n)(P)

2

2

+ P.V. faD

(e)

(HSg)±(P)

'

± 2 g(P) - p.v. fdD IH (Q)F(P-Q)Ij(Q)da(Q)

+

aXl (Sg )j

(d)

III g IIL2(aD,aa)

aPi



F(P - Q) g (Q) d a (Q)

2 g(P) + P.V. faD {H(P)r(P-Q))g(Q)da(Q) ,

where (H(X)r(X-Q))ig = ni(x) aXIP (X-Q) - Sil qf(X-Q)nj(X). J

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 L20D, da) of the operator 2 I+K, where Kg(P) - P.V. faD {H'(Q)T'(P-Q)jg(Q)da(Q). As in previous cases, it is enough

to show (3.2.4)

I2

I-K*)2L (aD,da)

+I(2

I+K*) 9

L2OD,da) 1

179

ELLIPTIC BOUNDARY VALUE PROBLEMS

This is shown by using the following two integral identities.

LEMMA 3.2.5. Let h be a constant vector in Rn, and suppose that Du = Vp, div u = 0 in D, and that u,p and their derivatives are suitably small at co. Then,

f

hFnPaS

1

a

Sdo=2 J

hFaX F

1

aD

da-2

f

PnS

h

fa

da. F

3D

aD

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

fhr

J hFnFp2da=2 aD

aNpda-2

3D

+2

fhrA i

aNda+

3D

ass fhns_.da. our

3D

The proofs of 3.2.5 and 3.2.6 are simple applications of the properties of u,p, and the divergence theorem. Choosing h = e3, we see that, from 3.2.6 we obtain

COROLLARY 3.2.7. Let u,p be as in 3.2.5. Then,

fP2do
aD

where C depends only on M. A consequence of Corollary 3.2.7 and Lemma 3.2.5, is that, if av

aN

-

then we have

180

CARLOS E. KENIG

COROLLARY 3.2.8. Let u,p be as in 3.2.5. Then, IN, 2 aD

dam aD

fVt2da I fnS2da, l aD

i

where the constants of equivalence depend only on M. Proof. 3.2.5 clearly implies, by Schwartz's inequality, that f D 1Vu 12dor < 2

C f,9D ICI do. Moreover, arguing as in the second part of the Remark 2

after 3.1.5, we see that 3.2.5 shows that

f pul2da
aD

aD

I

aD

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

('

CJ

aD

1 /2

1 /2

Vu-1 2do

fins '

j2da)

(9D

3.2.8 follows now, using 3.2.7 once more.

To prove 3.2.4, let u = S(g). By d) in 3.2.3, Vtu and ns !2!!! are 1

continuous across 3D. 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

(Au = Vp in D (3.2.9)

divu=0 in D ((pu+Vut)N-p-N)1j aD = f e L2(3D,do)

.

This problem is very similar to (3.1.2). Using the techniques introduced in part (a), together with the observation that if Au = Vp, div u = 0 in

D , the same is true for each row v of the matrix [Vu V t - p-11, we have obtained

THEOREM 3.2.10. Given f c L2(3D,da) there exists a unique solution (u ,p) to (3.2.9), which tends to 0 at -o, and with N(V ) e L2(3D,da). Moreover, u(X) = S(g)(X) , with g e L2(3D, da) . DEPARTMENT OF MATHEMATICS UNIVERSITY OF CHICAGO CHICAGO, ILLINOIS 60637

REFERENCES [1] [2]

A. P. Calderon, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sc. U.S.A. 74(1977), 1324-1327. R. R. Coifman, A. McIntosh and Y. Meyer, L'integrale de Cauchy definit un operateur borne sur L2 pour les courbes lipschitziennes, Annals of Math. 116(1982), 361-387.

[3] [4]

R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. AMS 83 (1977), 569-645. B.E.J. Dahlberg, On estimates of harmonic measure, Arch. Rational Mech. and Anal. 65 (1977), 272-288.

[5]

, On the Poisson integral for Lipschitz and CI domains, Studia Math. 66(1979), 13-24.

[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]

[8]

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

CARLOS E. KENIG

G. David, Operateurs integraux singuliers sur certaines courbes du plan complex, Ann. Sci. del'Ecole Norm. Sup. 17 (1984), 157-189. [10] , personal communication, 1983. [11] E. Fabes, M. Jodeit, Jr., and N. Riviere, Potential techniques for boundary value problems on C1 domains, Acta Math. 141, (1978), [9]

165-186.

[12] E. Fabes and C. E. Kenig, On the Hardy space H1 of a C1 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. [141 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 C1 domains, University of Minnesota preprint, 1980. [161 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.

[181

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

, 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. [19]

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.

, Boundary value problems of linear elastostatics and hydrostatics on Lipschitz domains, Seminaire Goulaouic-MeyerSchwartz, 1983-84, Expose no. XXI, Ecole Polytechnique, Palaiseau, France. [231 , 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,

[22]

1979.

[251 O. 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 Au + 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, 1, 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 Cl, with a view to finding techniques which might generalize

to Cn. These are (I)

Exploit symmetry of the domain;

Use differential forms and Stokes's theorem; (III) Use functional analysis.

(II)

Discussion of f. Let A = (z EC : Jz j < 11. For n c Z, define rlnlein9 Then direct calculation shows that

n(0)

fnd0, all n

On(Ceie) =

.

0

Now if f is harmonic on a neighborhood of A, then

f has an L2

convergent Fourier expansion 00

f(reie)

=n=`w an0n(reie)

(1.2)

By linearity, (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 Ke Ying greatly simplified the task of writing this paper.

185

186

STEVEN G. KRANTZ

f(ei0)d0 .

f (0) =

(1.3)

f21T 0

Of course formula (1.3) holds in particular for holomorphic f ; then we rewrite (1.3) as 2n f (O)

Tir-i

fC(C0

f eie- 0 (iei0d®= o

d

(1.4)

Y

where y(6) = ei0, 0 < 0 < 2rr.

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 r A is fixed then the function

0(0_oz(0=i +zC

,

called a Mbbius transformation, has the following properties: (a) 0: A -. A is biholomorphic (i.e. holomorphic, one-to-one, and onto, with a holomorphic inverse). (b) 0(0) = z

(c) 0, 0-1 are smooth on a neighborhood of A. If f is holomorphic on a neighborhood of A, define g(e) = f oO(e).

Then (1.4) applied to g yields

f(z) = g(0) = 21 £g(e)de= 1

f Y

Y

Change variables by e _ 0-1(C) =

f( z) =

00(e) de

'ri f f(O 1

Y

Then

171 _(o

1d C

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

187

and logarithmic differentiation of 0-I gives

f(z) =

27i f 00

1

-z

+

z 1=z

dC

(1.5)

y

dC 27ri = 1 fCQ + -Z

fr

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

function of C which vanishes at 0. By (1.4), the second integral vanishes. Formula (1.6) now becomes

f( z) _

2ni

f

Cf(C)

-z dC

y

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 2n

f(z) = Zn

y,

f(ei9) 110_ z2 2 d le -z10

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 [321.) Thus approach I has serious limitations in CI . In Cn 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 Cn and gain some new insights.

188

STEVEN G. KRANTZ

Discussion of H. We need some notation. Recall that in real differential analysis on R2 we use the basis ax , for the tangent space (i.e. all ay

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

=<$,dy>=1, =<1,dx> =0. In complex analysis, it is convenient to define differential operators

a -1 (ax a 2

az

a

lay

a _1 (a a . , aj_2c+iay)

The motivation for this notation is twofold. First, a 3j7z

z=1,

z= -z=0

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

-fi

and

ay=-1,

(1.7)

//

= 0 means that

which is the Cauchy-Riemann equations. Thus ho lom orph ic.

We also define

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

<,dz>

=

1

,

_ < a,dz> = 0. az

f

is

189

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

An arbitrary 1-form is written

u(z) = a(z)dz + b(z)dz

(1.8)

and we define the exterior differentials au = ab dz A dz , X = . dz A dz

(1.9)

.

Clearly du = au + au. Recall STOKES' THEOREM. If ci CC Rn is a bounded domain with smooth

boundary and u is a smooth form on ci then

fu = an

fdu. sZ

In our new notation, if ci C CI ti R2 and u is a 1-form as in (1.8), (1.9), then Stokes' Theorem becomes

fu= fau+u ale

11

(1.10)

(t

as

j)dzA.

a Now we can prove

THEOREM. If SZ C C is smoothly bounded and f is holomorphic on a neighborhood of ci then

f(z)

=

2ni

f.!2dC, all zci -z aci

.

190

STEVEN G. KRANTZ

Proof. Fix z e it. Let E < distance (z, aft). Define D(z,E) = 1C E C : IC-zI < El and it = it\D(z,E). We apply Stokes' Theorem to the 1-form u(C) _

f(t -z

dC on the domain it (note that u has smooth coefficients on a

ti

neighborhood of the closure of it, but not on all of it ). Thus, by (1.10),

fu(i= fdu=_j(2)dAd. ti

ti

ti ti

This last is 0 by (1.7). Thus, since aft = dl2 U 3D(z,e) (with suitable orientations), we have

fu() = aft

J

u(O

3D(z,E)

Parametrizing 3D(z,E) by C = z + Eeie, 0 < 0 < 21r, we obtain

f-!d= J

21r

('

f(z+Eei0)id9 2aif(z)

o

as E -. 0+. This completes the proof. 0 REMARK 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

dC A dC. Thus the proof of the

vanish; instead it equals theorem yields

f(z)

2ai

ff

C -z

2ni

J

f(/a -z

A dJ.

This formula, valid for all f c C'(KI), will be valuable later on.

(1.12)

191

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

REMARK 2. As it stands, the method of Stokes' theorem will not generalis holomorphic with an ize to Cn. Namely, we used the fact that isolated singular point at z . In Cn , n > 2 , holomorphic functions never have isolated singularities. A more sophisticated approach will therefore be needed.

Discussion of III. If Q C C is a smoothly bounded domain and if E > 0, define S1E _ {z a iZ : dist(z, 00) > E 1. If E is sufficiently small, say

0 < E < E0, then 0E will also be smoothly bounded. Define

H2(fl) = f holomorphic on 0:

f If(C)I2ds(C) < 4

sup O<E<E

0

.

do E

(Here ds is the element of arc length.) That this definition is unambiguous is a technical matter (see (31, Ch. 81). We shall need PROPOSITION ([311). Each f e H2(fl) has associated to it a unique f e L2((3Q). The Poisson integral of f is f. Also

sup 0

f

If(()I2da( )I/2

I

II f

L2

(f)

E

If ITE: 09 E

i3f is normal projection then (f I

au

E)

=

(17 E)-I' f in L2 (do)

ffda an

for f,g a H2(SZ) .

(1.13)

192

STEVEN G. KRANTZ

BASIC LEMMA. if K C Sl is compact then there is a constant C = C(K)

such that

all f E H2(0) .

sup If(z)I < C1101 H 2s

(1.14)

zEK

Proof. Fix z E K. Let Et < U (distance (K, 311) ). Then

1f(z)1 °

1

z dC

2m

I

3D(z,el)

I

(Stokes) 1

dC

2771

C -z a1Z

el

< --L-

ffIds

- 2nE1

aloe 1

(Schwartz) 1/+

< C(E1)

J

If(y S)l2ds(OI/2

< C(K) IIf1IH2 .

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

z c fl and define the functional Oz :H2(11), C

fF-f(z).

193

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

Then the lemma, with K = {z#, shows that ¢z is continuous. The Riesz Representation Theorem yields a unique kz f H2(il) such that

95z(f)=, all f eH2. In other words,

f(z)

ff())dsQ) ,

all f fH2(Sl), z eSl.

(1.15)

ail

Formula (1.12) is called the Szego formula and kz(c) = S(z, C) the Szego kernel (see [31, Ch. 11 for further details). REMARK 1. Formula (1.15) has the advantage of working on any smoothly

bounded domain (even in Cn ), and the disadvantage of being non-

explicit. As an exercise, check that when Sl = A C C and z = 0 then S(z,0 _= 21r. Then use Mobius transformations to calculate

for

any z f A. You will rediscover the Cauchy formula! §2. The Cauchy -Fanta ppie Formula

Now we begin to consider integral formulas in Cn. For purposes of differential analysis we introduce the notation

1 !a

a

=2

'

aa

1

2

(1)"ij

dzj = dxj +idyj , dzj = dxj - idyj,

j

It is easily checked that

T =1,

a_ zj =

<-l, dzj> = <-, > = 1 ,

j

and all other pairings are 0.

j = 1,...,n ,

194

STEVEN G. KRANTZ

If a =

are tuples of non-negative

integers (mufti-indices) then we write

dza = dza, A ... A dzak ,

di- 13

= dz16A ... A d Q

1

A differential form is written

u=Iaa,6 dzaACZ /3

(2.1)

a,f3

with smooth coefficients a

a/.

(If 0 < p,q e Z and the sum in (2.1)

ranges over jai = p, 1131 = q only, then u is called a form of type (p,q) .) We then define

au =

t dzj n dza F. dz

ru =

a# dzj A dza Adz c3i

a calculation (or functoriality), du = au + au. A C1 function f is called holomorphic if Ju = 0. (Note: this means that df = 0, (9zj

so f is holomorphic in the one variable sense in each

j

variable separately.) Finally, we introduce two special forms: if w = is an n-tuple of smooth functions then we define the Leray form to be 77(w) = I (-1)l+lwj A dw1 A

A dwj-1 A dwj+1 A - A dwn .

j=1

Likewise m(w) =dw1 A... Adwn.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

195

We define a constant

W(n)= J

W(C)AW(C)

B(0,1)

Here B(z,r) = IC a Cn : IC -z I < r 1.

Now we may formulate a generalization of approach II in Section 1.

THEOREM (The Cauchy-Fantappi(; formula). Let Sl C Cn be a smoothly

bounded domain. Assume that w = wj

cC°°(SZ x SZ\A),

and n

I wj(z,4)'(Cj-zj)-1 on iZxSZ\A.

(2.2)

jj=1

If f e Cl(1) is holomorphic on 0, then for any z e 11 we have

f(z)

_W (n)

f

1(i) i w) A

W (C) .

do

Before proving this result, we make some detailed remarks. REMARK 1.

In case n = 1, then w = wl =

1

4-z 1

rl(w)_---,

nW(n)

C-z

The Cauchy-Fantappie formula becomes

f(z)

m

f do

which is just Cauchy's formula.

f(C) dC, z

(of necessity). So

_2ni 1

(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

1-z1

(wi(z,0,...,wn(z,0)

w(z,0 _

.,

_

Cn-n

.

I.

Let us calculate what the theorem says for this w in case n = 2. Now r!(w) A c(C) _ (w 1dw2-w2dw 1) A dC1 Ad C2

-

2dc1 + w I

w1

aa1

- w2

1

dC2

aC2

w2

a 1

1A

1

dC, A dC2

(X2

which by direct calculation

- 2z IC -z

q dc-1 + IC _z q

A dC1 A dC2

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

f(z) = z(2)

4=z2)

5 f(O

IC-zI

A dC1 Ad

2

for f c C 1(Sl), holomorphic on Q.

Now we turn to the proof of the Cauchy -Fa ntappie formula. For sim-

plicity, we restrict attention to n = 2 . Let 2

a=(a1,a2) cC00(SlxSZ\O): 1 aj(z,c) (cj-zj) =1 j=1

.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

197

If al,a2 E `.f then we define

B(a',a2) =det(a',dal)

_I

E(a)ao(I) A

aES2

We claim that B has three key properties:

(1) B(a', al) does not depend on aI ; (2) c3CB(a1, a2) = O ; (3)

If a I, al, RI, R2 E f then B(a1, al) - B((31, R2) 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 wI

aI =a2

c EJ

=

(C2-z2)/K zI2

we have (observe that B(at,a2) = q(w) )

J f(())7(w)AW(C)=

ff()B(ata2)

A w(C)

do

ag

Note that, by (2),

dC(f(() B(at , a2) A W(C)) = a.f(C) A B(a1, a2)

=0+0=0.

A Ev(C) + f(C) dCB(a1, a2) A W(C)

(2.5)

198

STEVEN G. KRANTZ

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

=

f(C)B(a1,a2) A 0)(C)

J

aB(z,E)

f f(C)IB(al,a2)-B(j61, 62)1 Atd(C)

(2.6)

a6(z,E)

+

B(j6 1,62)AW(C).

J aB(z,E)

But the first integral

J

=

f(()dA A W(C)

aB(z,e)

(for some A, by (3))

= f d(f(() A A A W(C)) 3B(z,e)

(2.7)

=0,

by Stokes' Theorem, since a9B(z,E) = 0. Thus (2.5)-(2.7) give

f Al which, by (2.4),

f f(C)B(j61,,62) A W(C) 3B(z,E)

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

=

A W(C)

J

I C-z l4

3B(z,E)

=

1 E4

f (C) rl(- z) A W(C) .

J 3B(z,E)

=13B(z,E) J f (z) rK -) A 0)(t) + ((E) 4

(Stokes)

4f(z)

= 64

2((-z)Aw(C)+O(E)

J

B(z,E)

= 4f(z)E4

2(a(J)Au(C)+0(E)

J B(0,1)

= f(z) 2W(2) +

O(E)

Letting E - 0 yields the desired result. 0

We conclude this section by proving (1)-(3). For (1), we have

B(a1, a2) = det

1

(C1-z1)(C2-z2)

det

199

STEVEN G. KRANTZ

200

which, by adding row 2 to row 1,

det

1

(C1-zl)(C2-z2)

aC ((C2-z2)a2)

((C22)a2

_

(Cl

z1) ? a2

This calculation is correct for C1

z2. The full result follows

z 1 , C2

by continuity.

For (2), imitate the proof of (1). For (3), use (1) to write B(a1, a2)

- B((31, p2) = B(a1, a2-P2)

ai (ai-bi) = det a21 'j (a 2-b2

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

A = det a2

a2

2

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.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

201

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 = ,SI f j(z) dz j , we

seek a function u such that au = f . Notice that since 0 = d2 = a2 + ;72, linear independence considerations yield that '2 = 0. + Hence au = f necessitates of = 0. A simple calculation shows that, for n > 2 , this compatibility condiaft

n=1

af k

all j, k. Notice, however, that when

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

tion is equivalent to

=

,

ence can be explained in part by the fact that the equation au = f is

really n equations

f

one unknown (namely u ). For n > 1

1 the system is then "over-determined" and a compatibility condition is necessary. For n = 1 the system is not over-determined. The three basic considerations about a PDE are existence, uniqueness 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 3 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 of = 0 does not by itself guarantee existence of u. EXAMPLE. Let S1 C C2 be given by 11 = (B(0,4)\B(0,2)) U B ((2,0), 2/

202

STEVEN G. KRANTZ

Let U = B ((1,0), 2) and V = B ((1,0), 4) as shown. Let 0 e C° (U) satisfy

--1 on V. Finally, let

Then f is smooth and j 'closed on Q since

zl 1

is well-defined and

holomorphic on supp (drl) fl 11. If there existed a u satisfying Al = f

then the function h =_ u

zt-1

would be holomorphic (Jh=0) on

cl\IB (1,0),41 n 1zt=11 . 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 B(0,4)\B(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).

C1

Now that we know that du = f is not always solvable, let us turn to an example where it is useful to be able to solve the a equation.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

203

EXAMPLE. Consider the following question for an open domain it CC Cn :

(3.1)

If w = fl fl 1zn = OI' 0 and if g is holomorphic on co (in an obvious sense), can we find G holomorphic on ft such that GIC0 = g ?

1 is the unit ball, then the trivial extension g(zi,,zn_1,0) will suffice. However if 0 = B(0,2)\B(0,1) C C2 then g(z1,0) = 1/z, is holomorphic on a) but could not have an extension G (else the Hartogs extension phenomenon would be contradicted). o If

THEOREM. Suppose that w C Cn is a connected open set such that whenever f is a smooth a-closed (0.1) form on fZ then the equation au = f has a smooth solution. Then the answer to (3.1) is "yes."

Proof. Let u: Cn - Cn be given by (z1,..,zn-1,0). Let B = ;z f l : nz / col. Then B, w are disjoint relatively closed subsets of fZ, so there is a C°° function 4 on fZ such that 0 = 1 on a relative neighborhood of co and = 0 on a relative neighborhood of B. ti Define F on fZ by ti

c(z) f(n(z))

if

0

else.

z e supp q

F(z) _

ti Then F gives a C°° (but certainly not holomorphic) extension of f to Q.

204

STEVEN G. KRANTZ

We now seek a v such that F + v is holomorphic and F + vIw = f. With this in mind, we take v of the form zn u and we want d(F+v) = 0 or

=0. Now f o n is holomorphic on supp (b and zn is holomorphic so all that remains is

(f°n) - zn' 3u=0 or

n) zn

(3.2)

The critical fact is that, by construction, 4 = 0 in a neighborhood of n f1

{zn=0# so the right-hand side of (3.2) is smooth on Q. Also it is

easily checked to be 3 closed. Thus our hypothesis is satisfied and a ti u satisfying (3.2) exists. Therefore F - F +v has all the desired properties.

Our two examples show that solving the 3 equation is (i) subtle and (ii) useful. Thus we have ample motivation to prove our next result. LEMMA. Let (he CS(C) , k > 1 , and define f = (z)dz. Then u(z)

=

2i

dC A dC

J J ; -z C

satisfies u e Ck(C) and 3u = f . Proof. We have

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

0 "U

az

205

a (-L ff 2ni

)g-

C

2ni

f(a/)(+z) do A dC

f C

1

2ni

II

w d6 (o d A dC

D (0, R)

where D(0,R) is a disc which contains supp 0. We apply Remark 1 of II in Section 1 to 0 on D(0,R) to obtain that the last line (C)dC 2ni

J

z

aD(0,R)

The integral vanishes since 0 = 0 on (91)(0,R), hence au = f . Observe finally that u e Ck by differentiation under the integral sign. o REMARK. In general, the u given by the lemma will not be compactly supported. Indeed if f f 0 0 and if u were compactly supported (say u C D(0,R) ) then a contradiction arises as follows:

0=

J aD(0,R)

udC=

J D(O,R)

The supports of solutions to the 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.

STEVEN G. KRANTZ

206

THEOREM. Let n > 1 and let D(z) _ q5 1dz 1 +-.. + (kndz 1 be Xclosed on Cn and suppose each (kj CC(Cn). Then for any 1 < j < n the of

function 1 uj(z) _ - 217i

ff

dC A dC

(kl(z

C_zj

C

satisfies uj a Ck(Cn) and aut = 40. Moreover

uj = of for all j,

Proof. Fix 1 < m < n. We need to check that auj fim m = j then the result follows from the lemma. d0j

compatibility condition

=

Sam

-m azj

= (kj

uj(z)

rn

<

m < n. If

If m ' j then use the

to write

j (zl'...,zj-iX'zj+l,...,zn) '3z_

M-

1

C-zj

dC A dC

C

4m 2ni

ff

dCAdC.

C

By the lemma, this last equals
component of c(U supp (kj). It also shows that there is at most one j=1

207

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

compactly supported solution to au = f . In the present case, one exists and is given by an integral formula. §4. The Hartogs 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 fZ C Cn, n > 1 ,

be a bounded, connected open set. Let K C 9 be compact. Assume that Q\K is connected. If f is holomorphic on Q\K then there is a holo-

morphic F on 9 such that F Ia\K = f . Proof. Choose (k F C°°(S1) such that (k = 1 in a neighborhood of an

and (k = 0 in a neighborhood of K. Define ti F(z) =

O(z) f(z)

if

z FQ\K

0

if

zcK.

ti Then F is a C°° extension of f to 9, but it is not in general holomorti phic. We now seek v such that F + v is holomorphic on Q (and ti F +vjQ\K = f ). Thus we seek v satisfying a(F +v) = 0 or

49 ((k f)+av=0 or

0'v = (-) f since f is holomorphic on supp 0. Now (k - I

(4.1)

near M so (-

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 aa. Thus F + v is

f

208

STEVEN G. KRANTZ

holomorphic and, near ail, F?+v = V _

f = f. By analytic continuation, F + v = f on U and the result follows. o 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 Cn , 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 a small hence, by the Hartogs phenomenon, on B(P,2E), and hence at P. That is a contradiction (ii) A holomorphic function f in Cn, 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 C Cn is open, E C 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 f j(il\E)ne ranging over all two dimensional complex affine spaces QC Cn.

Now we return to discussion of the d operator. There are essentially four aspects to this matter: (1) Existence of solutions; (2) Support of d data and d solutions; (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 av = 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 0 E C° (B,(P,r)) satisfy B (P, 2) . Let

E Cc 1B

u =95-v and f = d(¢ v) =

= 1 on

2satisfy c = 1 on B (P, 4/ . Define v/+c

g.

When Ju = f.

``Apply

the theorem

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

209

of Section 3, decomposing f as f = (V J0 v) + (1 i)( 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 au = f then also a(u+h) = f for any holomorphic h. Given f , one cannot expect all u + h to be nice (i.e. bounded, or L2 , 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 au = f which

is orthogonal to the kernel of a, 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 Cn. (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.)

is a bounded pseudoconvex domain in Cn and f = ifi dzj is a a-closed form with all fj e L2(f), then there is a u e L2(f) with See [20]. (Exercise: the U = f . Also lull 2 < C(f) i 2

(a) If

SZ

dA

L

j

1L

canonical solution also satisfies this estimate.) (b) If Q C Cn is smoothly bounded and pseudoconvex and f = ifi daj is a a-closed (0,1) form with all fj a C°°(1) then there is a u e C°°(Q) satisfying au = f. See [26]. It is not known whether the canonical solution has this property. (c) If Q C Cn is strongly pseudoconvex with C2 boundary and if f = ifjda] is a a-closed (0,1) form with bounded coefficients, then < C11lf]1IL°.. The there is a u satisfying au = f and lull L°°

210

STEVEN G. KRANTZ

canonical solution has this property. See [11], [17], [22], [16], [35]. Sibony [39] has shown that there are smooth pseudoconvex domains on which uniform estimates for d 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 -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 d satisfies a subelliptic estimate. Many times an estimate tells us how to choose the right solution to du = f. We conclude this section with an example of how estimates can be useful.

DEFINITION. Let SZ C Cn be a domain and P r 30. A holomorphic function f :

St

C is called singular at P if for every e > 0, f Inns(P.F)

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 of P and a holomorphic function on u fl a 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 Sl C Cn be a domain on which the d operator satisfies uniform estimates. If P E 30 and there exists a local singular function at P which is bounded off any B(P,e) then there exists a global one.

211

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

Outline of proof. Let g : U no - C be a local singular function at P. Let V be an open neighborhood of P such that V C U. Let k eC°°(U) satisfy q = 1 near P and ¢i e 0 off V . Set f = ¢i g + u and solve a a problem to find a bounded u. Then f is a global singular function at P

Fix a strongly pseudoconvex domain Q. We shall prove later that (i) uniform estimates for the a operator hold on ct and (ii) local singular functions satisfying the hypotheses of the lemma exist for each P e Al. By taking a suitable root of the local singular function and applying the lemma, we may construct for each P e act a singular function Fp at P which is in L2(1Z). Now we will prove that there is an L2 holomorphic function F on ct that cannot be holomorphically continued past any boundary point. This shows that ct is a domain of holomorphy and essentially solves the Levi problem (see [311). For the construction of F , let IPili° I be a countable dense set in act. Let Hij be the L2 holomorphic functions on ct U B (p11

j = 1,2,... . Let A2(1l) be the L2 holomorphic functions on ct. Consider the restriction map yij : Hij A2(cl). Define Xij = image yij C A2(ct). Because FP exists for each i , Xij A2(cl) for all i, j . We claim that i U X A2(9). Assume the claim for now. Take F e A2(ct)\ U X...

'

i.i This is the F we seek. The claim now follows from-

i.i Il

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 to R. Huff for this proposition.) o

212

STEVEN G. KRANTZ

§5. Convexity and pseudoconvexity

Let Q C RN be an open set. Then fZ is called geometrically convex if whenever P, Q E Q and 0 < t < 1 then (1-t) P + tQ ( fZ. In calculus, however, a C2 function y = f(x) is called convex if f"> 0. How are these ideas related? If 11 has smooth boundary, then we may think of f1 as given by

0 =}x(RN:p(x)<01 for a smooth function p with V/p 0 on d11 (Exercise: use the implicit function theorem). The function p is called a defining function for f1.

If P f aQ, let Tp(au) = (al,...,aN) E RN

:fax

(p)

0} )1)

J

Then Tp((At) is the (real) tangent space to we say that f1 is convex at P E all if

I n

j, k=1

We say that

f1

c3

at P. If

SZ = }p< 0}

a2

dx dx J

(P) a] ak > 0

Aa E TP(dul) .

(5.1)

J

is strongly convex at P if strict inequality obtains in

(5.1) when 0 a c TP(dfZ). A domain is convex (strongly convex) if each boundary point is. EXERCISES (see [31 ]):

For a smoothly bounded domain, geometric convexity is equivalent to convexity. (ii) If 1 is smoothly bounded and convex, then iZ can be written as an increasing union of strongly convex domains. (iii) The above definitions are independent of the choice of p. (i)

In order to understand the role of convexity in complex analysis, we need to discuss inner products. If z, w (Cn , we define the Hermitian inner product

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

213

n

H = I zjwj j=1

and if we identify

z = (zl,...,zn) =

(x1+iy1,...,xn+iyn) ti (x1,y1,...,xn,yn)

(t 1,...,t2n)

and likewise w = (wl,...,wn) ti (S,,---,s 2n)

then we define the real inner product 2n

Re = I tjsj

.

j=1

Notice the following facts:

(1) Re =Re(H) ; (2) If Sl = (z a Cn : p(z) < 01 has smooth boundary, P e an, then

Tp(a1l) = Ia ( Cn : Re =

01 .

(3) With Tl, P as in (2), let 5' p(9) = {a a Cn : H = 01. We

call `.11(atZ) the complex tangent space to c%) at P. If a e J'p(9) then is e `.'p(atZ). Also .`I p(9S2) C Tp( ) and it is the largest subspace of Tp(atl) which is closed under multiplication by i . Now if Il = #z a Cn : p(z) < 01 is smooth and convex and P e at1 then

lies on one side of Tp(ai2). Thus if we define fp(z) = H then the zero set Z(fp) of fp lies in P + Tp(af2). In particular Z(fp) n S c ac (and if is strongly convex then Z(fp) n aft = 4PI). I

Tl

Thus 1/fp is singular at P. Also Re fp < 0 on t2 so we may choose 0 < N e Z such that 1/(f p)1 IN is holomorphic on 0 and in L2(1l).

214

STEVEN G. KRANTZ

Thus each P e afl has an L2 singular function. By the argument at the end of Section 4, fl supports an L2 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 0: A C given by '(z) = (z+3)3. What is needed is a new notion called pseudoconvexity: DEFINITION. If fl =;z a Cn : p(z) < 01 is smoothly bounded we say that fl is (Levi) pseudoconvex at P e afl if

!mar

i, k=l

a2p aZ.aZk

0

Vw e 3'p(on) .

(5.2)

7

We call fl strongly pseudoconvex at P if strict inequality holds in (5.2)

for all 0 / w e fp(afl). The domain is pseudoconvex (strongly pseudoconvex) if each boundary point is. EXERCISE. If P e afl is strongly pseudoconvex prove that if A > 0 is sufficiently large and p(z) _ (eAp(z)-1)/A then

I aZ az k i,k=l

(P)w]`k > ClwI2, VP E afl,Vw E p(afl) .

(5.3)

1

See [311 for details.

The rather technical notion of pseudoconvexity is vindicated by the following deep theorem (see [311):

THEOREM. If U C Cn is smoothly bounded then the following are equivalent: (i) fl is pseudoconvex (ii) fl is a domain of holomorphy (iii) the equation au = f, f a 3 closed (p,q) form, is always solvable.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

215

This theorem 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 iZ C Cn is smoothly bounded and P e all is a point of convexity then P is a point of pseudoconvexity.

Proof. Let p be a defining function for Q. Let a e J-p((3il). Writing the definition of convexity in complex notation we have n

d2

(P)aj ak 1,k=1

j

k

l

j,k=1 n

+12

,

2

ap

k=1 1

dzjdzk

(P)aa jk >0

But a similar inequality also obtains for is e fp(dil). Adding the two inequalities yields the result. o §6. Solutions for the d problem

We briefly describe the Hilbert space setup for Hormander's L2 theory of the d problem. We fix a smoothly bounded ft C Cn and introduce the notation

L2(fl)

o o =T L20

o

1)(1)

L(o

1)(ft)

11

H1

H2

H3

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

216

STEVEN G. KRANTZ

(since C' is dense in L2 ). It is easy to check that T, S are closed; also S o T= 0 so if F= ker S then Range T C F. An existence theorem for the a equation amounts to proving that Range T = F . Moreover, it is an exercise in functional analysis to check that this is equivalent to proving an inequality of the form 1IYIIH2 < CIIT*YIIH

,

dY c5) * n F . T

(6.1)

1

See [201 for details. Rather than prove (6.1), it is more convenient to study the symmetric inequality {IYIIH2
Notice that when y E F

dy c T* n1DS.

(6.2)

(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 [251.) Hormander's idea [201 was to work not in Euclidean L2 but rather in

L2 of the measure space a-Odx. If 0 is chosen to have certain convexity properties and to blow up rapidly at ail, then the boundary is effectively suppressed ( f 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 SZ is bounded) and one obtains an existence theorem in L2(i2,dx). A leisurely exposition of all these ideas can be found in [311. We now formulate a version of HSrmander's result, which we will use freely in what follow: THEOREM (Hormander). If SZ C Cn is smoothly bounded and pseudo-

is a a closed (0,1) from on SZ with L2(Sl,dx) coefficients, then there exists u E L2(fl,dx) such that i9u = f. convex, and if

f

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

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 [311 for details.

Our next main goal is to obtain an integral formula for a solution to the 0 problem on a convex domain. We first need:

THEOREM (Bochner-Martinelli). Let fl be a domain in Cn with CI boundary. Let f c C I (1) . Then for all z c fl we have

f(z) =

1

A W(C)

P077

nW(n)

an

-

ff()A7J Q

Proof. Apply Stokes' theorem to the form

µ(0 = f(i)n

-z A W(C)

on the domain fl\B(z,e). Imitate the proof of the Cauchy Integral Formula (or see [311). Now fix 1 a bounded, C2, convex domain. Choose e > 0 so small that 1 == l z c Cn : dist (z, 11) < e I is convex (hence pseudoconvex). Let f be a smooth, 0-closed (0,1) form on fl. By Hormander's theorem, there is a smooth u on fl such that du = f. We apply the Bochner-Martinelli

formula to u (which is certainly in C(fl) ). Thus I

STEVEN G. KRANTZ

218

u(z) =

Z2

fu(017

WW(n)

au

nW(n)

nW(n)

fu()1?

z2

f

z2

An

Ate(()-n 1

A)(t)

(6.3)

-z{

ff(4)A rl

Z-a2

A.W(C).

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 0 =(p<01, let ap

-ap

Xn (D(z, 0

with 4(z,

4)(z,

I

d (C) (zj -Cj) and observe that j=1 Oxj

H(z) = -1

fu(xw(z))

A

is well defined and holomorphic in z (because w is). Now let v(z) = u(z) - h(z). Then certainly 3v = du = f and, by (6.3),

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

1 v(z) = nW(n)

{fu

n

/

A

- J uw«-z

12

ale

asp

nW(n)

f

=z2

A ri

z

Aw(C)-I-II.

_z

Q

Let G = SZ x [0,1] and define g(z, C, A) = (1-A)

z-z

+ Xw(z,0 to be a

K z I2

form on G. Then I=

n

W(u) full(s)

A 0)(C)

aG

By Stokes' theorem, applied on G, this

1

nW n)

f d(u(C)rl(b) A G

But d(u(C)rl(g) A

au A ?7(g) A W(C)

+ u(C)d(n(g)) A W(C)

= f A-J(g)AW(C)

(this last equality takes advantage of the special algebraic properties of Cauchy-Fantappie forms). So we finally obtain HENKIN'S INTEGRAL FORMULA. 11 f = Ip < 01 is C2 and convex and f is a smooth, a-closed (0,1) form on a neighborhood of 12 then the

function

220

STEVEN G. KRANTZ

v(z) - n .

5 f(c) A n(g) A W(C)

W(n)

-J

-z

A 17

a

satisfies dv = f on 11. Here g(z, , X) _ (1-A)

- z`

2+ Aw(z, 0 ,

VIP

(D(z,

and

( )(z-

(P(z

)

.

The standard reference for Henkin's work is [181; see also [311 Similar formulas were derived by Grauert-Lieb [111, Kerzman [221, and (6vrelid [35].

Now we assume that i1 is strongly convex, and show how to use Henkin's formula to obtain uniform estimates for solutions to the

problem. For simplicity we work in C2 only. So a f

,(g) Aa )f ^ [g1

dg2

dA-g2

dA

AdC1

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

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

a 1

1

(C)(r,2-z2) + 4'(z, S)I4_zI2

all

+f2( )

A (f

1

1

2W(2)

(C)(1-z1)

2

v(z) = 2W(2)J

2W(2)

221

I

n

f1(b) (_I= 1) +

d1S n d Cz n dCl ^ d s2

K z I4

f

ndC1 AdC 2

as

+ 2W(2)

( f2(C)A2(z,

n dC1 ^ d62

ag

+ 2W(2)

J f(C)B1(z,

n

dC1 ^ dC2

a

+ 2W(2)

J

f2(C)B2(z,C)dz1 n dZ2 n dC1 ^ dC2

Q

In order to prove an inequality of the form

10 L,o < CIif11Lw

it suffices to check that

(6.4)

STEVEN G. KRANTZ

222

flAj(z)idcr()
j =1,2

(6.5)

j=1,2

(6.6)

190

and

J IBj(z,C)jdV(C)
with the estimates uniform over z e 11. (Here d or is area measure on By symmetry we check only j = 1 .

.)

For B1 , choose R > 0 such that B(z, R) D t1 for all z e Q. Then

f IB1(z,C)IdV(() <

r

C

dV(C)

B(z,R)

Sl

R


r3dr=CR<- . r3

For A 1 , we must work harder. We need to know something about the degree to which the denominator 4D(z, C) IC-z i2 of Al vanishes. (This is where strong convexity plays a role.) Think of z r St as fixed. Notice

that, writing the Taylor expansion for p about t in complex coordinates, we have

p(z)=p(()+2Re

dp

ap

(C)(z2-C2)

+ (quadratic terms) + (error terms)

> 0 + 2Re (D(z, C) + C lz- l2 , C>0.

The last estimate is by strong convexity - see Section S. Thus

223

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

IRe (D(z,S)I > CIIz-CI2 + Ip(z)II

Let nz be the normal projection of z to Al. Let t1 be a coordinate in the complex normal direction at nz (that is, normal direction) in ail. Recall that 2

ap

(C) a j= 0

i

times the real

a c `.`P(R)

and 2

Re 5

dp

(C)aj = 0

a ETp(SZ).

j=1

It follows that 2

2

Im I

(Nzj-cj) = Im I

=1

j=1

7

z)(zj-Cj)

+ 00Z-0)

J

measures (essentially) the complex normal component of z - C at 1Z. As a result

IImtI >C1t1I.

Introducing two additional coordinates t2 , t3 centered at uz , which span the complex tangential directions at nz , we conclude that I(D(z,C)I >2 (IRe w(z,C)I + IIm(z,C)I)

>C 1t1+t2+t3+Ip(z)I1 provided C is near nz, say K-rrzI
fAi(z)da(i)= an

J B(nz,r0)lao

+

J as\B(rrz,r0)

.

224

STEVEN G. KRANTZ

The second integral is trivially bounded since when iz-Cl > r0 then Al is bounded. The first is majorized by ,/t2 +t2 +t3 +p(z)2 dt ldt2dt3

C

(t2I +t22 +t23 +p(z)2) (it,1 I + t 22 + t 23 + Ipz)I)

J ti+t2+t3
=C IpI+ItlI
IpI+It11>t2+t3

tl+t2+t3
ti+t2+t3
Y1+Y2 Now

1

IY11
I

(t2 +t3)

IPI+I t1 I
J

< C

t2+t3

1

t22 + t23

dt2dt3

dttdt2dt3

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

225

dtldt2dt3

IY21 < C

t22 +t23 (it,1 I + IPI)

IPI+It1I>t2+t3 ti+t2+t3
I


1

t2+t

2

(Ilog(t2+t2)I+llog(Crdl)dt2dt3 3 2

3

t2+t3
C r0


f

(Ilog rl + log CI)rdr

0


f JAIIda
.

an

The estimates for A2 , B2 follow by symmetry, as already noted. We have proved THEOREM. With fZ strongly convex and f, v as in Henkin's integral

formula for solutions of the a problem, IV II

L°°(tt)

< C Y-IIfjIl L,

.

REMARKS. 1) To eliminate the hypothesis that f is defined on a neighborhood of ft we observe that for E > 0 small enough the domain 06 = fz Ef : dist (Z, 3f) > e y

226

STEVEN G. KRANTZ

is smoothly bounded and strongly convex. Moreover, the estimates in the theorem on 1lE depend boundedly on E (by a calculation). Thus estimates can be obtained for f a smooth form on 11 with bounded coefficients by applying a limiting argument to the solution uE of d(*) = f on QE.

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 it CC Cn be a

strongly pseudoconvex domain with C2 boundary. Then there is a neighborhood SZ of _Q, a k > 0, a C2 strongly convex domain U C Cn+k and a holomorphic imbedding F : c Cn+k such that (i) F(12) C U (ii) F(6\5) C Cu

(iii) F(4) C au (iv) image F is transversal to X. The upshot of this theorem is that the Henkin singular function (D which we know how to construct on U can be pulled back to Q. The construction of the Henkin solution to the a equation and the uniform estimates follow just as before.

3) The singular function c can be constructed more directly on a strongly pseudoconvex domain fl as follows: first write it = { p < Of where

dw (Cn,

(C)wjwk>CIwl2 J

k

(see (5.3)). For C E 9Sl fixed we define 2,

L(z, )

i=t

J

J

y

aP

_Cj) +

k-t

J

J

k

k

227

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

The function L, called the Levi polynomial, has the property that there a neighborhood Uc of C such that sz n UC n #z : L(z, t) = 0} _;c } . One can modify L, by solving a a problem (see [181 or [311) to obtain

4i

n

such that 5 n {z : d>(z, c) = 01 = I I and d>(z, C) =I Pi (z, 0 (z)-Cj) j=1

with Pi holomorphic in z . Henkin's program may be carried out using this and w(z, C) = (-PI(z, 011b(z, 0, ..., -Pn(z, 0/(D(z, 0) Notice that, by the discussion in the preceding paragraph, 1/L( , C) is a local singular function at C. 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 CI . However the construction goes through for domains in Cn once one has the basic lemma, and that follows in Cn from the Bochner-Martinelli formula. We leave the details of the basic Szego theory in Cn as an exercise. Recall that the Szego kernel for a domain 1Z is the reproducing kernel for H2([Z) . Now fix z e Q. By construction, S(z, ) a H2(fl) . By the reproducing property, it follows that

S(z,4) =

fS(w)S(z.w)da(w) aQ

=

f ag

= S(C,z )

=S(C,z)

228

STEVEN G. KRANTZ

Thus the operator

S:f i+

has the following three properties: (a) S : L2(dQ) - H2(fl) (b) S is self-adjoint

(c) S is idempotent. Therefore S is the Hilbert space projection of L2(8l1) onto H2(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 [81, [361), the Henkin operator

H:fi.nWn)

w

J

f(4)r!(w)A

produced from the Fornaess theorem as in Section 6) maps

L2(afl) onto H2(C1). Also H is idempotent. Now H is not quite selfadjoint, but it is nearly so. To see this, one needs to write (7.1) in the form 1

nW(n)

f

N(z, pn(z,

4)

da

where da is area measure on d fl. This is a straightforward but tedious calculation (see [231). It turns out that N is real. It also turns out that (D(z, ) - I(4,z) vanishes to higher order at z = 4 than does (P. (Try this when Il is the ball to see how this works - details are in [231.)

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

229

As a result of the preceding observations, the kernel N(z, C)

N((,z)

n(z, 0 is a kernel which is less singular than the original Henkin kernel. Thus H - H* , rather than being a non-isotropic singular integral operator (as is H ), is a smoothing 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 L2. We may resubstitute the equation into itself as follows: S = H + SA = H + (H+SA)A

=H+HA+SA2 = H + HA + (H+SA)A2

(7.2)

-H+HA +HA2 + SA3 .. = H + HA +

HAk + SAk+1

Now we know that each of the operators HA, HA2, -- are smoothing. If

we apply both sides of (7.2) to a sequence Oj E Cc '(4) such that ) -, S

230

STEVEN G. KRANTZ

in the weak`* topology on 91, we obtain an equation relating S(z, C) and H(z, C). 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 SZ CC (:n and define

A242) = f holomorphic on

SZ: J If(z)I2dVol(fl) <

(Notice that, for SZ smoothly bounded, H2(fl) is a proper subspace of A2(fl)

- Exercise.) Define

=

ffdv ft

[fil = f If 12 dV' /2 Q

for f,g a A2(fl). The basic lemma in this context, sup If(z)I 5 CKIIf II K

for K CC [Z, 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, C) = K(C,z). Thus the associated operator

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

B:f H

231

ff()K(zs)dV() 12

is self-adjoint. It maps onto A2 (by construction) and is idempotent. So B : L2(1l) - A2(1l) is Hilbert space projection. A remarkable construction of S. Bergman (see [311) is as follows: note that, for z E it fixed,

K(z,z) = J K(z,w)K(w,z)dV(w)

I K(z,w)

12

dV(w) > 0 .

Therefore we may set 2

gij(z) _

il

l 0-T

log K(z,z)

By a calculation (see [31]), the matrix (glj(z)) gives a non-degenerate Kfihler metric on iZ (called the Bergman metric) which is invariant under biholomorphic mappings. In particular it holds that if ( : flI - fl2 is biholomorphic then

distBerg(Z,w) = distBerg(c(z), NW))

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 integral 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 Q. However, for z c fI fixed, Henkin's kernel 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],

[331: for each fixed z c fl, let N(z,i (Dn(z,

)

S) Easl

Now construct a smooth extension Tz of 1Az to 9. The Cauchy-

Fantappie formula is still valid with Tz replacing qz (since the integral takes place on the boundary where Tz = Oz ). Thus Stokes' theorem can be applied to the new Henkin formula containing Tz. The resulting solid integral operator on L2(il) can be compared with the Bergman integral via the program of Kerzman and Stein (details are in [331). The result is that K(z, C) = 9'z(C) + (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 aQ 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 it :

(a) As Q 3,z - a11, the Bergman metric curvature tensor at z converges to the constant Bergman metric curvature tensor of the unit ball. The convergence is uniform over M. (,B) The kernel and the curvature vary smoothly with smooth perturbations

of M. (y) fl, equipped with the Bergman metric, is a complete Riemannian

manifold.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

233

Now we conclude this paper by coming full circle 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), (,3), (y). Now we use those to derive concrete information about symmetries. If Q C Cn is a domain, let Aut a denote the group of biholomorphic selfmappings. If two domains f11 and a2 are biholomorphic we will write

a1 1 Q2' THEOREM (Bun Wong [41]). If a CC Cn is smoothly bounded and

strongly pseudoconvex and if Aut Q acts transitively on fl, then Q ti ball.

Proof (Klembeck). Let P0 a fl be any fixed point. Let IPi I C 0 satisfy Pj a g. By hypothesis, choose (k] a Aut a such that (k](P0)=pi . Then the holomorphic sectional curvature tensor K for the Bergman metric

satisfies K(P0) = K(cbj(Po)) = K(PH) - (constant curvature tensor of the ball).

(*)

Thus the Bergman metric curvature tensor is constant on Q. 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 f, C Cn is smoothly bounded and if

is C°° sufficiently close to the unit ball B then either (i)

0tiB

or

(ii) Q

B and Aut a is compact and has a fixed point.

234

STEVEN G. KRANTZ

Proof. Step 1. If (1 B then Aut 1Z is compact. For if Aut 0 is not compact then a normal families argument [12] implies that for P0 E 1 3 Oj E Aut f such that iij(P0) 3(1. As in the proof of the preceding theorem, it follows that Q ti ball. Step 2. Recall the following result of Cartan-Hadamard (see [241): 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 (B) implies that this statement holds for domains t which are C°° sufficiently close to B. By Step 1, Aut iZ is compact. So the result follows from (y) and Step 2. Now we turn to a conjecture of Lu Qi-Keng (see [34]):

CONJECTURE. If 0 C Cn is a bounded domain then the Bergman kernel never vanishes on a x 12.

On the disc and the ball this conjecture is correct; for the ball in Cn one can calculate (see [31]) that K(z, ) = nn! 1 nn (1-z. )n+1

However it turns out that in C1 the conjecture is true if and only if Q is simply connected (see [40]). From this it follows that the conjecture is not always true in Cn either. To see this, let C2 ) ci = disc x annulus. Then the uniqueness of the Bergman kernel easily implies that the kernel for 1Z is the product of those for the disc and annulus. Now 0 = UGj where fj C fj+1 and each 0i is smooth and strictly pseudoconvex (see (311). By a theorem of Ramadanov [37], Koj - K11 normally.

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

235

By Hurwitz's theorem [31], KQi vanishes for j large enough. So there exist smooth strictly pseudoconvex domains with vanishing Bergman kernels. Thus we have the MODIFIED Lu QI-KENG CONJECTURE. If fZ CC Cn is smoothly

bounded and diffeomorphic to the ball, then K0 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°°

topology. Then we have (see [12], [13]): (i)

If

= In: Kfl never vanishes l then a is closed.

(ii) If 'U = In: Kfl is bounded from 01 then `U is open.

Statement (i) follows from Hurwitz's theorem. Statement (ii) follows

from (f3). Facts (i) and (ii), together with the fact that e, 11 are nonempty (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 noCC Cn is a smoothly bounded

strongly pseudoconvex domain and if

fZ

is a sufficiently small smooth

perturbation of no then (i) Aut iZ

Aut 00

C_

subgroup

(ii) 34): 0 -. no a diffeomorphisrn such that Aut 0 3 a H 4 o a -q5_1 E Aut Q0

is a univalent group homomorphism.

Sketch of proof. We may as well suppose that no ball, else the result is straightforward. Then normal families arguments show that, for it sufficiently near S10, [1 ball. Thus Aut 0 is compact and, by averaging the Euclidean metric, one can construct a new metric y, smooth

236

STEVEN G. KRANTZ

across aU, which is invariant under Aut Q. By patching this metric with the Bergman metric, and modifying it near o5n, we can arrange that

Isom(y) = and that y is a product metric near ffi. Finally, we construct the metric double M of 0 equipped with y. Known theorems [5] about semi-continuity of isometry groups of deforma-

tions of a compact Riemannian manifold now give the result.

We introduce our final result by recalling a corollary of the Uniformization Theorem (see [2]):

THEOREM. If SZ CC C, P e 0, and the isotropy group IP of Aut 0 is infinite, then Q Z A, The generalization of this result to Cn would require new ideas since, in that context, there is no uniformization theorem. Also, on dimensional grounds, the infinitude of IP 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 IP has a compact subgroup H

which acts transitively on real tangent directions at P, then M is biholomorphic to either the ball or Cn.

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 Cn. This last is the heart of the argument: It involves analysis of geodesics and equivariance proper-

ties of the Bergman metric on B(P,r) and of the canonical solution to the 5 equation. 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. 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

INTEGRAL FORMULAS IN COMPLEX ANALYSIS

237

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 iZ depends on the geometry of ail. 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-compact automorphism groups. Another is to relate the dimension of Aut (St) as a Lie group to the rank of the Levi form on ail. I hope that the survey presented here will inspire some new people to consider these

questions. DEPARTMENT OF MATHEMATICS THE PENNSYLVANIA STATE UNIVERSITY

UNIVERSITY PARK, PA. 16802

BIBLIOGRAPHY [11

L. Boutet de Monvel and J. Sjostrand, Sur la Singularite des noyaux de Bergman et Szego, Soc. Mat. de France Asterisque 34-35 (1976), 123-164.

[2]

R. Burckel, An Introduction to Classical Complex Analysis, Birkhhuser, Basel, 1979.

D. Catlin, Necessary conditions for subellipticity of the a-Neumann problem, Ann. of Math. (2)117(1983), 147-172. [4] , Boundary invariants of pseudoconvex domains, to appear. [5] D. Ebin, The Manifold of Riemannian metrics, Proc. Symp. in Pure [3]

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 [7]

[8]

STEVEN G. KRANTZ

G. Folland and J. J. Kohn, The Neumann Problem for the CauchyRiemann Complex, Princeton Univ. Press, Princeton, 1972. G. Folland and E. M. Stein, Estimates for the ab complex and analysis of the Heisenberg group, Comm. Pure Appl. Math. 27(1974), 429-522.

[9]

J. 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 and die Gleichung of = a im Bereich der Beschrankten Formen, Rice Univ. Studies 56 (1970), 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 d equation, and stability of the Bergman kernel, Adv. Math. 43 (1982), 1-86.

[14]

, The automorphism groups of strongly pseudoconvex domains, Math. Ann., 261 (1982), 425-446.

-,

Characterization of complex manifolds by the isotropy subgroups of their automorphism groups, preprint. [16] P. Greiner and E. M. Stein, Estimates for the a-Neumann Problem, Princeton Univ. Press, 1977. [15]

[17] G.M. Henkin, A uniform estimate for the solution of the 3-problem on a Weil region, Uspekhi Math. Nauk. 26 (1971), 221-212 (Russ.). [18] , Integral 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, Izvestija Akad. SSSR; Ser. Mat. (1971), 1171-1183, Math. U.S.S.R. Sb. 5(1971),1180-1192.

[20] L. Hormander, L2 estimates and existence theorems for the a operator, Acta Math. 113 (1965), 89-152.

[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

[22] N. Kerzman, Holder and LP estimates for solutions of

c3u = 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 CauchyFantappie kernels, Duke Math. J. 45 (1978), 85-93. [24] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1, 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. Global regularity for c3 on weakly pseudoconvex mani[26] folds, 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 3-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 Au = 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.

[32]

, 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 projection and proper holomorphic mappings, preprint. [34] Lu Qi-Keng, On Kahler manifolds with constant curvature, Acta Math. Sinica 16(1966), 269-281 [Chinese]; Chinese J. Math. 9(1966), 283-298.

[35] N. Ovrelid, Integral representation formulas and LP estimates for the 3 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). [37] 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. Jour. Math. 78 (1978), 173-189.

240

STEVEN G. KRANTZ

[39] N. Sibony, Un exemple de domain pseudoconvexe regulier ou

1'equation u = f n'admet 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 Cn 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 Szego projection in some of these domains, and in estimating the size of approximate fundamental solutions to certain nonelliptic, hypoelliptic partial differential 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 1. Some classical theorems and examples

In order to motivate our later discussion, we begin by considering three examples of metrics: 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 and related 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 naturally 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 n

1/2

Ix-yI = ± Ixj-yj12 j=1

In this example, the important first order operators are just the partial The Laplace derivatives with respect to the n variables O-A 1

n

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 xn,y) = (x,y)Iy>01. We identify the Dirichlet problem for Rn+1 = {(x1 boundary of R++1 with Rn = Rn x {0}, and, given a function f on Rn, we want a function u(x,y) harmonic in Rn+1 such that u(x,y) - f(x0) 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 L1(Rn) + L°°(Rn) . Set

Pf(x.v) = P. * f(x) = c_v

dt

f (t)

I

+1

2

Rn 1kX-t12+y21

where n+1

cn=r (n21`/n 2 Then:

(a) Pf is harmonic on R++1 (b) Pf extends continuously to the boundary and takes on the boundary values f. (c) fRn IPf(x,y)ipdx < Iliii

for 1 < p < 00.

(d) Suppose u is harmonic on R++1 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 < -, let hp denote the space of functions u(x,y) harmonic on R++1

which satisfy

sup

I

Iu(x,y)Ipdx = Ilulihp < 00 if p < Do

y>o.f Rn

sup Iu(x,y)I = IIuII ,o<00 if p=c R n+i

h

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 xo t Rn define: ra(xo) = (x,y) F R++1I Ix-xo1 < ayl

Note that if B(x0, S) = lx r RnI Ix-xoI < 8l are the balls defined by the standard Euclidean metric, then ra(xo) = I(x,y) a R++'Ix a B(xo,ay)l

Thus the nontangential approach regions in Rn+1 are really defined in terms of the projection rr(x,y) = x of Rn+1 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, natural approach regions can be defined in essentially the same way.

VECTOR FIELDS AND NONISOTROPIC METRICS

245

We say that a function u(x,y) has a nontangential limit at x0 a Rn if and only if for all a > 0, lim u(x,y) exists as (x,y) approaches (x0,0) and (x,y) a ra(xp). In 1906 Fatou [4] proved: THEOREM 1.

For I < p < -, if u e 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 r L1oc(Rn) and set Mf(x0) = sup IBL-i f If(Y)I dy B

where the supremum is taken over all Euclidean balls B which contain x0. The basic estimates for the maximal operator are given in: THEOREM 2 (Hardy and Littlewood). For I < p < 00, there are constants Ap < 00 so that

(i)

IIMfIIp <ApllfIIp if 1 < p < 00

(ii) lix a RnlMf(x) > All < AiA-i llflll

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 Rn+i Thus for any

a > 0 and any v(x,y) defined on Rn+I set Nav(x0) =

sup

Iv(x,Y)I .

(x.Y)fra(x0)

For Poisson integrals, this non-tangential supremum is point-wise dominated by the Hardy-Littlewood maximal function of the boundary data:

ALEXANDER NAGEL

246

THEOREM 3 (Hardy and Littlewood). For a > 0 there exists a constant Ca < «, so that if f E L1(Rn) + L°°(Rn) and if u(x,y) = P,, * f(x), then

for all x c Rn Nau(x) < CaM'(x)

.

These are the two quantitative estimates which underlay the qualitative statement of Fatou's theorem. Complete proofs of 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 from these two theorems.

Let p < oo and let u c hp.

If

s > 0 and if we let fs(x) = u(x,s),

then

sup

Iu(x,y+s)I = Na[P(fs)I(x0)

(x,y)Era(xp) < CaIIM(fs)](x0)

.

Therefore if A > 0 I}x0fRnI

Iu(x,Y+s)I > A}I sup (x.Y)fra(xo)

< I}x0ERnI M(fs)(xo) > Ca'A}I < [CaA-IIIM(fs)IIp]P

< [CaAp011fs Iip]P [CaApA-1 IIu IIh JP P

Since s > 0 was arbitrary, we obtain for any u c hp I;xocRnINau(xo)>,'}I < [CaApA-1IIuHIhp]P

.

(1)

VECTOR FIELDS AND NONISOTROPIC METRICS

247

Now let u of hp be real valued, and set Slau(xo) = lim sup u(x,y) - lim inf u(x,y)

where the limits are taken as (x,y) approaches (x0,0) and (x,y) E Fa(xO). Then the following facts are easy to verify: (a) fau(x) < 2Nau(x) (b) Sla(u+v)(x) < Slau(x) + Slav(x)

(c) Slau(xo) = 0 if and only if u has a limit within Fa(xo)

(d) Slau(x) - 0 if u = Pf and f is continuous. Now let un(x,y) = u (,y

+

n)

.

Then

Slav = SZa(u-un + un) < Sla(u-un) + Sh(un)

= fla(u--un) < 2Na(u-un)

Hence, by inequality (1), we have: Ilx ERnlfau(x)>A}I < Ilx ERnI Na(u-un)>A/21l < [2CaApk-Illu-un11hp]p

Since p < o, llu_unllhp

0 as n - oc, and since A > 0 is arbitrary, it

follows that Ilx ERnjjlau(x)>011 = 0

.

By taking a countable sequence of a's which increase to infinity, we obtain a proof of 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

ALEXANDER NAGEL

248

balls are also involved in studying the fundamental solution for the Laplace operator. An important fundamental solution for A is given by the Newtonian potential:

N(x) =

where con = 2nn/2/P(2) /r( . Then AN = S as distributions. In particular, if

C U (Rn)

c(x) =

(2) Rn

and if fN(x_Y)(Y)dY

'/r(x) =

(3)

Rn

then Ar/r = 95.

Proofs of these facts can be found in Folland [5], Chapter 2. A great deal is known about the operator

f

N * f(x) =

r N(x-y)f(y)dy . Rn

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 Holder continuity condition of order a, 0 < a < 1 , then f * N is of class C2 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 operator f -, N * f depend on size estimates of the kernel N(x,y) _ N(x-y) which can be written: ,32IB(x,8)I-I

IN(x,y) < C

(4)

IV N(x,y)I + IVyN(x,y)I < C sIB(x, s)I-1

where S = Ix-yI. 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 2

than f . Using formula (2), this means that for all i, j ,

9256

should be

as smooth as A S6. Thus we are led to the study of the operator which

carries A-0 to

32 . i

There are two ways in which we can think of

j

this.

Formally differentiating equation (2) we see that

(x) _ 5 kij(x-y)A S(y)dy

(5)

Rn

where kij(y) = cn

2 ylyl

IyI-n

for an appropriate constant cn

0. We note

IyI

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 kij(x,y) = kij(x-y) satisfies the following estimates in terms of the Euclidean metric:

250

ALEXANDER NAGEL

Ikij(x.y)I
IVxkij(x,y)I + Ioykij(x,y)I <

CS-1IB(x,

S)I-t

where S = Ix-yl. It is well known how to combine estimates of this type with the Calderon-Zygmund decomposition of L1 functions to prove 2

LP(Rn) boundedness of the operator 0¢ -

provided that one J

knows that this operator is bounded on L2(Rn) (see for example, Stein [161, Chapter 2). Boundedness on L2(Rn) follows from the cancellation property

f kij(x)dx = 0 , a
and can also be established by studying the operator AO terms of Fourier transforms. Define:

(e) =

r Rn

so that

-O(x) =

f

e

Rn

if, for example, 0 E C o (Rn) . Then:

(A) (e) = _4V2 lei 2 (e) (e) _ -4n2 eiej Vi(e) u&i/

in

VECTOR FIELDS AND NONISOTROPIC METRICS

251

so

ei el

a295

lei I

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 L2(Rn); Thus we have THEOREM 4. For 1 < p < 0 there are constants Ap < 00 so that if 95 E C p (Rn) (9 II

2o IILp < ApIIok IILp

§2. Spaces of homogeneous type In our discussion of the Laplace operator, we emphasized the role of the Hardy-Littlewood maximal operator and the Calderon-Zygmund decom-

position of Ll 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 Coif man and Weiss to what they call "spaces of homogeneous type." A locally compact space X is a space of homogeneous type if there is a continuous map p : X xX [0, 00) and a non-negative Borel measure

dµ on X so that: (1) p is a pseudometric; i.e., for all x,y,z E X , (a) p(x,y) = 0 if and only if x = y (b) P(x,Y) = P(Y,x)

(c) p(x,z) < K;p(x,y)+p(y,z)].

ALEXANDER NAGEL

252

(2) The measure µ has the "doubling property" relative to the family of falls B(x,5) = IyeXJp(x,y)<5I: There is a constant A so that for all

x eX, 5>0

g(B(x,25)) < AW(B(x, 5)) .

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 e Lloc(X,dµ) , define Mf(x0) = sup IBI-1

f

If(y)Idsx(y)

B

where the supremum is taken over all "balls" B which contain x0. We prove: THEOREM 5.

There is a constant Al so that if f e L1(X,dtt), p(xeXIMf(x)>hI < A1A-1II01l

.

Proof. For X > 0, let E = IMf >X1, and let F C E be any compact subset. If x e E, since Mf(x) > A there is a ball Bx containing x so that (Bx 1-1

I

If(y)Id(y) > A

Bx

The balls IBxlxel cover F, and since F is compact we can find a subcover, which we call B1, , BN. Suppose Bj = B(xj,5j) so Bj has "radius" 3 J.. Choose BI so that 5i1 > S i for all j. We can then l

inductively choose Bil, , Bik so that (1) B.

is disjoint from B.

`k

(2) S ?a for all `k

B.

11

>

j

`k-l

is disjoint from B

such that B

't

>

B'k-1

In this way we get a finite subsequence Bt , , Bt m which are disjoint. 1

VECTOR FIELDS AND NONISOTROPIC METRICS

253

Suppose B is a ball from our original finite sequence which does not

appear in the subsequence. Then there is a first k so that B; fl B 0. k

> &.. Let xik be the center of

By property (2),

Sik center of B1. Then if z < Bik fl Bl ,

and xj the k

P(xikxj) < K[p(xik,z) + P(z,xj)l < 2KSi k

If y c Bl , then p(xi,y) < Si , and so P(xik.Y) < K[P(xik,xj) + p(xj,y)l

< K(2K+1)S.

.

k

Let B k = B(xik,K(2k+l)Sik). Then Bi C B* N

and so

1m`

L.t C U Bj C U Bi k j=1

k=1

m

Hence µ(F) < Y p(Bi ). But by property (2) of spaces of homogeneous k

k=1

type

*

P(Bih) < A

1+1092K(2K+1)

µ(Bik)

=A1p.(B'k).

Thus µ

Al I u(Bik) < Ala 1 k=1

k=1

J B. 'k

If(Y)I dY < A 1X 1 IIfII1

ALEXANDER NAGEL

254

since the balls (BikI are disjoint. Since I was an arbitrary compact 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

I n

3t

a2

'=

a x

j=J

Rn+1, where we use coordinates

(x,t). Unlike the Laplace operator A on Rn , L is not elliptic. Nevertheless there is a remarkable fundamental solution for L. Define on

if

t>0

if

t<0.

E(x,t) = 0

Then E is C°° on Rn+1\1(0,0)L, and LE = S in the sense of distributions. (See Folland [5], Chapter 4.) Thus if 0 c C°°(Rn+1) 0 + n

r00

(k(x,t) =

J (4rrs) 2 e-IYI2 /4s I(x-Y,t-s) dy ds

J

Rn

0

=

f J

where L = -

t

2e--2/4(t-(Y,s)dyds S (4(t-s)

- A 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), S) denotes the ball in the standard Euclidean metric, the estimate JE(x,t)l < C 321B((x,t), 8)l-1

with a=

8-n+1

(Ix12+t2)1 /2

is false. To see this, let t =x12 with lxi small, so E(x,t) =CIxt-n, while S =(Ix12+Ixl4)1/2 ti IxJ, so S-n+l ti xj-n+1 Thus we cannot 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 nonisotropic. For A > 0 define SX(x,t) = (Ax,A2t) .

It is easy to check that E(SA(x,t)) = E(Ax,A2t) = A-nE(x,t) .

We associate to the family of dilations a pseudometric P((x,t),

(Y's))

_

(lx-y14 + (t-S)2)1 /4

so that P(Sa(x,t), Sx(Y,s)) = kp((x,t), (Y's))

The corresponding family of balls BP((x,t), S) = {(Y,s) a Rn+1 jp((x,t), (Y,s)) < SI

are now ellipsoids of size S in the directions of x1, ,xn, and of size S2

in the direction of t. Thus IBP((x,t), S)1 ti

8n+2

ALEXANDER NAGEL

256

Moreover, it follows from the homogeneity of E that we now have: (a) IE(x,t)I < C 821Bp((x,t), 6)1-1 (b) IV E(x,t)I < C6IBp((x,t),6)1-1

(7)

(c) IdE (x,t) < C 1Bp((x,t), 6)1-1

(d)-

(x,t)I < C IBp((x.t),

6)x-1

where 6 = p((0,0), (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 as acting like a second order operator, so in equation (7c) operator

we loose two powers of 6 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 < oc there are constants Ap < oo so that if

0 e Co(Rn+1) then

Ilp

< Ap IIL0IIp

a20 p

A p IIL0li p <-

The operator L is the sum of squares of the first order operators a minus the operator , which we shall count as an operator

(9 1

a

of order two. We shall later see how the appropriately weighted vector fields i-, , _ , and as give back the nonisotropic metric p. 1

n

VECTOR FIELDS AND NONISOTROPIC METRICS

257

§4. The Siegel 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 [171. 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 {(z1,...,zn'zn+1)

H=

n

= (z,zn+l) c

Cn+1 I1mzn+l

> I izjI2 = IzI2 j=1

n+1

Recall that U is the image of the unit ball B = (w1,-',wn+1)I

jwj12<1

under the biholomorphic map wk

zk =

l+wn+1

1
zn+1 = I l+wn+l

The boundary of H is the set an = I(z,zn+1) c Cn+1IImzn+1 =

= Iz121

I(z,t+ilz12)Iz c Cn,t c RI

Thus we can identify afl with Cn x R by associating to the point (z,t+ilz12) c dfl the point (z,t) c Cn x R. We can make Cn x R into a group Hn, the Heisenberg group, by defining

(z,t) (w,s) = (z+w,t+s+2Im) n

where = I zjwj. This definition can be motivated and the group j=1

ALEXANDER NAGEL

258

properties verified by the following considerations: to each point h = (w,s) f Cn x R we associate a holomorphic map Th : Cn+1 Cn+1 by the formula

Th(z,zn+1) _ (z+w'zn+1 +

+2i)

.

Th is clearly holomorphic. Moreover, if hj = (wj,sj), j = 1,2, then a simple calculation shows that Th2 ' Th1(z,zn+l) = Th3(z,zn+l)

where h3 = (w1+w2, s1+s2+2Im<w2w1 >)

= h2

h1 .

From this it follows that Hn is a group, and Th2

Thl = Th2,hl .

Let p(z,zn+1) = lmzn+l - Iz12 be the "height function" for f1 so that (z,zn+1) f ,f1 if and only if p(z,zn+1) > 0, and (z,zn+1) f c3S1 if and only if p(z,zn+1) = 0. If h = (w,s) f Hn we compute p(Th(z,zn+l))

=

Im(zn+l+s+ilwl2+2i) - (z+wl2

= Imzn+l - Iz I2

= p(z,zn+l) '

Thus Th carries i1 to itself and

to itself. If h = (w,s) or Hn, we have identified (w,s) with the point (w,s+iiw)2) f d2, and this is the same as Th(0,O). Thus under the identification, Hn acts on (90 as follows: if (z,t) fan and if h = (w,s) f Hn , then f3I1

VECTOR FIELDS AND NONISOTROPIC METRICS

259

Th(z,t) = ThT(z.t)(0,0) = Th.(z,t)(0,0) _ (w,s)

(z,t)

(z +w, t+s -2 Im< z,w >)

We now make Hn (or ail) into a space of homogeneous type. Define d : Hn x Hn -+ [0, c) by

d((z,t); (w,s)) _ [Iz-wI4+It-s+2ImCz,w>I211/4 ,

or, if we identify Hn with rail, d((z,zn+1); (w,wn+l)) _ [Iz-wI4+Re(zn+l-wn+l-2i)I211 /4

The function d is invariant under the action of H

Thus if h = (u,r) r Hn , easy algebraic manipulations show that n.

d(Th(z,t), Th(w,s)) = d((z,t), (w,s))

If we define the balls B((w,s),8) _ I(z,t)Id((z,t), (w,s)) < 81

then this invariance property means that B((w,s),8) = T(w,s)(B((0,0), 8))

We claim that d is a pseudometric. For if z = (z,t), w = (w,s) and

u = (u,r) are in Hn with d(z,w) < 8, d(w u ) < 8 then

1z-w( < 8

It-s+2lmI < 82

1w-ul < 8

Is-r+2lm<w,u>I < 82

260

ALEXANDER NAGEL

Hence

Iz-u I < Iz-w I + Iw-u I < 28

,

and

It-r+2ImI < It-s+21ml + is-r+2Im<w,u>I

+ 12Im(--<w,u>)I < 232 + 21Im1

< 432 = (2S)2 . Theref ore

d(z u) < 2 m a x (d(z,w ), d(w,u )) < 2 [d(-Z-, w ) + d(w, u ) I .

What do the corresponding balls B(i,S) look like? We see that w c B(Z,S) is essentially equivalent to the pair of inequalities: Iz-wI < S )Re (zn+i-wn+1-2i)I < 82

Fix z = (z,zn+1) a asl. The complex tangent space to asl at z is given by the equation (''n+1

- Zn+1 - 2i<w,z > = 0 .

If w = (w,wn+1) a afl, the distance from w to this complex tangent plane is essentially Iwn+i-zn+1-2i<w,z >1

=

IRe(wn+1-zn+l)+i(lw12+Iz12-2Cw,z>)I

= IRe (wn+l-zn+1-2i)+i lw-z 121

ti lw-zI2+(Re(wn+1-zn+1-2i)l

ti< S2

VECTOR FIELDS AND NONISOTROPIC METRICS

261

Thus the balls B(z,S) are essentially "ellipsoids" of size S in the directions of the complex part of the tangent space to df at z , and of size S2 in the orthogonal real direction, and hence in particular B(- Z,

ti 82n+2

Thus the doubling property of the balls is verified, and di1 (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 Q. This problem was first studied by Koranyi [9] for domains like 1, and was later generalized by Stein [17] to general smoothly bounded domains in Cn. 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 it. Let n : SZ K1 be the projection n(z,zn+1) = (z,zn+l-ip(z,zn+1)) -

For a> 0 and w = (w,s+iIwl2) c dSl let Aa(w) = l(z,zn+l)cg!n(z,zn+l)EB(w,ap(z,zn+l) where of course B(w,S) 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 Fa(xo) in Rn+1 Now (z,t+ilzl2+iy) f Aa(w) is essentially equivalent to the following pair of inequalities: Iz-wI I < ay

.

As above, It-s+2ImI is essentially the distance from (z,t+ilz12) to the complex part of the tangent space at W. Thus Aa(w) allows tangential approach to w of order two in the complex directions at w,

262

ALEXANDER NAGEL

but requires nontangential approach in the complementary real direction, and so the sets (Aa(w )1 are essentially the admissible approach regions introduced by Koranyi [9]. The invariance of the balls B(w, S) 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 H. Thus it is easy to check that if h = (w,s) c Hn, then (z,zn+1) t Aa((0,0)) if and only if Th(z,zn+t) E Aa(w,s+iIwl2). Let F be a function defined on 11, and for C c 3O , set sup

NaF(Co) =

IF(z,zn+1)I

(z,zn+l)tAa(y SO) If f c Lloc(3t1) , define

Mf(Co) = sup IBI-1

J

lf(C)I da( )

B

where the supremum is taken over all nonisotropic balls B containing

Co, and do is surface area measure on Al. We now have the following analogues of Theorems 2 and 3, which were

used to prove Fatou's theorem. THEOREM 7. For 1 < p < oo there are constants Ap < oc so that

(i)

(ii)

IIMf IIp <_ Ap IIf lip

1 < p < cc

I < A1A-lllfil ,

p =1.

if

THEOREM 8. Suppose u is continuous on iZ and pleurisubharmonic on

1. For a > 0 there is a constant Ca (independent of u) so that for all Ct do Nau(C) < CaMu(C)

.

VECTOR FIELDS AND NONISOTROPIC METRICS

263

Theorem 7 of course follows from Theorem 5 and the Marcinkiewicz interpolation theorem (see Stein [16], Chapter I), since we already know dfl is a space of homogeneous type. Before proving Theorem 8, we point

out some of the consequences of these results.

For 1
sup

< °o

= IIFIIH

y>o

if

p < 00

p

an

sup IF(z,zn+l)I = IIFIIH

p = 00 .

<

COROLLARY. For 1 < p < oo and a > 0. There are constants Ap a so that if F f Hp(11) (i)

INaFNLP < Ap,aIIFIIH

< 00

for '
(ii) IlCE3f INaF(C)>AII < A1,aa-1IIFIIHI

if

p=1.

Proof. Put FE(z,zn+l) Then FE is continuous on fl, and pleurisubharmonic on 0, so by Theorem 8 =F(z,zn+l+ie).

NaFe(C) < CaMFE(C) .

Since IIFEI{LP < 1F

for 1 < p < oc we see that

IINaFE>,'MI < AICaa-1IIFIIHI

and

if

IINaFEIILp _< ApCaIIFIIH P

But NaFE(() ,' NaF(C) as

a

0, and

P>1 .

ALEXANDER NAGEL

264

INaF>Al = U INaFE>AI E<0

so the corollary follows by the monotone convergence theorem and the regularity of the measure do. We can now apply exactly the same argument we used for Fatou's theorem to prove THEOREM 9 (Koranyi). If F c Hp(Q) , then F has admissible limits at

almost every point of caf2.

It remains to prove Theorem 8. For simplicity, we deal with the case n =1 . Since the approach regions Aa(C) 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: 2

If

Iu(O,iy)1 < -I Ay

2

IS12+It-yI2< Y)

f211 <

If

2 Ir2y2 Is

u(N/tei0,s+it)Id9dsdt

I2+I t_Y 12<(2)

y

3y

2

2

2 rr

0

2

f

r

0

Iu(V/teie,s+it)Idtdsd9.

Letting r = V/t , and interchanging the order of integration, we get

VECTOR FIELDS AND NONISOTROPIC METRICS

Iu(O,iy)I <

4

u(re10, s+ir 2) Irdrd gds

- n2y2

<

4 Try2

Z

265

f

B((0,0),2\)

J

IB((0,O),2,Vy-)I-1

Iu(C)Ida(C)

.

B((0,0),2 J ) Thus, by translation invariance, we see there is a constant A so that for (z,zn+l) E lZ

Iu(z'zn+1)I <_ AIBI-1 J lu(C)Ida(C) B

where

B=B(rr(z,zn+l),2p(z,zn+1)1/2),

Now let (z,zn+l) E Aa((0,0)) . Then ir(z,zn+1) E B((O,O), ap(z,zn+1)1 /2) so

d(n(z,zn+1), (0,0)) < ap(z,zn+1)1/2 If Cc B(rr(z,z n+1),2p(z ,zn+1)1 /2) it follows that d (C, (0,0)) < (2a+4) p(z,zn+1)1 /2

But now using the doubling property of the balls, it follows that there is

a constant A so that if (z,zn+1) E Aa(0,0)

Ju()1da()

I°(z,zn+1)I < A'IBI-1 B

(8)

266

ALEXANDER NAGEL

where B = B((0,0), (2a+4)p(z,zn+1)112), and this last average is certainly dominated by A'Mu(0,0). We remark that estimates of type (7) are actually false for Poisson integrals of harmonic functions. The nonisotropic balls on dV are also useful in studying singular integrals and fundamental solutions. For example, the Cauchy-Szego projection of L2(ail) onto H2(SZ) is given by the operator

S((z,zn+1)

=

asp

where =cn[i(wn+t-zn+1)-2]-(n+1)

S((z,zn+1),(w,wn+i))

with cn = 2n-1n!/nn+1 (see Nagel and Stein [11], page 23). In particular,

for z = (z,t) and w = (w,s) on dQ we have, .

S(z,w) =

Thus IS(z,w )$

cn(i)-(n+1)[(s-t-2Im)-ilz-wl2]-(n+1)

IB(i,8)I-1 where 8 = d(i,w ). It is also easy to check

that derivatives of S yield estimates with corresponding negative powers

of S. 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 Kor£nyi and Vagi [101). Finally, we consider fundamental solutions. On Hn let Xj =

+2YjYj =

-2xj. , T

where we write zj = xj + iyj. These vector fields form a basis for the left invariant vector fields on Hn. Put

(Xj-iYj), Zj =

Zj =

2 and consider for a c C.

(Xj+iYj) 2

VECTOR FIELDS AND NONISOTROPIC METRICS

267

n

'a=-2 1 (ZjZj+ZjZj)+iaT j=1

This second order operator arises in the following way: if we identify Hn with asp, the vector fields Zj annihilate the boundary values of holomorphic functions. Thus, in analogy with the operator a, we consider n

Zjdzj

abf =

on functions,

j=1

and we extend this in the usual way to (0,q) forms on Al. In L2(Hn) we can define a formal adjoint (ab)*, and the Kohn Laplacian is then

°b=ab+ab

.

On q forms, obq) acts diagonally, and is given by the operator '2a where a = n - 2q. The operator ob 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: _jn+a

`` Y'a(Z,t) = (IZ I2-it)

2

a

(1212+jt) -(n 2

THEOREM 10 (Folland and Stein). '2acba = cab in the sense of distribu-

tions, where ca is a constant, and ca

0 if a ±n, ±(n+2), ±(n+4),

etc.

Thus except for the exceptional values of a, c Oa is a fundamental a solution for 2a , and it is easy to verify that I'ka(z,t)I <

C52IB((0,0),5)I-1

where S = d((0,0),(z,t)). One also obtains corresponding estimates for derivatives of Oa, so that again there is a complete analogy with the estimates (4) for the Newtonian potential.

ALEXANDER NAGEL

268

In the case of the Heisenberg group, the basic vector fields are

X1,"',Xn'Y1,"',Yn, 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 [71, 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 of vector fields have also been studied by Fefferman and Phong [4a], by Folland and Hung [5a], and by SanchezCalle [15a 1.

Let Q C RN be a connected open set, and let be C°° real vector fields defined on a neighborhood of Q. We associate to each vector field Yj an integer dj = d(Yj) > 1 which we call the formal degree, and we make two fundamental assumptions about this collection of vector fields. span RN. (1) For each x c Q, the vectors

(2) For all j, k, we can write [Yj,Yk' =

Y-

d1
ck . (x)YE where 1

cek c C°°(Q). Here [X,Y] --- XY -YX is the commutator of the two J

vector fields.

There are several basic examples to keep in mind.

(A) Let N = q = n , let Yi =

and let dj = 1 for 1 < j < n

.

In this

case we shall recover the standard Euclidean metric on Rn.

(B) Let N=q=n+1, let Y

and let d1 for 1<j
let do+1 = 2 . In this case we shall obtain the nonisotropic metric on Rn+1 appropriate for the study of the heat operator

VECTOR FIELDS AND NONISOTROPIC METRICS n

n+l

I

269

012

I c3xj

(C) Let N = q =2n4-1 and let Yj

1<j
+2yj

with dj = 1 Yn+j =

Yj

- 2xj

1<j
with den+1 = 2

Yen+1 = [Yj,Yn+j] _ -4 at

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 Y3 = [Y 1,Y 2] = a

X

2

0x2

with dl=d2=1 and d3 = 2 . This example leads to a metric 2

appropriate for studying the hypoelliptic operator See Grushin [6a]. (E) (This example generalizes (A), (C), and (D).) Let real vector fields on Sl. Let X(1)

=

2

+ x2 2 1

(2 2

be C°°

iX1,...,xp1

x(2) = ; . . . , [x1 x I, ...1 X(3) = #..., [Xi, [XJ,Xk]], 1, etc.

so that X(k) is a vector whose components are all the commutators of length k. We say that the vector fields of

are of Hormander type m or finite type m if at each x e 1, the components of X(1) ,X(m) span RN.

ALEXANDER NAGEL

270

Let

be some enumeration of the components of

and if Yi is an element of XU) we set d(Yi) = j. Then property (1) follows from the assumption of finite type, while property (2) follows from the Jacobi identity for commutators. This example leads to a metric appropriate for studying the hypo-

elliptic operator Xi + + Xp and its variants. (F) (This gerreralizes (B).) Let Xo,Xl,.",X, be C°° real vector fields as in (E) of finite type, but this time let d(X3) = 1 for 1 < j < n while d(X0) = 2, with appropriate weighting of higher commutators. This example leads to a metric appropriate for studying the generalization of the heat operator Xi+Xi Xp-X0. Our object now is to construct natural metrics out of the family of vector fields Y1, .,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 of 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 xp,xl c f2, and say p(xp,xl) < S if and only if

there is an absolutely continuous map 0: [0,1] -+ fl with c(j) = xj , j = 0, 1 , so that for almost all t c (0,1)

VECTOR FIELDS AND NONISOTROPIC METRICS

271

q

(t) _ I aj(t)Yj(cS(t)) j=1

where laj(t)I < S

d(Y.)

PROPOSITION. p is a metric on Q. For every compact set F CC i2

there are constants C 1, C 2 so that if x0,x 1 < F 1

C11x0-x11
where 1x0-xII 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 ca(t) by 0(1-t). (3) Given x and y , there is a smooth curve joining them, so p(x,y) < (4) The triangle inequality: Suppose x,y,z e Q. Given e > 0, there are curves 0, 0: [0,1] -.1Z with 0(0) = x, b(1) = y , 0(0) = Y, l'(1) = z q

4'(t) =

q

a j(t) Y -(O(t)) , &'(t) = F b j(t) Y j('(t))

, with

1 d(y,z).+E)d(Yi)

Iaj(t)I < (d(x,y)+e)d(Y1)

Ibj(t)I <

Define 0: [0,1] - 9 by

0
0(at)

0(t) = at-1 as -1

where a = 1 + q

d(y,z) + E d(x,y) } E

1
a

Then 0(0) = x, 0(1) = z and 0'(t)

E cj(t)Yj(0(t)) where j=1

Icj(t)I < (d(x,y)+d(y,z)+2E)d(Yi)

.

,

This shows, since a is arbitrary, that d(x,z) < d(x,y)+d(y,z). Now let F C it be compact. Then there is a constant C = C37 so that if x,y c F , there i s a smooth curve cb joining x to y with l,0'(t)I < Clx-yl. Since span RN at every point, we can q

write

(t) = I cj(t)Yj with Icj(t)I < CI0'(t)I

.

But then:

j=1 1

C[Ix-yId(Yj)]d(Yj)

Icj(t)I < CIx-Y1 =

<

C[Ix_yIm]d(Yj)

and so p(x,y) < CIx_yIm

Conversely, if x,y c I and p(x,y) = 8, there is a curve q

x to y with 0'(t) = F aj(t)Yj(c(t)) and

Iaj(t)I < (28)

joining

d(Y ) i

almost

j=1

everywhere. But then 1

O'(t)dt

J0 1

q

I laj(t)I IYj(O(t))Idt

< 0

j=1

< C8

since the lengths of the vectors (Yjl are uniformly bounded on 0. We now give two other possible global definitions of metrics. DEFINITION 2. Let x0,x1 E Q and say p2(x0,x1) < 8 if and only if

there is a C°° curve

:

[0,1 ]

U with (j) = xj

j = 0,1 , and

,

d(Y)

q

q'(t) = I ajYj(c(t)) where aj c R and Iajl < 8

I

.

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 p2(x,y) is

finite for any two points of Q. For example, if 9 is not convex,

N=q=n, Yj =

and dj =1, 1 <j
only if x and y can be joined by a straight line contained in SZ 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 p2 is locally equivalent to the standard Euclidean metric. Now suppose we are in the case of vector fields

of

finite type m. DEFINITIQN 3. Let x0,x1 E Q and say p3(x0,x1) < 3 if and only if

there is an absolutely continuous map 0: [0,1] - 9 with q(j) = xj, a

j = 0,1, with 0'(t) = E aj(t)X.(c(t)) and Iaj(t)I < S almost everywhere. j=

Note that in this definition, we only allow the derivative 0'(t) to which may not be all of RN. belong to the span of It is again easy to verify that p3 satisfies the triangle inequality, but it is not a priori clear that p(x,y) is finite for any two points of Q. 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 R2. Let the coordinates be x and y and let X1 = Note that these vectors fail to span R2 along x = 0, but X2 = xk &,

[XI,X2] = kxk-1

,

[X1[x1,X2]] =

k(K-1)xk-2

[X1,

[x1,x2] ]] = k!

19

O

13Y

I...

vy

where the commutator is of length k + 1 , so (X1,X21 are of finite type

k+1. What points belong to the ball centered at (x0,0) of radius 3 ? We shall consider two special cases: x0 = 1 and x0 = 0.

ALEXANDER NAGEL

274

Case 1. If we let

¢1(t) _ (1 +St,0) so ci(t) = S and

952(1) = (1,St) so c?(t) = S

(1)k

then this shows that the points (1 +5,0) and (1, S) belong to the p3 ball centered at (1,0) of radius S. In fact, this ball is essentially the Euclidean ball of radius S centered at (1,0). Case 2. Again, starting at (0,0) we can go to (5,0) by using the vector field (3 , 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 .0 1(t)_(St,O)

0
so

95i =SX1

QSZ(t) = (S, Sk+lt)

0
so

952 = SXZ

c63(t)=(S-St,Sk+1)

0
so

95g=-6X1

This shows that the p3 distance from (0,0) to (0,8k+1) is at most 38. In fact the p3 ball with center (0,0) is an "ellipsoid" of size S in the x direction and size Sk+1 in the y direction. Thus we have given three possible definitions of a metric or pseudoIt is also clear metric in terms of the family of vector fields from the definitions that we have the following inequalities: P(x,y) < P2(x,y)

p(x,y) < p3(x,y) (when p3 is defined). In fact, these three quantities are locally equivalent. One can prove that for every x0 E (I there is a neighborhood U of x0 and constants C1,C2 so that for all x,y c Q, x 4 y , and j =2,3.

275

VECTOR FIELDS AND NONISOTROPIC METRICS P3(x,y) 0
§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 (1, we let TpiZ denote the tangent space to SZ defined near at p. Suppose we are given C°° vector fields

p which form a basis for RN at p. Then we can construct a map from a neighborhood of 0 c T 1 to a neighborhood of p in [ as follows: N

every tangent vector v at p can be uniquely written as

I j_1

J

j

(p)

=

v

N

E RN, and soj=1 I

with

J

J

a smooth vector field defined

near p. We can flow along the integral curve of this vector field for unit time if IlajI is sufficiently small, and the result is by definition N

exp

jS (p) , the exponential map of V. J Given vector fields

we can identify T f with RN via

N 1Y'

ajSj(p), and then the exponential mapping

(al,...,aN) .exp (ajSj) (p) introduces a coordinate system centered at p , the so-called canonical The Jacobian of this coordinates relative to the vector fields the volume of the exponential map at 0 c RN is just "parallelopiped" spanned by S1, ,SN. It is important to remember however that this exponential map from TP( to [ depends on the choice of vector fields S1, SN. in Now we return to the general situation of vector fields

1 C R.N On each tangent space TxI there is a natural notion of length.

ALEXANDER NAGEL

276

If V_ E Tf) we say

Nx(v)
v =

ajYj(x) j=1

d (Y )

-.

J <5 i . Of course this representation of v need not be unique. Nevertheless questions about the set where

Ia j

IV ETxSZINx(v)<6

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 11, 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 TxQ of the set { vlNx(v)<SJ ? This amounts to the following be vectors in RN which span, and consider problem. Let the map

q

ajyj

©(a 1 ,...,aq) = j=1

What is the volume of the image under ® of the box QS = 1(a1,...,aq)(RgI IajI<S

For any N-tuple I =

d (Y .)

let d(I) = d(Yi )+ ... + d(Y. ), and let 1

AT = det (Yi1,.. ,YiN).

I?

N

VECTOR FIELDS AND NONISOTROPIC METRICS

277

LEMMA. There are universal constants C1,C2 so that 0 < C1 < Ip(QS)I/ I IXII Sd(I) < C2 < 00. (Here the sum is taken over all N-tuples I.) I

Proof. For each N-tuple 1, the image 0(Q3) contains all vectors of d(Yi) N ajY1

the form

and this is just the image under

where Iaj I <

i=1

a linear map`from RN to RN with determinant

AI.

Thus

1A11 sd(I) < 1®(ns)1

5d(I) IXII

< (;l)

We must now prove the reverse inequality. Pick an N-tuple 10 so that IAI01 ad(I0) > 1A J I ad (J)

for all N-tuples J. By renumbering, we may assume 10 = Since AI0 0, we can write N

1<j
Yj=4 bjkYk k=1

and using Cramer's rule, we see that A,

Ijk bjk

X1

0

where Ijk is obtained from 10 by replacing Yk by Yj. From our choice of I0 , it now follows that Ib jkI _ 5

d(Yk)-d(Y j)

.

ALEXANDER NAGEL

278

q

Now let v c O(Q8), so v -_

I

j=1

ajYj with Iajl < 8 d(Y I) . Then

N

9

±" ajbjk Y k

= k-

V

j=1

and q

<

1: ad(Y

J)Sd(Yk)_d(Yj) = q Sd(Yk)

j=1

(gS)d(Yk) Hence

d(I°) IB(Q8)I < q

I

d(1

°)

!)L1

0

< qmN

IA118d(I)

This lemma suggests the following further definitions. For each let x c 12 and each N-tuple I = d(Yi

N

BI(x,$) =

YcHIY=exp(I ajYi) (x) with jajI <8 1

Clearly for every

)

I

BI(x, 8) C (y c 12Ip2(x,Y)<8j

B2(x, 6) .

Hence U BI(x,6) C B2(x,S). We also have I

B2(x,8)C B(x,6) = iycI p(x,y)<81

since p(x,y) < p2(x,y).

VECTOR FIELDS AND NONISOTROPIC METRICS

279

It is now reasonable to conjecture that for each x c 11 and each 6 > 0, if we choose an N-tuple 10 so that IA1

o

8d(J) (x)I ad(10) > IAJ(x)I

for all J

then for this particular 10 B(x, S) C Bl (x, ca) 0

where C is a constant independent of x and 8. This would of course show that p and p2 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 f B(x,6). Then there is 0 : [0,1] - t with 0(0) = x , 6(1) = y and q

bj(t)Yj(o(t))

0,(t) _ j=1

Sd(Y'

almost everywhere. Now assume without loss that and let denote canonical coordinates near x I0 = relative to the exponential map using Then the curve c(t) is given in canonical coordinates by (uI(t),...,uN(t)) and our object is

with lbj(t)I <

d(Y.)

to show that there is a uniform constant C with Iuj(1)I < (CS) 1

uj(1) = uj(1) - uj(0) =

r

[uj(t)]dt

0 1

q

f Y, bk(t)Yk(c(t))(uj)(t)dt k=1 0

Now since IY11...IYNI span near x we can write

.

But

ALEXANDER NAGEL

2$O

N

Yk =1 akYP =1

where ak (C"°. Thus q

u (1) _

Lr I k(t)ak((b(t))(Ypu))(t)dt b 0

P=1

k=1

We now need to make two kinds of estimates: (1) Iak(c (t))I < Cad(Yprd(Yk) (2) IYPuj(t)I <

Combined with the estimate Ibk(t)I < which is what we want.

Sd(Y k),

this gives Iuj(1)I < CO

d(Y

Now estimates (1) are obtained by using Cramer's rule to write ak(y) _ A J(y)/A1 (y) as in our earlier lemma. At x , our choice of Io 0

shows that IAJ(x)I/IAI (x) I <

Sd(Yp)-d(Yk) 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 , Ypui = S)p, 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 I CC it there are con-

stants C1 and C2 so that C1 (IIXi(x)I6')) < IB(x,6)I < C2`I I IAI(x)Isd(I)/ I

In particular, the balls B(x, S) satisfy the doubling property, and so Sl 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 ue 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 hypo-

ellipticity of b. Suppose

are C°° real vector fields on a C RN of finite type m. Hormander showed that the second order operaor are hypoelliptic. As in the work tors of Folland and Stein [6] on b , 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 on this difficulty by proving their "lifting theorem." Given iZ C RN (with coordinates of type m, they show that one E Rs and form new vector fields can find additional variables S

X =X+

ajf(x,t)

.

f=1

These new vector fields will again be of type m on a neighborhood of iZ x 101 C iZ x RS , and in addition they are free up to step m , so that one can model by vector fields on a free nilpotent group of step m

with p generators. Rothschild and Stein are able to construct a parametrix for 'X-2 +... +

Xp on 9 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

Ik((x,t),(y,s))I <

CS21]j((x,t),6)I-t

where S = p((x,t), (y,s)), and p is the metric constructed from the vector fields They then define a restriction operator

Rk(x,y) =

fk((xiO)(Yis))(s)ds RS

where 0 E C°°(Rs) is supported near s = 0, and if k((x,t), (y,s)) is the R2p, then D(x,y) = Rk(x,y) is a parametrix for parametrix for Xi +... + X2 X. Using the properties of the metric p constructed from

X 1

one can now prove, for example:

THEOREM 11. Let D(x,y) be the Rothschild-Stein parametrix for

Then, if N>3 ID(x,y)I <-

C821B(x,3)I-t

and it N>2,
where S = 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 [15a]. One also gets estimates of this type for

the Rothschild-Stein parametrix of the operator X1+X2+ +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 metrics to boundaries of domains in Cn+t (Some of these results were announced in [12J.) Thus let p c

C°°(Cn+l),

VECTOR FIELDS AND NONISOTROPIC METRICS

283

with dp 4 0 when p = 0 and let it = Iz cCn+llp(z)<01

so that fl is a domain with smooth boundary. If we fix a point Co c ait, then near Co we can find n linearly independent tangential holomorphic

Ln. For example, if ' (Co) 0, we can take

vector fields

n+1

Lj

a

(9P

(9P

Nn+I dzj - Nj

a

We write Lj = 2 (Xj-iXn+j), 1 < j < n, so that X1, ,X2n are C°°

real vector fields defined near Co on aft . Since the real dimension of sit is 2n + 1 , near Co we can find an additional real tangential vector field T, so that is a basis for the real tangent space to

aft near Co. Again if

ap az

(CO)

0, we can let

n+1

TaP [fin+i

a Wzn+l

_

dp

a

()n+l °Wn+1

The notion of type of a point was introduced by Kohn [8]. If C c aft

is near Co, C is of type m if every commutator of of length at most m - 1 at i lies in the span of 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 and T, but is really a biholomorphic invariant. We say that the domain it is of finite type m if every point

c sit

is of type < m. In this case of course, the vector fields viewed as being defined on an open set in R2n+1 , are of finite type as defined earlier. In particular, we then have a metric defined on sit, and various equivalent descriptions of the associated family of balls.

ALEXANDER NAGEL

284

Our first object is to show that we can define an equivalent family of balls on ail in terms of a single exponential map using the vector fields For each finite sequence of integers with 1 < ij < 2n, we can write the commutator [Xik_I,..., [Xi2'XiII...11

Xil'...'ik = [Xik,

Ai1'ik T (mod X1''**,x 2n)

where Ai,,---,ik c C(il) (near CO), since

is a basis for

the tangent spaces near Co. Let 5k be the ideal in C°°(l) generated with f < k. It is easy to check that this by the functions ideal does not depend on the choice of the vector fields Also (A11'...,ik) - Aii...... c 9k Xrk+i

or T.

k+1

One can verify this claim by

so in particular, Xjk+1 (Ai1' ik ) E 9k+i

T, and also Xli'...,1k+1 _ r1' 'in+1 and then expanding this commutator.

noting that X.li,...,rk+i - A. [Xrk+l,XiI,,,,,ik1

DEFINITION. For Cc ail near CO, set

Ak(0 =I IAil,...'if(S )I and

m

Aj(t)8j

A(C.8) = j=2 2

where the first sum is over all the generators of 9k. Note that Ak(O is a function whose size measures how much T component the commutators of XI,---'X2, of length < k can have. In particular, since c3il is of

type m, then Am(0 > µ0 > 0.

VECTOR FIELDS AND NONISOTROPIC METRICS

285

We are now in a position to define balls in terms of a single exponen-

tial map. For

E Al set 2n

ajXj+yT (c), where

77co-QIrl=exp 1

IajJ<S, 1<j<2n, and byl
.

Thus B(C,S) is the image of a box of size S in the "complex direcand of size A(C,S) in the complementary tions" given by real direction. Our first result is then:

THEOREM 12. The "balls" B((, S) are equivalent to the balls B(C, S) i.e., there defined in part fl in terms of the vector fields is a constant C so that B((, S) C B(C, CS) and B((, S) C B(C,C&).

We sketch only part of the proof. Let q c B((,3), so that

n=exp

tajXj+yT (0

It 1

M

with lajJ <S, Iyj
it follows that

2

A.(C)SI. Since A.(C)=F1ai1... M

y=IbjAI 2

where IbjI < SJ and Ij is a j-tuple of integers. Hence m

2n

Ibi X1. + I .6I.,EXE

yT = 2

Q=1

where 0,j,f c R and X1 is the jth order commutator defined earlier. Hence we can join C to q with a curve 4(t) (the integral curve of 2n

q

ajXj+yT) with 4'(t) _

r) E B(c, CS).

III

ajYj and JajI < CSdIYi. 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 H 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 t; , and if g(a1,..., a2n, Y) = f (exp

(

ajxj+YT) (c)

then g has a formal Taylor series at the origin, and it is given by: 00

g(at,...,a2n,Y)

^' I

k!IajXj+YT\\)

k

flo

k=0

where (lajXj+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 n A(n,8) is essentially constant on B((,8); i.e. there is

a constant C so that for n e 4(C, 8), IA(n,6)I < CIA((,S)I

A typical term in IA(71,8)I is SklAill,,,lik(n)I . We can expand

Ail ,,,,,ik(n) about C in canonical coordinates. Any term involving

applying T to this Ail,

.

k

is certainly of the right size since

Iyl < A(C, S). On the other hand, if we differentiate

ik with one

this gives us an element of 9k+1 of the vector fields again we get the correct estimate.

,

so

§9. Balls in terms of a polarization We now want to find a description of the balls on the boundary of a domain Il C Cn+1 directly in terms of inequalities involving the defining function for U. As before, let

VECTOR FIELDS AND NONISOTROPIC METRICS

287

SZ =(z ECn+llp(z)<01

where p : Cn+1 - R is C"° and dp 0 when p = 0. A polarization of p is a function R : Cn+1 x Cn+1 - C of class C"° satisfying: (a) R(z,z) = p(z)

(b) JzR(z,w) vanishes to infinite order on the diagonal z = w. R(w,z) vanishes to infinite order on the diagonal z = w. (c) 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. If

P(z) = I as

zap

then

R(z,w) = S` as za W13

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 e d 'Q

and S > 0 set B'(w,8) = 1z CaQ 11z-wI <8 jR(z,w)j
surface tangent to c3i at w . Thus B#(w,8) is essentially the set of points z e 31 within Euclidean distance 8 of w, and within Euclidean distance A(w, 8) of Vw. Let us quickly calculate what all this means for the Siegel upper half

space discussed in part I, where p(z) = n

Imzn+1

n

R(z,w)=I zjwj-Zi(zn+l-wn+l). j=1

.

Then

288

ALEXANDER NAGEL

Also, on the Heisenberg group [Xj, Xn+jI _ -4T so A(w, S)

32 for all

w E M. Thus in this case 1

B#(w,S) = z -F ul Iz-wI <6,

n

zjwj 2i (zn+1 -wn+1) <32 L j=1 14

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'a(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 a = SZ0 = 1(z1,z2)EC2IIm z2>'b(z1)'

where 0: C

R is a polynomial of degree m. Here the defining

function is

P(zl'z2) = b(zl) - 2i (z2-z2) .

It is easy to check that if L _ a - 2i a", (zl) a (9f1

azl

az2

then L globally spans the space of tangential antiholomorphic vector fields. We also set

T= a2+

a

02

We want to investigate the geometry of the balls near a fixed boundary

point, say (0,ij(0)). Since all our families of balls are invariant under biholomorphic mappings, we can choose a special coordinate system to study the geometry. Let

O(z)_Cz)-MO)-2Re(

(0)zl}

VECTOR FIELDS AND NONISOTROPIC METRICS

289

If we make the change of variables:

zl=zl j

m

z2 = z2 - i c (0)+2 1 i j=1

1

dzi

(0)zi

our original domain now has the form 1(zi,z2) f C211m z2 > 0(z, )I

and moreover 0 now satisfies j a-o

chi

00 00

1<j<m

We shall work in this coordinate system near (0,0) f a g. We identify 311 with C xl{ so that (z, t+iq (x)) corresponds to (z,t). Then our basic vector fields are: L=

+i'(z)N,

5j

(97

(Z)

Jt

T=?. We can calculate the various functions XiI", ik using the vector fields L

and L (instead of Re L, Im L ). Thus:

[L,L]=-2i'! 2

so the ideal

92

is generated by

O or Ac . Thus aza

A2(z,t) = IoO(z)I

ALEXANDER NAGEL

290

alo

Ak(z,t) = I a+,B1

aza&-#

To investigate the exponential map exp (aL+uL+yT) (0) with a e C, y e R, we must find a curve CxR

[0,1]

with

q (t)=a G

iff

12 (t) = y + is

(01 (t))

with (¢1(0),2(0)) = (0,0). Thus ¢1(t) = at and

q2(t) = yt + i

f (a J0

CV

(as)

a

0

(as)) ds.

And so exp (aL+aL+yT) (0) = a, y+i

(as) - a

We can estimate the integral by expanding a and aj

(as)\ dsl

about 0. Since

(0) = 0 1 < j < m, all the terms will be dominated by the correspond-

Jz-j

ing Aj(0), so I

f (aas)

_a

(as) ds

I2A2(0) +...+ JalmAm(0)

0

= A(O, lab

VECTOR FIELDS AND NONISOTROPIC METRICS

291

In particular, we see that the image under the exponential map of the box I(a,y) e C xRI IQI < s, Iy) < A(O,6)1

is essentially {(z,t) a CxRI Izl < s, Itl < A(0,8)1.

On the other hand, the polarization of our defining function is R((zIz2)(WIw2)) _

!1Sia_zQ+a5_

a'

(0)

ziw - 2i (z2 W2)

and

R((zlz2), (0,0)) = -2i1 z2 = -1 2i (t+iV'(z1)) if

(zt,z2) = (zI, t+ici(z1)) c aft. Thus the inequalities IR((zIz2). (0,0))1 < A(0,8),

Iz1I
are equivalent to the inequalities

ItI
IzII<8

and so the families of balls are equivalent. §10. Estimating pleurisubbarmonic functions We continue our study of domains of the form "= 1(z1,z2) EC2IIm z2 > q(zi)1

,

but we now make the additional hypothesis that L4 > 0; i.e. di 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:

ALEXANDER NAGEL

292

THEOREM. Suppose u is continuous on G and pleurisubharmonic on fl . Let n : iZ Al be the projection onto the boundary. Then if (z 1,z2) a fl

lu(zl,z2)I


f lu(C)Ida(() B

where B = B(n(z1 z2),AS) and A(n(zi,z2),S) = Im z2-c(zl). Here A,C are constants independent of u and (zi,z2). As before, by making a holomorphic change of variables, we can

assume Co) _

(0) = 0, 1 < j < m, and it is enough to

(0) _ i

estimate u at the point (0, iy). We begin as we did in part I:

lu(0,iy)I <

4

u(0,s+it)I ds dt

.

(8)

7ry2

I s 12+l t-y 12<(v /2 )2

Now for each s + it we want to imbed an analytic disc into fl, whose boundary is mapped to A) and whose center is mapped to (0,s+it). To do this, we try to find S > 0 and a function G(4), continuous for 141 < 1 , holomorphic on

1 , so that the holomorphic map

(84,G(4))

has the required properties. This means we want Im G(e'0) _,(Sei0)

0 < 0 < 2n

and

G(0)=s+it. Let 0&(ei0) _ O(Sei0). Let P[0S] be the Poisson integral of (kS , and let Q[958] be the conjugate Poisson integral, with Q[0,51 (0) = 0. Then

VECTOR FIELDS AND NONISOTROPIC METRICS

293

G(() = Gs t(o = s + i[P[gS](o + Q1081 (01 has the required properties provided that S is chosen so that

t

2n

f .0(8 ei0)d9

.

(9)

0

But by Green's theorem 2n

dS 2

f 0(S ei0)de

=

r

2 775

0

AO(x,y) dx dy > 0

X2+Y2<5 2

so the function t(S) = Zn

fo21r

j(S ei0)d0 is a monotone increasing func-

tion of S . Thus given t, there is a unique S = S(t) so that equation (9) holds. We thus obtain 3Y /2

Iu(0),iy)1

< na Y

22

J

Iu(s(t)e

-Y /2

0

s+

Y12

dt dOds

We want to make the change of variables r = S(t). Let

A(S) =

1

nS2

ff Ad!(x,y)dxdy x2+y2<52

so that t'(S) = SA(S). We get: 2

ALEXANDER NAGEL

294

Y/2

u(O,iy)<

2s 32

Lff 2

J

77Y

-Y/2 0

S(I) 2

We now need the following result, which is where the hypothesis is used:

0

LEMMA. There are constants C1, C2 which depend only on m so that

for S>0, (i)

0 < C1 < A(0,S)/t(S) < C2 < 00

(ii) 0 < C1 < A(0, 8)/82A (8) < C2 < 00 (iii) IWS(eie)I < C2t(S) We defer the proof for a moment, and return to the estimate for u(0,iy).

In our last integral, when r is between S 2 between

and 2 .

and S

,

t(r) is

2

It follows from the lemma that in this range,

2 A(r) ti y/6(y)2 , and it follows from part (iii) of the lemma that the range

of s integration in the integral is contained in

{ Is I < Cy I

for some

constant C. Thus: VT

Iu(0,iy)I <

C

yS(y)2

ff

IsI
0

CS(y)

f Iu(rei0,s+ic(reie)Irdrdeds 0

But finally, rdrdOds is essentially surface area measure, and on the surface r7lZ we are integrating over a region centered at (0,0) of size y

in the "real" s direction, and of size S(y) in the "complex" directions. But this is exactly our nonisotropic ball B(0,5) where A(0, S) = y and since IB(0,8)I .ti yS(y)2, we have shown: there are constants A and C so that

295

VECTOR FIELDS AND NONISOTROPIC METRICS

J

Iu(0,iy)I < C`B(0,A8)j-I

ju(J)jda(()

B(0,AS)

where A(0,5) = y . Thus we have managed to estimate u along the "radius" from (0,0). Just as in part 1, we now easily extend this result to obtain th'e theorem. We finally turn to the proof of the lemma. The inequalities t(S) < CA(0, S) and S2A(S) < CA(0, S) follow easily by expanding c or Aq 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 there is a constant Am so that if J q(z) is a real polynomial of degree at most m with a

(0) = 0,

az

0 < j < m , and if AO(z)>O for JzI <8 D then for S < S0

r

2n

('

t(S)=2n

0(Sei0)d9>AmL L

J0 f I

0

2n

as+(30

azar}i

(0) I Sa+R = AmA(0, a)

S

f

Ab(rei0)rdrde> Am

0

llazaaza

(0) Sa+1i = AmA(0, S) .

In fact, if we can prove this for S = So = 1 , then given c , we apply the

result to t#(z) = q(Sz), and the result for general S follows, so it suffices to study S = So = 1 . But now we let b(z) real polynomials of degree < m such that J

j

and

(0)=0, 0<j<m; 66(b(z)>0 if ,zj<1

I

+(3

o (0) 1(gzaaZo as

1

;

296

ALEXANDER NAGEL

It is easy to see that I is a compact set of polynomials and that the maps

0

fe10o 0

1 71

L7T r

(reie) rdr d9

J0

are continuous on 1. Moreover, these functions are strictly positive since AO > 0 and AO' 0, on JzJ < 1 , and so the functions are bounded below by a constant Am > 0. The general result now follows by dividing

a general polynomial 4 by I laza(90 ( 0)1

.

Finally, in order to show sup IQ.0g(ei9)J < Ct(S)

when 0 is a polynomial of degree < m, it again suffices to check this for S = 1

.

But sup IQ.08(e1O)I = 111.0111

0

is a norm on this space of polynomials. Hence III(bPHH < CA(0,1) < Ct(1).

As an easy corollary of the theorem we obtain an estimate for the Szego kernel S(z,C) for the domain 1 on the diagonal z = C. Recall

that the orthogonal projection S of L2(af) onto H2(1Z) is called the Szego projection, and formally, this projection is given by integrating against a kernel: Sf(z) =

ff()S(zs)da() 0 il

,

z e iZ .

VECTOR FIELDS AND NONISOTROPIC METRICS

297

In fact S(z, J) = E 0j(z)0j(C) where (0j! is a complete orthonormal basis for H2(H), and this series converges uniformly on compact subsets of SZ x Q. (See Krantz [l0a], Chapter 1, for further details.) Now it is easy to check that for z f SZ S(z,z)1 12 = sup IF(z)I

where the supremum is taken over all F e H2(1) with by our theorem, if F E H2(SZ),

IF(z)I <

2 < 1 . But

IIFII

H

r IF( )I da(C)

CIBI-1

B

< CIBI-1 /2

r IF( )I2 da(C)

1 /2

aJiZ

=

CIBI-1 /2IIFIIH2

where B = B(n(z),S) is the ball centered at the projection n(z) of z , and A(n(z), S) = - p(z) . Thus we obtain: COROLLARY. If z E SZ

S(z,z) <

CIB(n(z),8)I-1

where A(n(z),S) = Im z2 - O(z1).

§11. Estimates for the Szego kernel on aQ As a final application, we show how one can make estimates of the Szego kernel S(z,C) on the boundary, at least for certain very special

domains Q. Thus, let f' ='(zt,z2)EC2IIm z2>q(zi){

ALEXANDER NAGEL

298

where 95 is a subharmonic, non-harmonic polynomial of degree m. We

also make the very restrictive assumption that e95(z) = AO(x+iy)

is actually independent of y . Our approach is the following: if

-2ic3z

L= 1

1

(z )a

2

then L is a global tangential antiholomorphic vector field, and H2((qf2) =If fL2(a[l)jLf=0 as distribution)

,

so we identify H2 with the kernel of the differential operator. When we identify dfl with C x R in the usual way, and write z = x + iy , this operator becomes: L

07 Tt 2[aao

+2 (I-

49

We now make a change of variables on C x R D(x,y,t) = (x,y,t-A(x,y)) where x

A(x,y) _ - f

(t,y)dt .

0

Then if we put x

b'(x) = f A95(t)dt 0

VECTOR FIELDS AND NONISOTROPIC METRICS

299

and

L

+i

=

r

+ b'(x)

a

it is easy to check that L(f ocb) = 2(Lf) o (D.

Thus our object is to obtain estimates for the orthogonal projection onto the kerpel of

L=

X +i

+b'(x)

it] 11 10i 11 where b"(x) > 0. Our finite type hypothesis is now E IbW)(x)l > go > 0, i=2

and we want to obtain estimates in terms of balls defined by the vector fields and 22- + b'(x) in R3. dly

-t d

If u = u(x,y,t) is a reasonable function we define partial Fourier transforms by `.fu

f

= u(x,rl,r) =

e

R2

so that

u(x,y,t) = ff e21ri(Yri+tr)u(x,i7,r)di7dr R2

is an isometry of L2(R3) and

Luf-1ifu where Lu . e2"(nx+b(x)r) dx (e 2rr(rix+b(xp)u)

Let q,(x,rr,r) = e 2rr(nx+b(x)r)

ALEXANDER NAGEL

300

and let M.g(x,rl,r) = O(x,r1,r)g(x,71,r) .

Then

Lu=f

0- 1-iA,fu. dx

Now

Mq,: L2(R3, dx di7dr) - L2(R3, e4n(r/x+rb(x))dx d77 d,) and

MO-1 : L2(R3, e477(71x+rb(x)) dx drldr) -+ L2(R3, dx dr/dr)

are isometries. Thus L is similar to the operator N acting on functions which satisfy

f

g(x,r1,r)12e4r7(t7x+rb(x))dxdr7dr

< +- .

The kernel of this operator thus consists of functions g(77, r) so that

fJ'g(iir)I2[j'e4T'dx] drldr < 00 . Let

Y

=

(r),r)ER2l fe477(?7x+rb(x))dx < o

Then since b"(x)> 0, 1 = {(71,r)lr<0). For (>),r) a 2, the space

VECTOR FIELDS AND NONISOTROPIC METRICS

L2(e4rr(rlx+rb(x))dx)

301

contains the constants. Let P. be the projection

of L2(e4n(r)x+rb(x))dx) onto the constants, if (,l,r) a E, and let otherwise. Let

P'7,7

=0

P: L2(e4n(rlx+rb(x))dxdrldr) -, L2(e4n(rlx+rb(x))dxdrldr)

be defined by Pg(x,rl,r) =

where g,7 r(x) = g(x,rl,r). It is then easy to check that P is an orthogonal

projection whose range is precisely the null space of dx on L2(e4n(rlx+rb (x)) dx drl dr) . Thus if we set P

.` -IM

PMtA

then P is the orthogonal projection of L2(dxdydt) onto the null space

of L. Now if (rl, r) e

1 Pr1,r g = < 1,1 > 00

r g(r)e4n(rlr+rb(r))dr 00 00

r e41r(Tlr+rb(r))dr

Thus 00

P77,rg(x) = J

g(y) It 7,r(Y)dy

ALEXANDER NAGEL

302

where

Kll,r(x,Y) =

Thus if f f L2(R3, dx dy dt)

Pf(x,y,t)

If

f(r,s,u)S((x,y,t); (r,s,u))drds dy

where

S((x,Y,t); (r,s,u)) =

ff K,?,r(x,r) dry dr

00

fe

00

e21rq((x+r) + i(y-s))

27rrt(b(x)+b(r))+i(t-u)]

foe

0

00

f

e41r(nr-rb (r)) dr

-00

This is the kernel we have to estimate. We begin by estimating the inner integral. For r > 0, set 00

e21r-q(,k+it)

F A+it r -00

r e4n[r7r-rb(r)]dr -00

d

di

dr.

303

VECTOR FIELDS AND NONISOTROPIC METRICS

and 77 by r) + rb'(2) it follows that

Then replacing r by r + 2

F(A+it, r) _ I_

2rrr 2b e

(2)

(`]

+itb'

00

e2ni77td77

f+

/1

00

-00

J

2(2)1J

2(r+)+rbe 47rl77r+rrb ' (-rb LLL

-00

Let G(r) = rrb'(2) - rb (r+2) + rb(2)

.

Then

G'(r) = rb'C2) - rb'(r+2)

G"(r)= -rb"(r+2) Since G(0) = G'(0) = 0, we have m

G(r)

1 b())

r j=2

1

( -) r)

Hence

[2b(3) + itb'(2)1

277r

F(a+it,r) = e

L

JJ

00

e2rrjr7td,7

f J

-00

00

4n nr-r

m

f

1 b (j) ( ) r7

Now choose p = µ(A, r) so that

I In

j=2

1 b(j)(a>rµj12 2

=1

1!

and in the last integral, make the change of variables r - gr,

77

77.

ALEXANDER NAGEL

304

Then

F(a+it, r) =

e2ar [2b()

+

e2ai7,(

itb "%]

rl

A-2

M

477 r)r-

.j

e

-00

j

where ai = 1

dr

00

Ia.2 = 1 . and hence Ym 2 7

2

We now make two observations. First, in terms of size, m µ(X, r)

1 ti Y z

Ib(j) ( l \2/ I rl

/i

This is clear from the definition of µ . Second, the collection of functions f4a r - a i m 2 e

6a (n)

dr

m

where a =

am), Y-IajI2 = 1

,

and

r

'I' air) is convex, is a com2

pact set of functions in the Schwartz class S(R). Thus e2rrr L2b

F(A+it, r) =

rj'l

+ itb (2/] t1o, r)-1

\/

.

ea 1

From this, one can make estimates on the size of F(A+it,r) and its derivatives. Finally we have 00

S((x,y,t); (r,s,u)) 5 0

e-2ar[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))l <

CIB((x,y,t),6))-I

where S is the nonisotropic distance between (x,y,t) and (r,s,u). ALEXANDER NAGEL DEPARTI4IENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WISCONSIN

REFERENCES [1] [2]

Bets, L., John, F., and Schechter, M., Partial Differential Equations, Interscience Publishers, John Wiley and Sons, Inc., New York 1964. Caratheodory, C., "Untersuchungen fiber die Grundlagen der Thermodynamik," Math. Ann. 67 (1909), 355-386.

[3]

Coifman, R. R., and Weiss, G., Analyse harmonique non-commutative

sur certain espaces homojenes, Lecture Notes in Math. #242, 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.

Folland, G. B., Introduction to Partial Differential Equations, Mathematical Notes Series, #17, Princeton University Press, Princeton, N. J. 1976. [5a] 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. [5]

[6]

complex and Folland, G. B., and Stein, E. M., "Estimates for the analysis on the Heisenberg group," Comm. Pure Appl.Math. 27 (1974), 429-522.

[6a] Grushin, V. V., "On a class of hypoelliptic pseudo-differential operators degenerate on a sub-manifold," Math. USSR Sbornik 13(1971), 155-185. [7]

H6rmander, L., "Hypoelliptic second order differential equations," Acta Math. 119 (1967), 147-171.

[8]

Kohn, J. J., "Boundary behavior of d on weakly pseudoconvex manifolds of dimension two," J. Diff. Geom. 6 (1972), 523-542.

306

ALEXANDER NAGEL

[8a] Kohn, J. J., "Boundaries of Complex Manifolds," in Proceedings of the Conference on Complex Analysis, Minneapolis, 1964; 81-94; Springer-Verlag, New York, 1965.

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 spaces and some problems of classical analysis," Ann. Scuola Norm. Sup. Pisa 25 (1971), 575-648. [10a] Krantz, S., Function theory of several complex variables, John Wiley and Sons, New York, 1982. [9]

[11] Nagel, A., and Stein, E.M., Lectures on Pseudo-differential Operators, Mathematical Notes Series, #24, Princeton University Press, Princeton, N.J. 1979. [12] 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.

"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. [15a]Sanchez-Calle, A., "Fundamental solutions and geometry of the sum of squares of vector fields," Inventiones Math. 78, 143-160 (1984). [16] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J. 1970. [17] , Boundary behavior of holomorphic functions of several complex variables, Mathematical Notes Series, #11, Princeton Univ. Press, Princeton, N. J. 1972. [13]

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 turn to oscillatory integrals of the second kind suggested by twisted convolution on the Heisenberg group and the theory of Radon singular integrals. 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.

Oscillatory integrals of the first kind, n = I We are interested in the behavior for large positive X of the integral b

1(X) =

r e"(x),A(x)dx a

where q5 is a real-valued smooth function (the "phase"), and i' is

complex-valued and smooth; often, but not always, one assumes that has compact support in (a,b).

,Ji

*The reader will note that there are several related topics not touched on in this survey. Chief among them is the subject of oscillatory integrals 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.

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

309

The basic facts about I(A) can be presented in terms of three principles. (a) Localization : The asymptotic behavior of I(A) is determined by those points where q'(x) = 0, (assuming that t has compact support in (a,b) ). More precisely, PROPOSITION 1. Suppose 0 e C 0 (a,b), and q5'(x) Then I(A) = 0(A-N), as A ao for every N > 0.

0 for x in [a,b].

The proof is very simple. Let D denote the differential operator dx , and let !D denote its transpose, -Df = d Df = Then clearly DN(er40) = erA¢ for every N , and integration by parts shows that

fe "'0 0 dx = fDN(e),dx = (-1)N fe"(Wtdx. Thus clearly jI(A)I < ANA-N, and the proposition is proved. I

(b) Scaling : Suppose we only know that dk'(x) > 1 for some fixed k, dxk

and we wish to obtain an estimate for f b a

which is independent

of a and b. Then a simple scaling argument shows that the only possible estimate for the integral is 0(A-Ihk). That this is indeed the case goes back to van der Corput.

PROPOSITION 2. Suppose 0 is real-valued and smooth in [a,b]. If I0tkl(x)I > 1 , then

b

f a

holds when

ea(bi"ldx


E. M. STEIN

310

(i)

k>2

(ii) or k = 1 , if in addition it is assumed that 0'(x) is monotonic. Proof. Let us show (ii) first. We have b

b

e'4dx

=f

a

b

r e1X4t-D(1)dx + i1

D(eiAO)dx = -

b e

ix 4v a

a

a

The boundary terms are majorized by 2/A, while b

r ei

b

r eat i-j1 x (-1) dx

t_D(1)dx

'0,

aJ

a

a

b

<1

I

_ W

d

1

dx a

by the montonicity of 0'. The last expression equals 1 Iq

(b) 0'(a)1

which is dominated by 2/A. This gives the desired conclusion with c1 = 4.

We now prove (i) by induction on k. Let us suppose that'the case k is known, and assume (taking complex conjugates if necessary) that 0(k+1)(x) > 1. Let x = c be the point in [a ,b] where 14,(k)(x)l takes its minimum value. If O(k )(c) = 0, then outside the interval (c-S,c+S), we have that I.0(k)(x)l > 5, (and of course 0'(x) is monotonic when k = 1 ). Write

c-S

b

= a

f a

}f c+S

r

J

c&

b

f

+d

c+S

311

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

By the previous case < ck/(XS)l /k 0

Similarly b

L

/k

e'4 dx

Clearly however

f

c+s

e'kO dx

<2S.

C-S

Thus b

f

e'40 dx <

2ck (xs)l /k

+25.

a

If j(k)(c) 1 0, and so c is one of the end-points of [a,b], a similar argument shows that ck/(tb)1 /k + S is an upper bound to the integral. In either situation the case k + 1 follows by taking a = 'k-1 /k+1 , which proves (1.1) with ck+l - 2ck+2 . COROLLARY. Under the assumptions on -0 in Proposition 2, we can

conclude that b

(1.2)

f 0

e"W(x)0(x)dx

<

Ck,\-1 /k

J

Jv(x)Idx

.

E. M. STEIN

312

This follows from (1.1) by integrating by parts an estimate of the form z

f

eaO(x)dx


a

r b e'AOV

(c) Asymptotics : We already know that the behavior of

Ja

dx is

determined by those points x0, where 0'(xo) = 0, (the "critical" points of (k ). Assuming that the support of i/r is so small that it contains only one critical point of j , the character of the asymptotic expansion then depends on the smallest k so that 0(k)(x0) 10, and is given in terms of powers of A in a way which is consistent with Proposition 2. PROPOSITION 3. Suppose * is real and smooth, k > 2, and d(k-1)(x0) = 0, while 0(k)(xo) j 0. If Co and j(x0) the support of 0 is a sufficiently small neighborhood of x0, then _,k'(x0)... _

00

(1.3)

1(X) =

f eix(b(x)q,(x)dx

A-1/k I ajX-j/k

,

j=o

in the sense that for any non-negative integers N and r N

(1.3')

(1E (A) -

as A, 00

ajA-jO(a-(N+1)/k-)

j=0

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)

eiXX2 x xPe x z

J

-,o

dx - 1 /2-Q/2

00

,

cj(Q) -j j=0

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

where

F.

313

is a non-negative integer; in the case a is odd the integral

vanishes. In fact the left-side of (1.4) is j e-(I-1A)C2xedx which by a change of variables equals (1-v1)-`/=-P12

L a "2xedx.

However, when

A > 0, (1-ik)-y'4/2

where we have fixed the z-'/r4/2 in the plane slit along the negative halfprincipal branch of axis. The power series expansion of (w-i)-`/'`e/2 (which holds for lwl < 1 ), then gives the desired asymptotic expansion (1.4). Step 2. Observe next that if -q e Co and a is a non-negative integer,

then 00

f

(1.5)

eax2xe77(x)dx < AX-1/2

/2

CIO

To prove this let a be a C°° function with the property that a(x) = 1 for lxl < 1 , and a(x) = 0 when lxl > 2 , and write

J

eikx2xe>)(x)dx = feiXx2xfrl(x)a(x/E)dx + J eikx2xeri(x)(1-a(x/E))dx

The first integral is dominated by CEe+1 . The second integral can be written as

5e2(tr)N [xf-q(x) [1-a(x/E)]]dx with tDf =

1

dx

f

A simple computation then shows that this term

.

is majorized by CN ,\N

f

lxle-2N dx

= C' \-NEe-2N-1 N

lxl>E

if

e - 2N < - 1 . Altogether then the integral in (1.5) is bounded by

.

E. M. STEIN

314 CN#E/+1 +A-NCQ-2N+1 I

(with N > e 21) A similar

,

(/but

and we need only take e = .1-112

to get the conclusion (1.5). simpler) argument of integration by parts also shows

that

fe2ex)dx =

(1.6)

0(A-N), every N > 0

whenever e c 8, and 6 vanishes near the origin. Step 3. We prove the proposition first in the case q(x) = x2 . To do this write

fe2r(x) dx = feiAXe x2(ex2 is a Co function which is each N, write the taylor expansion where

1

on the support of Vi. Now for

N

eX2c

(x) _ Y` bjxj + xN+IRN(x) = P(x) + XN+1RN(x) jj=0+

Substituting in the above gives three terms 00

f

(2)

eiAX2 e_

C2

xi dx

00

00

(b)

(C)

J00

fe2P(x)e2(1_(x))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 exeiAx2

c/i(x)dx . pansion for f Let us now consider the general case when k = 2. We can then write q(x) = c(x-x0)2 + 0(x-x0)3 with c 4 0 and set q(x) = c(x-x0)2[1+e(x)] , 1 where a is a smooth function which is 0(x-x0), and hence when x is sufficiently close to x0. Moreover, q'(x) 0, when x x0 but x lies sufficiently close to x0. Let us now fix such a neighborhood of x0, and let y = (x-x0)(1+e(x))I12. Then the mapping x y is a

diffeomorphism of that neighborhood of x0 to a neighborhood of y = 0,

and of course cy2 = 4(x). Thus

fei'(o)clr(x)dx = reilcy2

(y)dy

e Co if the support of 0 lies in our fixed neighborhood of x0. The expansion (1.3) (for k=2 ), is then proved as a consequence of the special case treated before. with

t

REMARKS:

(1) The proof for higher k is similar and is based on the fact that

f

eixxke xkxedx = ck,Q(1- i'\)-R+I)/k

.

0

(2) Each constant aj that appears in the asymptotic expansion (1.3) depends on only finitely many derivatives of and Vi at x0. Note \r1-T(-i0'(x0))-I"2Vi(x0). Similarly e.g. that when k = 2, we have a0 = the bounds occurring in (1.3') depend on upper bounds of finitely many

derivatives of 0 and Vi in the support of t,, and a lower bound for 0(k)(x0).

sli ,

the size of the support of

316

E. M. STEIN

References : The reader may consult Erde1yi [8], Chapter II, where further

citations of the classical literature may be found. 2. Oscillatory integrals 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 (k defined in a neighborhood of a point xa

in R° has x0 as a critical point, if (Vg)(xo) = 0. PROPOSITION 4. Suppose cu c Co(Rn), and 0 is a smooth real-valued

function which has no critical points in the support of Vi. Then

I(A) =

r

O(A-N), as A

for every N > 0 .

Rn

Proof. For each xo in the support of ci, , there is a unit vector 6 and a small ball B(x0), centered at x0, so that (e,vx)o(x)> c > 0, for x c B(xp). Decompose the integral fek'0(x)t4(x)dx as a finite sum

I fe(')clFk(x)dx n

where each c/ik 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 xt,x2, ,xn so that xI lies along 6. Then

fe('c)ci/k(x)dx

'...,xn)Vk(xt,...,xn)dxI

=f(fe"(x

I

dx2,...,dxn .

But the inner integral is 0(A-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 S. Suppose 0 t Co , q is real-valued, and for some multi-index a, at > 0,

(d)a

(x),>1

throughout the support of t# . Then

(2.1)

f eikO(x)tp(x)dx < Ck('A)' A-I /k (IIqj .L00+ II174IIL1) Rn

with k = ial , and the constant ck(¢) is independent of A and t/i and remains bounded as long as the Ck+1 norm of 0 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 {xa}1a`=k is a basis for this space. However it is not difficult to see that there are d(k,n) unit vectors 1` (1), (2), , e (d(k,n)) so that the homogeneous polynomials ( (J) x)k , j - 1, , d(k,n) , give another basis. This means that if

0(x0I axa

>

for some lal = k , there is a unit vector e = ;(x°), so that I(e, px)kO(xO)l > ak, with ak> 0 . is bounded we Moreover since we can assume that the Ck+1 norm of can also conclude that I(e, p,)(k(x)l > ak/2 whenever x c B(xo), 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+1 norm of (b.) Next choose an

318

E. M. STEIN

appropriate covering of Rn by such balls of fixed radius, and a corresponding partition of unity, 1 = F 173(x), with 0 < ijj < 1 , {pnj{ < bk , and each rlj supported in one of our balls. So k

fe'Acj'dx=

feuic

feuicci7j dx

lij dx.

To estimate Jei-k'ipj dx , with a determined as above, choose a coordinate system so that x1 lies along C. Then

fe'1c1rjdx

xn)dx1) dx2,...,dxn

=J W

For the inner integral we invoke (1.2) giving us an estimate of the form -1 /kA-1 / ckak

a

f'

(x1,...,xn) dx1 L

J I1

f

(x1,...,xn) dx1 I

A final integration in the other variables then leads to (2.1).

REMARK. Let us note that in R2 if q(x) = x 1 x2 , the above proposition gives no better than a decrease of order 'k-1 /2 , while the asymptotics of the proposition below shows that the true order is 0 . Let us go back for the moment to the case of one dimension. If

95

has a critical point at x0, and q' does not vanish of infinite order at x0, then after a smooth change of variables (b can be transformed to a simple canonical form , with '(x) = ±xk (for x near 0 ). There is 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 4 has a critical point at xO. If the symmetric n x n matrix is invertible, then the critical point is said to be nonr7x (((

degenerate. It is an easy matter to see by the use of Taylor's expansion

that if xO is a non-degenerate critical point, then in fact it is an isolated critical point. PROPOSITION 6. Suppose q(xo) = 0, and (A has a non-generate critical point at x0. If i/r a Co and the support of q1 is a sufficiently small neighborhood of x°, then

e"(x)0(x)dx

(2.2)

A-n/2

ai X-j , as

A

j=o Rn

where the asymptotics hold in the same sense as (1.3), (1S).

Note. Again each of the constants aj appearing in the asymptotic expansion depends on only finite many values of derivatives of (A and 0 n / (-iµj)-1/21 O(x0), where at x0. Thus e.g. ao = (an/2 R i=I

1A1' µ2' "' µn are the eigenvalues of the matrix

a2(A(x0)

Similarly

2 axk 1

each of the bounds occurring in the error terms depend only on upper

bounds for finitely many derivatives of 0 and 0 in the support of Vi , the size of the support of

and a lower bound for det

a2(b(xo)

xk

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 xn, where 0 < m < n , with m fixed. Q(x) The analogue of (1.4) is 00

(2.3)

r eikQ(x)e Ix12xedx s%

Rn

ti

A-n/2-JeJ/2 5" c)(m,e)),-i

j=0

E. M. STEIN

320

with P =

a multi-index, Pl =

El

and

P

xn ; also note that if one Pi is odd then (2.3) is identically zero. To prove (2.3) write it as a product xP =

n

(f°°

('

eax2x_x2 J Pdx

J j=1

n

and expand the function

11

j=1

e

2

P x J dx

(1

-oo

-lh-P./2

(1/X+i)

(for large A ) in a power

series in 1/X . The analogue of (1.5) is the statement that

f

(2.4)

ei,\Q(x)xPrl(x) dx <

AX-n/2-IP!/2;

if r, a C0(Rn)

Rn

To prove it we consider the two-sided cones T i defined by Ci = { s Ixil2 > tsince Then

i

j=1

2n

Ix12 y, and the smaller cases hI _ {xi Ix)I2 >

1

J

= Rn we can find functions

n

Ix121

i = 1, , n, each

lZ , J

homogeneous of degree 0, and Co away from the origin, so that n

1=

fI.(x), x = 0, with ft. supported in F.. Then we can write

j=1

fe V Q(x)xPn(x)dx

fe'AQ(x)xP,r(x)SZ(x)dx

= i

In the cone C'i one uses integration by parts via pie"\Q(x) = eixQ(x) with Di(f) _- I Of NT T 1

7

This, together with the fact jxii > 1 Ixl in F., and I (tD.)Nj .(x)1 J2n

.

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

CN Ixl-2N

321

allows one to conclude the proof of (2.4) in analogy with that

of (1.5). A similar argument also show that whenever

e 8 and e vanishes

near the origin, then

(2.5)

1

e 4Q(x)e(x)dx

=

O(A-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 c (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 cb(xo) = 0, (V )(x°) = 0, and the critical point is assumed to be non-degenerate, there exists a diffeomorphism of a small neighborhood of x0 in the ym x-space to the y-space, under which cb is transformed 2 Y2. ym+1 yn' Observe that the index m is the same as that of the form corresponding to

1 .92`b(xO) 2 oPx j(?xk

-

References. For a proof of Morse's lemma see Milnor [201, §2.

D. Fourier transforms of surface-carried measures Let S denote a smooth m-dimensional sub-manifold of Rn (not

necessarily closed). We let do denote the measure on S induced by the Lebesgue measure on Rn, and fix a function 0 in Co (Rn) . Consider now the finite Bore] measure dµ = O(x)do 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 µ, i.e. dµ(e). 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 x0 be any point of S,

E. M. STEIN

322

and consider a rotation and translation of the underlying Rn so that the point xO is moved to the origin, and the tangent plane of S at xD becomes the hyperplane xn = 0. Then near the origin (i.e. near x0 ) the ,Xn_1) with ¢ e C*, surface S can be given as a graph xn = and 0(0) = 0, (%)(0) = 0. Now consider the (n-1) x (n-1) matrix 1

a2d?(o)

xk

_

are called the princi-

Its eigenvalues

/

1
pal curvatures of S at x0, and their product (= det 2 r°) is the Gaussian curvature at x . ` THEOREM 1. Suppose S is a smooth hypersurface in Rn, with nonvanishing Gaussian curvature at each point, and let dµ = tfida as above. Then (3.1)

AIfI-(n-1)/2

J(dµ)^(()I <_

.

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 tfi (and since when the surface is given as a graph we can reduce then da = 1+IVbl2 xn+1 = matters to showing that

f

(3.2)

<

AA-n/2

Rn

where 45(x,'1) = x181 +x2'22 +... + xnnn + (b(x1 ?".'xn)rln+1

,

with

n+1

det

(a)

iF1'1 =1, and A>0; also c'(0)=(42)(0)=0,

0, and the support of

tri

is a sufficiently small neighbor-

hood of the origin.

We divide consideration of the unit vector n into three cases.

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

20

when n is sufficiently close to the "north pole" nN = when n is sufficiently close to the "south pole" nN =

30

the complementary set on the unit sphere .

10

323

Let us consider 1° first. The function (¢(x, r,) has the property that (Vx(k)(0, 71N) = 0. We want to see that for each q sufficiently close to )IN there is a (unique) x _ x(,1) , so that (Vx) c(x(n), rl) = 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 ((VxpxqS) (0, RN)) 4 0; but this of course is our assumption of non-vanishing curvature. Notice that if the -q-neighborhood of 'IN

is sufficiently small, then det

(a2(J 10,77

l

4 0, so we can invoke

k

is suffiProposition 6 (with xe = x(rl) ), as long as the support of ciently small. This proves (3.2) when n is in region 1°. The proof when ri is in 2° is the same. So we now come to the region 30. Since c(x, r,) n

(rll,..., nn) + Howthen vxo(x, rl) _ ever ( 7 7 2+"'+ nn)1 /2 > c > 1 , since -q is in 3° and V (x) = 0(x) as x -, 0; thus IV I(x, q)j > c'> 0, if the support of 0- is a a sufficiently small neighborhood of the origin. Hence for the n in region 3° we may use Proposition 4 to conclude that the left-side of (3.2) is actually 0(11-N) for every N . The proof of Theorem 1 is therefore concluded.

i=1

l

4k(x)r/n+1 ,

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 dµ(e); its main term is explicitly expressible in terms of the Gaussian curvature at those points x c S , for which the normal is in the

direction 6 or -6. (2) We shall now consider the problem in a wider setting. Here S will be a smooth m-dimensional sub-manifold, with 1 < m < n-1 , and our assumptions on the non-vanishing curvature will be replaced by the more

E. M. STEIN

324

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 then write S as the image of mapping Rn, defined in a neighborhood U of the origin in R n. (To get a smoothly embedded S we should also suppose that-the vectors , ,

'

axI ax2

, NM

are linearly independent for each x, but we shall not need that assumpin Rn . tion.) Now fix any point x0 c U C Rm , and any unit vector does not vanish of We shall assume that the function infinite order as x --)x0. Put another way, for each x0 E U and each unit vector 17, there is a multi-index a , with 1 < lal , so that 71

(3)a (fi(x)

0. Notice that if (x, ii') are sufficiently close to

(x 0, rj), then also

(d)a(x.).,).I x=x

,

'0.

The smallest k so that for

each unit vector q then 3a , jal < k, with UJL

a

(O(x)

0

77)1

0 will

x=x

be called the type of (A at x0. Also if UI is a compact set in U,

the type of 0 in UI will be the least upper bound of the types for x0 in Ut . THEOREM 2. Suppose S is a smooth m-dimensional manifold in Rn of finite type. Let dµ = Vida, with t/i E Co(Rm). Then

du(e)l
(3.3)

,

for some e>0,

and in fact we can take e = 1/k, where k is the type of S inside the support of

t/i .

Proof. By a suitable partition of unity we can reduce the problem to showing that

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

J

ei (x)*

(x)dx =

325

0(IeI-I /k)

R`"

ti with (A as described above, and the support of 0- sufficiently small. Now we can write l; = AY), with Jill = , and X > 0 . Then we know that 11

there is an a , with tat < k, so that ()t2qs(x).?1

0, whenever x is

in the support of t (once the size of the support has been chosen small enough). Thus the conclusion (3.3) follows from (2.1) of Proposition 5.

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 < 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 and S is a sub-manifold of Rn (with some appropriate "curvature")

then there exists a p0 = p(S), p0 > 1 , so that every function in LP, 1 < p < p0 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

E. M. STEIN

326

1 /q

(4.1)

JSo

If(e)lgda(e)

< Ap,q(So) IIf tIp

holds for each f c 8, whenever So 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 0 and inequality (4.2)

e Co C. It will suffice to prove the

f

1 /2

o

AIIfII p

Rn

for PO = 2n + 32, and f e 5; the case 1 < p < p0 will then follow by

interpolation.* By covering the support of V1 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 Now with dµ = V'da we support o f Vi) as a graph: en have that

J )?(e)I2dµ = J f(e)f(()d, = J T(f)(x)13dx where (Tf) (x) _ (f * K) (x), with

K(x) =

fe

*In 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.

327

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

Thus (4.2) follows from Holder's inequality if we can show that (4.3)

IIT(f )Ilpo < AIIfIIpo

where po is the dual exponent to p0. To prove (4.3) we consider the function Ks (initially defined for Re(s) > 0 ) by

(4.4) Ks(x) =

e s2 1'(s,'2)

fe

1+s

2nix

)I

7

W(e')de

n(

Rn

by e'; we have set (C') =

Here we have abbreviated

+Iph(e')I2)I/2 , so that (C')dC'= dµ; also function which equals

ri

is a Co(R)

near the origin. Now the change of variables en -+ en + 0(') in the above integral 1

shows that it equals

('

e 2ni(x'-e1+xn0(e'))Yf(e')de'

s(xn)

= Cs(xn)K(x)

Rn-I with 00

2

Cs(xn)

1'(s/2) ./

e

2nix nenlfnl-l+s n(fn)den

-00

Now it is well known that 4s has an analytic continuation in s which is an entire function; also Co = 1 ; and I4 (xn)I < clxnl-Re(s), where Ixnl > 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

E. M. STEIN

328

(a) KO(x) = K(x) ,

(b) IK-n/2+it(x)I < A, all x e R, all real (c) IK1+it(e)I < A, all

t

£ Rn, 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 T5(f) _ f * Ks' From (b) one has IIT_n/2+it(f )IIL00 < AIIf1ILI , all real t

(4.5)

and from (c) and Plancherel's theorem one gets (4.6)

IITI+it(f )IIL2 < AIIfIIL2

,

all real t

,

An application of a known convexity property of operators (see [281) then shows that IIT0(f )II LP°, < AIIfJJ PO , with PO0 = 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 [121 has shown that then the corresponding results hold with 1 < p < 2k 2 , 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 = 2 alluded to above. For further details one should consult the references cited below.

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

329

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 po = po(S), 1 < po, so that S has the LP restriction property (4.1) with q = 2 , and 1 < p < po. (In fact if

the type of S is k, we can take po = 2nk/(2nk-1 .) COROLLARY. Suppose S is real analytic and does not lie in any affine hyperplane. Then S has the LP restriction property for 1 < p < po, for some po > 1 .

Proof. As we saw above, it suffices to prove (4.3). However Tf = f *K, and K(x) = dµ(-x), Theorem 2 tells us that IK(x)l < AIxI-11k, So according to the theorem of fractional integration, (see [26], Chapter V), where a=n-1/k, and this we therefore get (4.3) with __ = P -n,

0

0

relation among exponents is the same as PO = 2nknk1

'

Q.E.D.

Further bibliographic 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].

Oscillatory integrals of the first kind related to singular integrals A key oscillatory integral used in the theory of Hilbert transforms along curves is the following: 5.

00

(5.1)

P.V.

Jr

elpa(t) dt t,

E. M. STEIN

330

d

where Pa(t) is a real polynomial in t of degree d, Pa(t) = F a.O. j=o

It

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 ao,al,...,ad. The relevance of such integrals can be better understood by consulting Wainger's lectures [32]. We shall be interested 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)I < AIxI-n (i.e. K is bounded on the unit sphere); moreover, we assume the usual cancellation property: f x,l-1 K(x') d a (x') = 0. We let P(x) _

I aaxa be any real polynomial of degree d. Ial
THEOREM 5:

P.V.

(5.2)

f

eiP(x)K(x)dx < Ad

Rn

with the bound Ad that depends only on K and d, and not on the coefficients as . Nagel and Wainger observed that if K were odd, one could prove (5.2) from the one-dimensional 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) = F a tJ denote a real polynomial on R1 , and write also j=1

d

Pb(t) = F j=1

j

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

331

LEMMA 1:

Jipa(t)

(e

(5.3)

-e

1Pb(t) dt ) t

d

log`-II,


e

with Ad independent of (aj f ,

I bj {

and e > 0.

F aaxa be a homogeneous polynomial of

LEMMA 2. Let P(x) =

Iaj=d

degree d on R'. Write mp

=f

IP(x')I da(x')

.

Ix k=1

Then,

f

(5.4)

Ilog

(I P 1.)) da(x') < B

,

Ix'I=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 conse-

quence 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 jxj 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 in writing (5.4) we assume that P is not identically zero, i.e. mp > 0.

E. M. STEIN

332

THEOREM 6. Let P(x) be any polynomial of degree < d in Rn logIP(x)I is in B.M.O. and in addition

.

Then

11 log IP(x)1 IIBMO < Bd

where Ba 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 00

5

(eiat_eibt) dt
t

)

E

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 bt , 5 ;E 0. Thus we may assume that bd = 1 and Iadi < 1 . Now write 00

r (e1Pa(t)

- e iPb(t))

dt

J

=

5+ r J

t E

E

00

1

and we treat these two integrals separately. (If e'> 1 , we have only f"0

and that integral is estimate like

f00

.) Let us consider the second

integral. It equals 00

00

f eiPa(t) dt _ (' eiPb(t) dt J t t

f

Now since bd = 1, we see that (d/dt)d Pb(t) = d!, and hence

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

333

(' eipb(t) dt
1

by the corollary of Proposition 2 in §1. Next, by a change of variables t lad-1 /d t , the integral 00

t 1

becomes 00

e iVa(t)

dt

1

=

ladll/d

f +f ladll/d 1

where Pa(t) is a polynomial of degree d , with td having coefficient one. Again e"Va(t) dt t


while 1

<

r

dt_1

t d log l(Tl=a to

J

1

ladl1/d

since bd = 1 . Next

f(e Pa(t)

IPb(t)

i

E

a

d_ )t-J E

1

(e

)dtt+ 0('dt J t,

1Qa(t) e 'Qb(t)

E

E. M. STEIN

334 with d-1

d-1

Qa(t) = I ajtj ,

4 bjti

Qb(t) =

J=1

j=1

since IPa(t)-Qa(t)I < Iti and IPb(t)-Qb(t)I < Iti . However, by induction hypothesis (using (5.3) for E' = E , and E =1, and d - 1 ), 1

r (elQa(t) _ e iQb(t)

J

dt

Ad-1

t

Ilog

1+ J=1

E

Gathering all these terms together then proves (5.3). Armed with Lemmas 1 and 2 we can now prove Theorem 5. We may assume that P(x) = F axaa has no constant term, and using polar IaI
d

coordinates x = tx', t > 0, Ix'I = 1, we write P(x)

j=1

P-(x')ti

,

j

where Pj(x') are restrictions to the unit sphere of homogeneous polynomials of degree j . Let us also set mj = f x'I-I I Pj(x')I dv(x) , and write K(x) = t-n1Z(x'), with 1 bounded and f

x'I-11Z(x')dv(x') = 0.

Then to prove (5.2) it suffices to show that

el1(x)K(x)dx

with Ad independent of written as

('

J

E1 ,

E2 and P. The above integral can be

(fE2 e

Ix l=1


1FPj(x')tj

SZ(x")do(x')

E1

Since 0 has vanishing mean-value this integral may be rewritten as

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

f f

E

335

2

Ix'I=1 \ E1

However by Lemma 1 the inner integral is bounded by d +

Aa 1

m)

and so an appeal to Lemma 2 shows that fE2

exp (i

f Ix'1=1

mit) dt (sup IH(x')I)da(x') < Ad

Pj(x')ti) - exp (i

EI

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 Cm x R, i.e. Hm = #(z,t){, with z e Cm, t e R; the multiplication here is (z,t)(w,s) _ M

(z+w, t+s+), where = 2 Im II zjwj, with z = (zj), w = (wj). 1

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,p2t), and the appropriate quasi-distance (from the origin) is then (Izl4+t2)1 /4. 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 Cn+1 . This point of view, as well as related generalizations, is elaborated in Nagel's lectures [211.

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 R2m+1 . However the two conflicting types of dilations (and related metrics) occur in e.g. the solutions of Ju = f . (One sees this for example in Krantz's lectures [17], where in the formula of Henkin we have a kernel made of products of functions each belonging 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 (6.1)

Tf =f*K

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)S(t) .

L(z) is a standard Calder6n-Zygmund kernel in Cm = R2m, i.e. L(pz) _ p-2m L(z), L is smooth away from the origin, and L has vanishing mean value on the unit sphere. Here 6(t) is the Dirac delta function in the t-variable, and in an obvious sense is homogeneous 3(pt) = p-1 S(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 L2(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 TA defined by

*In particular the terms AI and A2 that appear in §6 of [17].

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

(6.3)

337

(TA)(F)(z) = i L(z-w)e1AF(w)dw Cm

(with the anti-symmetric form which occurs in the multiplication law for the Heisenberg group) is bounded on L2(Cm) to itself, uniformly

in A, -DO«t<

,

>_

B( , ) , and F = f. Then the operators TA have the form

(6.4)

(Tf)(x) =

K(x-y)eiB(x,Y)f(y)dy

I

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 -µ, 0 < g < n, smooth away from the origin, and with vanishing mean-value when p = n. (a) If B is non-degenerate, then the operator T given by (6.4) is

bounded on L2(Rn) to itself, for 0 < p < n ; when 1 < p < oo , the operator is bounded on Lp(Rn) to itself if -11 I < 2n I1

.

(b) If we drop the assumption that B is non-degenerate but require that

g = n, then T is bounded on LP(Rn) to itself for 1 < p < -o. 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 con-

sider first the L2 part of assertion (a) when n/2 < g < n. Suppose

ri

*For further details see Mauceri, Picardello and Ricci [19] and Geller and Stein [10].

E. M. STEIN

338

is a Co function, with 77(x)=l for IxI < 1/2, and ii(x) = 0, for 1x1 > 1 .

To is defined as in (6.4), but

We write T =

with K replaced by Ko = riK, and T with K replaced by K = (1-77)K. Observe first that since Ko(x-y) is supported where Ix-yl < 1 , estimating T0(f)(x) in the ball IxI < 1 involves only f(y) in the ball IYI < 2. We claim

5Ixl
(6.5)

ITo(f)(x)12dx < A

f

If(Y)I2dy .

IYI<_2

In fact when IxI < 1 ,

fKo(x_y)ei'cf(y)dy -J


Ko(x-Y)eis(y,Y)f(y)dy

Ix-YI-'`+'If(Y)IdY

,

Ix-YI
and thus the L2 theory for f

Ko (which is non-trivial only when

µ = n ) proves (6.5). While operators of the type (6.4) are not translation invariant they do

satisfy r_hTrhf = e1B(h,h)e1B(x,h)T(eiB(h*)f( ))

(6.6)

with rh(f)(x) = f(x-h). Applying this to To gives the following generalization of (6.5)

J Ix-hi
ITo(f)(x)I2dx < A

r Iy-hI<2

If(Y)I2dY

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

339

and an integration in h shows that as a consequence

f IT0(f)(x)I2dx < A2n J

If(y)12dy

Rn

Rn

We now turn to the proof of

J IT f(x)I2dx < A Rn

f

If(x)I2dx

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) = J e-iB(z,x-Y)1c(z-x)K.(z-y)dz

Now since K. is in L2(Rn) (here the assumption n/2 < µ is used), Schwarz's inequality implies IL(x,y)I < A .

that

We next integrate by parts in the definition of L(x,y), using the fact (Dz)Ne-iB(z,x-y) = e-iB(z,x-Y), where Dz = i(a,Vz)/Ix-yI , with

a = B-I

(Ix yI ` x-

,

and B denotes the matrix so that B(x,y) _ (Bx,y).

The result is IL(x,y)I < ANIx-yI-N, for every N > 0, and hence (6.7)

IL(x,y)I < AN(1 + Ix-yI)-N

,

N>0.

This shows that L is the kernel of a bounded operator on L2 proving the boundedness of T ,T and thus of T.. The proofs of the L2 boundedness when 0 < µ < n/2 (in part (a) of the theorem), and the L2 boundedness when µ = n but when B is not assumed to be nondegenerate, are refinements of the above argument.

E. M. STEIN

340

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 H1 ) 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 IeQ(x)I =

,

Q(x),

all x

where XQ 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) Ia(x)I < 1/IQI (iii) ,f a(x) Q(x) dx = 0

The space HE is then given by If if = I Ajaj , with each aj an E atom, and I IAil < ool. In a similar vein the function fE will be defined as

(6.8)

fE(x)

su IQI

I

!f -fQldx

,

Q

where

fQ

IQI

f

f U_Q

Q

and we take BMOE = IfIfE E L°°I .

Some of the basic facts about the standard H1 and BMO spaces* go through for HE and BMOE, and sometimes these come free of charge.

One such case is the following assertion: Suppose f c Lpo , 1 < p < p0 < oo, and fE f LP. Then f c LP and *The standard situation arises of course when eQ = yQ, all Q.

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

Lp

OilLp

(6.9)

341

To prove this we need only observe that (if I)# < 2f0 , and use the result (see [10]) for the standard # function. The point of all of this is that for operators of the form (6.4), there is a naturally associated HE and BMOE theory, and it is given by choosing eQ(x) = e

(6.10)

-iB(x,cQ) ,

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 u = n our operator T maps L°° to BMOE . Let us give the proof in the case (a). We may assume that ilf 11 °° < 1 , and

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)

1

Q,

f

JTf -yQ dx < A

Q

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 [91), since we must split f into three parts to take into account the oscillations of e1B(x,y) . Suppose Q = QS , has side-lenghts S , then write

f = fl+f2+f3, where

Q28' fl = 0 elsewhere, CQ25 n QS-1

,

f2 = 0 elsewhere,

SQ2S n cQ 5-1 , f3 = 0 elsewhere.

E. M. STEIN

342

(Note that f2 occurs only when S < x/2/2 .) We have F = T(f) =

F1+F2+ F3, where FJ =T(fj). For F1 we make the usual estimate, using the fact that T is bounded on L2. Next observe that IK(x-y)eis(x,Y)- K(-Y)I < cS

if

1

1

+

IYIn+1

IYIn-1

'

x cQs and y e'-Q2s. Thus if yQ = f K(-y)f2(y)dy, we get that for

xCQ3

IF2(x)-yQI
f

dy

dy

+S

l
IYI°-1

IYIn+1

Qs-1

Q2S

Finally

K(x-Y)eiB(x,Y)f3(Y)dY

F3(x) = I

= J (K(x-Y)-K(-Y))eiB(x,Y)f3(Y)dY + J K(-Y)f3(Y)eiB(x,Y)dY

For F3 we again make the standard estimates, and for F3 we use Plancherel's formula (which we may since we have assumed that B(x,y) is non-degenerate). The result is

5 IF(x)I2dx < J IF3(x)I2dx = A J 3 Q

Rn

Rn

den

IK(-Y)f3(Y)I2dy < Al J

IYI

cQ S-1

Combining these estimates proves (6.11), and hence the fact that T takes L°° to BMOE .

=

A5"-

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

343

We shall now state a generalization of this theorem which also includes the oscillatory integral result given in Theorem 5 (in §5). Suppose P(x,y) is a real polynomial on Rn xRn of total degree d . Consider the operator (6.12)

(Tf) (x) = p.v.

I

e'p(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 T given by (6.12) is bounded on L2(Rn) to 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
is homogeneous of degree - g, n - E < µ < 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) + Pl(y). (iii) One can replace K(x-y) in (6.12) by a more general "Calder6nZygmund kernel" K(x,y), a distribution for which the operator when P e 0 is bounded in L2 , and which in addition is a function (when x y ) which satisfies IK(x,y)l < Alx-yl-n, lpxK(x,y)l + IoyK(x,y)I <

Alx-yl-n-I

References. For the detailed proof of Theorem 7, other variants, and applications to the a Neumann problem see the papers of Phong and the author [22], [231.

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

fei(x)du(x) =

0(I6I-(n-I)/2)

as e

00

S

whenever 1 e Co , 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 Co function l by an Lr 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 L2(da) to L p (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. We fix a curve t -Y(t) = (t, y2(t)) , 0< t < 1, lying in R2 , with y(t) a C2 , and having non-vanishing curvature, i.e. Iy 2(t)I > c > 0. Con-

sider the transformation T, which maps function on the interval [0,1] to functions on R2 , given by

(7.1)

(Tf)( ) - f e4'y(t)f(t)dt

.

0

THEOREM 9. Under the assumption above T is bounded from Lp[0,1]

to Lq(R2), whenever 3/q + 1/p = 1 and 1 < p < 4. (Note that when p -. 4

and q .)

,

then q -. 4 in the above relation between p

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

345

r(7.2) f

Proof. Write

1

F() = ((Tf)( ))2

0

J0

e e-(Y(s)+Y(t))f(s)f(t)ds dt ,

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 , which divides [0,11 x [0,1] into the union of two regions R1 and R2 . We then consider the mapping of R1 R2 given by x = y(s) + y(t), i.e. x1 = S + t , X2 = y2(s) + y2(t). It is easy to verify on the basis of our assump-

tions that this mapping is one-one, and its Jacobian j satisfies IJI = Iy2(s)-y2(t)I > cls-tI . Therefore

(7.3)

r e'6'(Y(s)+y(t))f(s)f(t)dsdt

=

f e'C'xf(x1,x2)dxldx2

R1

R2

with f(x1,x2) = f(s)f(t)1J1_1

the quantity appearing in (7.3) then when-

So if we denote by

ever 1 < r < 2 , and 1/r'+ 1/r = 1 , we know that (7.4)

IIF111Lr(R2) <- cOf1ILr(R2)

However

IIfIILr(R2)

5f12t12 =

f


If(t)Ir If(t)I`IJI I-r ds dt

f

If(t)Ir If(t)Ir Is-tl I-rds dt

346

E. M. STEIN

To estimate the last integral we need to invoke the theorem of fractional integration in one dimension in the form

fg(s)g(t)(st)_1+adsdt < AIIgflu , u - 1 = a, 0 < a < 1

.

So we take g(t) = If(t)Ir, then Ilgllu = IIfIIp when p = ur. Then if we fix

a so that -1 +a =1-r, then

3r

2 rr = u , and

The limitation

0 < a becomes r < 2, and with q = 2r' we obtain from (7.4) that 1

X2/P

IIFiIILr (R2) < c' 0

with a similar estimate for F2(e) which is the analogue of (7.4), but taken over R2 . Since F = FI +F2 and F = (Tf )2 we obtain IIT(f )IILq(R2) <- Ai1fIILp(0,ii

Note that Q =_i=32r3=1-P , so 1 < r < 2 is equivalent with 1 < p = 3

q r

+P =1, and the limitation < 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 L2 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 A , A > 0, defined by

TA(f)(e) = J Rn

ei1'(x,4)&(x,e)f(x)dx

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

347

where 0 is a fixed Co (Rn x Rn) cut-off function; (D is a real-valued C°° phase function which we assume satisfies the assumption that its Hessian is non-vanishing, i.e.

axi°j det a2L(x, e)

(7.6)

0

PROPOSITION: (7.7)

[ITA(f)IIL2(Rn) <

A)C-n/2IIfJIL2(R')

.

COROLLARY: (7.8)

IITA(f )II

Lp (Rn)

< AAn/P IIfff

LP(Rn)

,

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 - oc. This decrease is consistent with the special case when t(x,6) 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 the M. 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 AK-n. T*TA is bounded by Now this operator has as its kernel the function KA(e,n) given by

(7.9)

J Rn

348

E. M. STEIN

Now since

L

(a,Vx)[c(x,n)-F(x,e)] =l _ _ a,17-e'\ + OIn-ej2 OXOV

we can find a = (a,,...,an), so that the aj depend smoothly on x and (a,px). IA(x,e,n) >- cIe-nI on the support of KA(e,n). Set Dx = 0 = ei\(((x,n)-D(x,e)) , we can Then since integrate by parts N times in (7.9) and obtain (Dx)NeA(t(x,n}-'P(x,6))

(7.10)

IKA(e,n)I < AN(1

+ale-nI)-N , N > 0 .

It follows from (7.10) with N = n+1 , that the operator TX*TA which has kernel KA has a norm bounded by AKn and the proposition is proved. We shall now formulate some theorems for oscillatory integrals of the form (7.11)

ei

(TAf)(e) = J

(t.4)c&(t,e)f(t)dt,

6 E Rn

Rn-I 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 cF are as follows: We consider for each fixed (t°,e 0) the associated bilinear form B(u,v) defined by B(u,v) = (v,Vt) (u, Ve) (c) (to,4 0), with u E v e Rn . Our first assumption is that (7.12a)

B is of rank n - 1

Rn-1

.

Thus there exists (an essentially unique), u E R n, IuI = 1 , so that the scalar function t - (u,V,(D(t,e)) has a critical point at (t°,e°). We shall also assume that this critical point is non-degenerate, i.e. we suppose the non-vanishing of the (n-1)x(n-1) determinant:

349

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

det (pt (G,Oe1)) (t°, e 0) - 0

(7.12b)

These assumptions will be supposed to hold at all (t°,6 °) in the support of qi(t,e), where i/i is a fixed function in Cc (Rn-1 xRn). THEOREM 10. Under the assumptions above the operator (7.11) satisfies IITx(f )IIq < Ak n/gllf IIp

(7.13)

with q = n-1 p',

(n+1)

p +p 1 1,

if 13, if 1
REMARKS:

(1) When

(D(t,6)=t161+t262+...+tn-1en-1+c(ti,...,tn-1).en

,

and

(V)(0) = 0, then the conditions (7.12) are near the origin equivalent with the non-vanishing Gaussian curvature of the graph tn 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 Hormander [15]. Since part (b) has not appeared before, we will outline its proof. This will also serve as a good review of many of 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)

LIT, (F)IIL2(Rn-1) < AX n/r'IIFIILT(Rn)

350

E. M. STEIN

with r = 2(n+

,

where

(Ta)(F)(t) = J e-iAD(t14)0(t,e)F(e)de, t e

Rn-1

Rn We can calculate

J

TX*(F)Tj*(F)dt

Rn-I and write as

KA(6,r!)F(e)F(n)d6drl

J RnxRn

with

(7.15)

KA(e,r1) = J

e'

Rn-1

It suffices therefore to see that K,\ is the kernel of a bounded operator from Lr(Rn) to Lr(Rn), with norm not exceeding AA 2n/r' Because of our assumptions on 4) we can construct a phase function on Rn x Rn so that the following holds: we will write x e Rn, as Rn-1 . we can construct will The (t,xn) with t = satisfy: (i) $(x,e)_O(t,e)+(Dp(6)xn

(ii) the determinant of the nxn matrix pxVe$ is non-vanishing.

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

351

In fact Vtpe(b already has rank n-1 by assumption (7.12a), and so we need only choose (Do(e) so that (u,V,)Io(e) 0 to increase the rank of px e to n . Now, as in the proof of Theorem 3 in §4, we form Ka defined by es2

f eia( (x,n

I'(s/2) J Rn

with dx = dtdxn , and where v is a Co function which equals

1

near

the origin. We easily verify (7.16)

since when x = (t, 0) .

(V(x,

Next (7.17)

Kx+it is the kernel of a bounded operator from L2(Rn) to itself with norm < AX -n/2.

This follows by applying the estimate (7.7) of the proposition above and

using the non-degeneracy of the Hessian of $(x,e). Finally we claim that (7.18)

I K-n /2+1 /2+it(e ,l)I < A A

To see this write KAs(6,rq) as KA(6,r?)v(k(4)o(rl)-(Do(6))) where 00

2

vs(u) = F(ss/2)

r eixnuv(xn)+xnl-1+sdxn -Do

.

352

E. M. STEIN

Then since Iv_n/2+1/2+it(u)1-< cluln/2-1/2 , as u

00 we see that to

prove (7.18) it suffices to show that (7.19)

IKX(e,q)I


In 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 = to , (for otherwise we can write it as the sum of

finitely many such terms). When we write I(t,ri) = I(t,e) = (VeO)(t,77) ('i-e) + 0(77-e)2 we see that these are two cases to consider as in the

proof of Theorem 1 in §3: 1° when the directions n - 6 or e - n 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). In the second case, we actually get 0(AI,7-e I)-N , for every N > 0 as an estimate, by Proposition 4. This completes the proof of (7.18), and shows that Kin/2+1 /2+it is the kernel of a bounded operator from L1(Rn) to L°°(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 §5. First let `f'd denote the linear space of polynomials in Rn of degree < d . We claim that there is a constant Ad , so that 1 /2

(8.1)

IQI J (1 Q

IP(x)I2 dx

< Ad

IQI

f IP(x)I dx

holds for all P c 'd , and all cubes Q. The space Td is invariant under translations and dilations, and so a moment's reflection shows that to prove (8.1) for all P t Td , it suffices to prove it for Q = Q0 , the unit cube centered at the origin. However

OSCILLATORY INTEGRALS IN FOURIER ANALYSIS

(f

IP(x)I2 dx)1

2 and f

Q0

353

IP(x)ldx are two (equivalent) norms on the

Q0

finite-dimensional space 5d , so (8.1) holds for Q = Q0, 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 A.. Examining the proof of this fact one obtains an r = r(d), 0 < r < co, and a constant Cd , so that (8.2)

(7Q,1 f IP(x)I dx

f

(fQ_o I

Q

1Jr

IP(x)I"dx

<< Cd

for all 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 = fIXI_1 I P(x)l da(x) = 1 , we can conclude that

f

(8.3)

I P(x)I-r da(x) < ca

.

IXI=1

However, when u > 0, Ilog ul = log+u + log+u < u + u -r. Therefore (8.3) implies (5.4) whenever mp = 1 , and so that result also holds in

i

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.

3S4 [3]

[4] [5]

[6]

E. M. STEIN

L. Carleson and P. Sjolin, "Oscillatory integrals and a multiplier problem for the disc," Studia Math. 44(1972), 287-299. M. Christ, "On the restriction of the Fourier transform to curves," Trans. Amer. Math. Soc., 287(1985), 223-238. R. Coifman and Y. Meyer, in these proceedings. R. Coifman and C. Fefferman, "Weighted norm inequalities for maximal functions and singular integrals," Studia Math. 51 (1979), 241-250.

[7]

[8] [9]

S. Drury, "Restrictions of Fourier transforms to curves," preprint. A. Erdelyi, "Asymptotics Expansions," 1956, Dover Publication. C. Fefferman, "Inequalities for strongly singular convolution operators," Acta Math. 124 (1970), 9-36.

[10] C. Fefferman and E.M. Stein, "HU spaces of several variables," Acta Math. 129 (1972), 137-193.

[11] D. Geller and E. M. Stein, "Estimates for singular 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 Integrale auf konvexen Korper. I," Monatsh. Math. 54 (1950), 1-36.

[15] L. Hormander, "Oscillatory integrals and multipliers on FLU," Ark. Mat. 11 (1973), 1-11.

, "The analysis of linear partial differential operators. I,"

[16]

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 #51, 1963, Princeton University Press.

[21] A. Nagel, "Vector fields and nonisotropic metrics," in these proceedings.

OSCILLATORY INTEGRALS 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.

, "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 of a convex set," Trans. Amer. Math. Soc. 139(1%9), 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. [29] R. S. Strichartz, "Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations," Duke Math. J. [23]

44 (1977), 705-713.

[30] P.A. Tomas, A restriction theorem for the Fourier transform," Bull. A.M.S. 81 (1975), 477-478.

, "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

[31]

sets," in these proceedings.

[33] 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(l) 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

I f f(x-y)dy

f(x) = limo IQ r

a.e.

(where Qr is the square, Q. =jxcRnIsupIxil
f(x) = lim 1

r

r-+0 JBr) J

f(x-y)dt

Br

(where B. is the ball, Br = {xI lxI
*Supported in part by a grant from the National Science Foundation.

357

a.e.,

358

STEPHEN WAINGER

Our first problems are the following: Problem IA: Does

1)

f(x-y)dar(y) = f(x)

lim

a.e.?

r-+0

aQr

Here aQr denotes the boundary of Q. and dar is n-1 dimensional Lebesque measure on aQr normalized so that dar(aQr) = 1 . Problem IB:

Does

lim J

2)

r-O

f(x-y)dp = f(x)

a.e.?

aBr

Here di 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-, L2,

or LI. In questions IA and IB, we are considering certain averages

3)

MQ f(x) = r

r

,J

f(x-y)dar(y)

aQr and

4)

MB

r

f(x) = J

f(x-y)dkr(y)

aBr

We are asking if 5)

MQr f(x)

f(x)

a.e.

AVERAGES AND SINGULAR INTEGRALS

359

MBrf(X) - f(x)

a.e.

and 6)

The standard approach to this type of problem involves consideration of appropriate maximal functions. We define the maximal functions )lIQf(x) = sup MQ (If1)(x) r>0 r

7)

and

8)

)RBf(x) = sup MB (IfI)(x) r r> O

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 < p < °o , if we could show 9fIB

9)

11XIM Qf(x)>AlI <--9 1'fI'p

for every f in LP, and 2) would follow if we could show 10)

I1xI)RBf(x)>x1I <-L IIfIIP

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°° functions. In our case this means the following;

Let E be a measurable set and XE its characteristic function. Then if (11)

I1xIRQXE(x)>a1I < C(a) I E I

360

STEPHEN WAINGER

where C(a) may depend on A but not on E, then 1) holds for every f in L°°. If 12)

I1xI911BXE(x)>a1I < C(A)JEJ

then 2) holds for every f in

,

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
[f(x+r)+f(x-r)]

MQrf(x) = MBrf(x) = 2

So if we take f(x) to be log L near x = 0, have compact support, Ix

and be in C°° away from the origin, we would have a function f in every LP class such that )IIQf(x) =)IIBf(x) = oo for every x . We can also see that 11) and 12) are false in one dimension. We just take EE = Ix 10<x<E} .

Then

J EEl

0 but

)IlQf(x) = 911Bf(x) > 1

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 [SI] of 9), 10), 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 < p < 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°° functions. See [SW]. Thus we obtain negative results in one dimension. Similar reasoning gives the same negative conclusion for question IA in any number of dimensions.

361

AVERAGES AND SINGULAR INTEGRALS

One need only consider

f

is as

above and h is a nice function. So there are no interesting positive results in problem IA. However as we shall see later there are positive results for problem IB 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 boundary 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 of 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 h J f(x-y(t))dt = f(x) a.e., for

f in

L'° or L2 or LI ?

Problem III: Let v(x) be a smooth vector field in Rn. Does h

lim

1

h-'0 h _

1

f(x-tv(x))dt = f(x)

a.e.

0

for f in

L°° or L2 or LI ?

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 a

13)

Hyf(x) =

J-a

f (x-y(t)) Lt .

(where sometimes we wish to think of a as finite and sometimes as and

00 ),

362

STEPHEN WAINGER

1

14)

Hvf(x)

f(x-tv(x)) dt

-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)

IIHyf 11

LP

< CPIIIIILP

for some p's ? Problem III': Can we have an estimate 16)

IIHyfIILp <_ CpIIfilLP

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

Tf = K*f

where K is a kernel having "standard homogeneity" that is 17)

K(Ax) = A-nK(x)

h>0

AVERAGES AND SINGULAR INTEGRALS

363

(K(x) is a function on Rn ) to the one-dimensional Hilbert transform

ff(x-t) dt

Hf(x) =

(f a function on R1 ). We will explain how the method of rotations can lead to problem II'.

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 K(Ax.X2y) = 3 K(x,y)

18)

X3

We wish to consider the LP boundedness of the transformation 00

00

Tf(u,v) =

19)

f(u-x,v-y)K(x,y)dx dy .

I

We now introduce parabolic polar coordinates into 19) x = rcos 0 y = r2sinO and find 00

20)

277

Tf(u,v) = J

J

0

0

f(u-rcos 0, v-resin 0)

K(rcos 0, resin 0)r2N(0)drd0

where r2N(0) is the Jacobian factor in the change of variables. N(0) is smooth and N(0+n) = N(0). By 18 we see that

364

STEPHEN WAINGER o0

27r

r

21) Tf(u,v) =

N(O)K(cos d,sin O)dO

0

r C f(u-rcos 0,v-rsin 0) dr 0

2rr

00

-J

N(O)K(cos(O+n),sin(O+n))dd f

0

0

=

i f(u-rcos 0, v-rsin(O)) dr

since K is odd. Thus 217

r

fTf(u,v)

i f(u+rcos O,v+rsin 0) dr

N(O)K(cos 6,s in 0) dd

0

0

since

N(6+n) = N(6) .

Finally 277

r

21A) Tf(u,v) =

00

N(O)K(cost,sinO)dO

r r f(u-rcos6;v-r2sinO)dr

Now adding 21) and 21A) we find that 2n

f

Tf(u,v) = 2

ao

N(O)K(cos O,sin 0)dd f -00

0

i (u-rcos6,v-r2sin6)dr f

If K(cos O,sin 0) is in LT of [0,2n] , we can apply Minkowski's inequality n

IITfIILp < C f de IIHIILp 0

where

AVERAGES AND SINGULAR INTEGRALS

Hef =

f

365

00

f(x-yg(r))

dt

,

-00

with

ye(r) = (rcos O,r2sin O) .

Now we prove IITfIILp
by showing IIHefIILP
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

MEf(x,Y) =

E2

ff

f(x-r,Y-s)

1

drd s

(is)(i+ E2

(") If K(x,y) were dominated by a decreasing, radial, LI function, the classical theory would imply lim MEf(x,y) = f(x,y) E-0

see [SWE]. However K(x,Y) =

1

(1+x2)(1+y2)

a. e.

366

STEPHEN WAINGER

so the smallest radial majorant of K is

1

which is not integrable.

1+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 Cn. 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 B1 and B2 of the same radius, r, intersect, then B2 , the ball having the same

center as B2 but having radius 3r contains B1 . To see how badly this property fails for our problems let us suppose we were considering averages h

1

ff 0

t2+E

f(x-r,y-s) dr ds

t2

over slightly thickened parabolas or balls BE,h = i(r,s)I0
.

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 r ).

We can also see the difficulty of using the Calderon-Zygmund theory to study Problem 11'.

Suppose y(t) is the parabola (t,t2) in R2 and

Tf(x,y) = J f(x-t,y-t2)dt Then we may formally write

Tf(x,y) =

fff(x-t,y-st2)-

= JJ

dsdt

f(x-t,y-v) . S (1- -1 dv dt

0)

Or

22)

Tf=K*f,

where 22A)

K(x,Y)=XS(1--X

),

368

STEPHEN WAINGER

The Calderon-Zygmund theory deals with convolution operators with

kernels K(x,y), but in their theory K(x+h,y+k)- K(x,y) should be much

less than 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 = x2 and (x+h,y+k) is not 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. The Calder6n-Zygmund Theory is based on 4) tools a) The Fourier transform (The Fourier transform is even used in the L1 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 Calder6n-Zygmund decomposition. Some progress in finding covering lemmas for related problems was made by Stromberg [Str] and [STRO] and Cordoba [COR1], [COR2], Cordoba and Fefferman [CF1], [CF2], [CF3], 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], [CS], 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 ,,

r 0

O(t,O)dt .

AVERAGES AND SINGULAR INTEGRALS

369

u is supported on a straight line, namely the x-axis, and 1

g(e'exel)?y)

=

r eietdt 0

which is independent of n and hence cannot decay at infinity along the Y7-axis. Now let us consider a measure supported on a parabola, 00

23)

v(q) =

r

e_t2rb(t,t2)dt

.

_ Then

v(e exei7y) 00

fe_t2eietet2dt. This integral may be computed exactly by completing the square, and it

is easy to see that e2

Ce 1+1771

Thus v(e,q) tends to zero at infinity. Another example is afforded by rotationally invariant Lebesgue measure on the n-1 dimensional sphere Ixl = 1 in Rn. If we denote

this measure by dµ, we have for e c Rn

= CnJ(lei) Ii

2

(v).

370

STEPHEN WAINGER

See [SWE]. Thus

^

_

Idu(4)I < Cn(1 + I.I)

(n

1

2

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.

VAN DER CORPUT'S LEMMA. Let h(t) be a real function. For some j , assume Jh(1)(t)I > A in an interval a < t < b. If j = 1, assume also that h'(t) is monotone, then b

r

exp (ih(t))dt

a

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

24)

du(O)

f

2

(k(t,t2)dt.

1

Then

2

el4telnt dt

du(e,i) = 1

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 t4 rt2 . First we use the fact that h"(t) = rt . Thus by Van Der Corput's lemma with j = 2 , we see

AVERAGES AND SINGULAR INTEGRALS

n

ldu(, ,

25)

C

1

(1 + Inl)I12

Now if Ir1I < igl,

Ih'(t)I = Ie+2,itI > e/8

.

So by Van Der Corput's lemma with j = 1 , 26)

n .I

I

if 161 > 8lnl

Putting 25) and 26) together we have 27)

C

for some 8 > 0. Stein pointed out in retrospect that we can already see from an estimate like 27) that du has interesting properties from the point of harmonic analysis - namely even though du is singular,

Tf=du*f maps LP into L2 continuously for some p < 2. For

371

STEPHEN WAINGER

372

f(Tf)2

=

=

ITf(6,r/)12 dtd77

f

Idu(e,17)12If(e, )I2 de d,1

=f

If( ,ii)I 2

+e 2 +1771)2,5

(1+e 2+I17I If(e,,7)12

(1+e2+1,,127

2q ) /q

J

If(e,rl)I

1 /q

(Ll+e2+'2')

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 e LP where

P +2q=1 II. The Hilbert transform along 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 of as a multiplier transformation Hyf(e) = my(e) f (e )

28)

where

29)

my(e) =

f

Lt

_,p

To see that 29) is true we may either substitute the formula

AVERAGES AND SINGULAR INTEGRALS

f(x)

373

je-e'x f(e)dC

into 13) or recognize the fact that

where D is a distribution

So

and b 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 L2 in 2-dimensions for the curve y(t) = (t, ltlasgnt) ,

a > 0.

So Fabes' proof consisted in showing that the integral

m((,q) =

r

exp(ite+iIt1a(sgnt)rt) Lt

-00

is uniformly bounded in 6 and q. 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

f 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,t2,t3), one would have to understand the zero set of

Real Part

uniformly in 61 , e2 and 63. So it is hard to imagine using the method of steepest descents, and for the curve t, t2, t3, t4, is it would seem close to impossible. Fabes' result was very important in that it gave the first clue that problems such as II and III 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 show that if y(t) = (t,tal,ta2'...'tan-1), 1
(here t 1 can mean either Itlal or Itlalsgnt ) then Hy was bounded on L2(Rn) [SWA]. Here we had to prove the boundedness of the integral 00

r

dt

J 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)

I

dt < C(n) .

E
We shall prove 30) by where C(n) does not depend on induction on n. By changing variables, replacing t by 1/n we

41

375

AVERAGES AND SINGULAR INTEGRALS

may assume en = ± 1 in 30). Then by using Van Der Corput's lemma with j = n , we find t

exp (iels + ... +

31)

ien-

sn-1 +

iSn)

< C(n) .

An integration by parts together with 31) shows that

32)

1 1<JtJ
Now

f exp (ieit+...+ien_ltn-1 +itn) dt

I exp (ieit+... +

ien-ltn-1) Lt

t

<

t

Jr to dt < C(n) . E
E
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 turn to the LP theory. We wish to emphasize how curvature and Fourier Transform are joining together to help us. So we shall compare the case of the parabola (t,t2) to the straight line (t,t). In the case of the parabola we are studying 34)

Hpf = DP * f

where DP is a distribution. For a test function 0

376

STEPHEN WAINGER

35)

Dpi

dt

In the case of a straight line we are studying

HLf=DL*f,

36)

where

37)

DL95 = J o(t.t) dt

00

38)

Dp(e,rl) = Dp(e1eXei"ly) = F

eieteirlt2 dt

-00

and

39)

DL(e,rl) = DL(e exe"7y)

f00ete'it. -00

t

We can calculate DL explicitly, and we find 40)

DL(e,rl) =c sgn((+n) .

We shall Notice that DL is discontinuous along the line show in contrast that Dp(e,rl) is continuous away from the origin. It is very easy to see that Dp(e,rl) is C°° away from the line i = 0 by complex integration. If for example rl > 0, we think of t as a complex z variable and integrate along the line lmt = Ret. Then the factor eirlt

decays as fast as e-c7ItI2 , and one can easily justify differentiation under the integral sign as long as it > e , for some positive E.

AVERAGES AND SINGULAR INTEGRALS

377

We shall now show that Dp(6,0 is continuous near 161 = 1 . What we must show is that l m0 Dp(4,-q) exists for 6 near ±1 . We shall show

41)

lim ,q-to

00

r

00

J

t

ei6t dt

ei 'tei77t2 dt _ t -00

Assume for simplicity that 71 > 0. Of course 1

1/3 00

77

42)

eiCt dt =

r

lim

r ei4t

J

7,-0

Lt

t

-00 ,17 1 /3

and

I ,71 /3

t

eiet(ei??t2_1) dt

43) 1

.771 /3

1

711r/ 3

< 271

J

tdt < 171/3 - 0

as 17 ->0.

0

In view of 42) and 43) we can show 41) by showing

44)

I

f

1/3
e'eteigt2 dt

o.

378

STEPHEN WAINGER

We shall prove 44) by using Van Der Dorput's lemma. By using Van Der Corput's lemma with j = 2, we see

f

t

eiese"gs

2

ds < c XFII

-2/3

(Let h(s) = es + qs2 , then h"(s) = 277 .) So an integration by parts shows

00

eieteir7t2 d t

45)

00

2/3

<

+

(' dt

1

77

,J

1/2

t2

77

77-

< C 77 2/3171/2

2/3

< C171/6

Note that if t < 77-2 /3 ie+217tl > iei - 2,71/3 So

-2/3 77

e tese l'7s 2 ds

1/3 77

is bounded if 6 is close to minus one. Hence an integration by parts shows 1

13 77

46)

f

00

t
eieteil7t2 dt

1

1/3 77

dt +77 1/3
AVERAGES AND SINGULAR INTEGRALS

379

45) and 46) together with similar estimates for negative t prove 44) and hence the continuity of Dp(e,-9) away from the origin. If one is a little more careful in the above argument, one can prove that Dp(C,.9) 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 2 . In fact if y(t) = (t,t2) one obtains 47)

IIHyfIILpCC IIfllLp,

3
For quite a while we tried to prove the range of p, 3 < 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

f

J=

IK(x+h)-K(x)Idx

R
where K is a kernel, in terms of K. One does this by using Schwartz's inequality and Plancherel's theorem.

f

J=

IxIaIK(x+h)-K(x)ldx I

la

R
<(

f

IXI2a 1

d)' x

IxI2aI(x+h)-K(x)I2dx)

(J

/

R
,_n

<

R

fI1ei2i

K( )12

2

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(e,n) very precisely by the method of steepest descents. We found 2

48)

Dp(6.77) = sgne + C

I77I

e1e2/n

`- 2/ + Better terms ,

where i/i is a Co function on R1 which is one near the origin. Hence,

the crux of the matter was to study the transformation Tf given by 49)

Tf (e,11) = m(e,rl)f((,rl) ,

where

50)

m(,n) =

1

Ir111/2

r)2

1

e

This suggests introducing an analytic family of operators in the sense of [SI] as follows: 51)

T (e,rl) = mz(e,rl)f(e,i)

where

52)

mz(e,rl) =

(171

2)

m(e,v)

Tzf is clearly bounded on L2 if Re z = -1/2 . So if we could prove that the kernel Kz corresponding to Tz for Re z positive satisfied a condition of the Calderon-Zygmund type, we could prove Tz was bounded in each LP if Re z > 0. Hence by Stein's interpolation theorem we would know that To = T was bounded in all LP, p < 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 Tz 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 To. T is essentially HY y = (t,t2),

and so the kernel K0 of To is essentially K0(x,Y)=X8(1-z )

from 22) and 22A). If we introduce "parabolic polar coordinates"

y =resin 0 x =rcos 0 , we see K0(x,y) = a S(9-90) r3

where

sin0 0 cos200

We might expect if Re z = e > 0 Kz to be

E

better than KO. So we

might expect Kz(x,Y) =

1

1

r3

Now we would like to explain why a

19-0011-e

1

10-0011-E

singularity is better than

a S(0-90) singularity. To see the situation more clearly, let us examine the analogous situation for the standard polar coordinates with 00 = 0. Suppose S3)

K0(x,Y) _

O r

where x = rcos 0 and y = rsin 0 , and

382

STEPHEN WAINGER

54)

KE(x.y) =

1

r 21011-E

for ordinary polar coordinates plays the same role as

(The factor

r

for parabolic polar coordinates.) We are trying to see whether

55)

jK(x,y)-K(x-h,y-k)j < C

1

r

.

x2+y2>c(h2+k2)

For either K = KO or K = KE . Let us take h = 0 and k = 1 and consider first K = Ko. Let us look at the contribution from y's which are very close to 0. We have

J J

r20

I-Dr2

)Id9rdr

r>C nearo

If y is very close to 0 161 '" r y r 1

0

y-1 r

So the left-hand side of 55) is at least fC i dr = 00 . So 55) can't

hold. Let us put the matter a little differently. If we consider the 0's with 0 > 0 where the difference K0(r, 0) - Ko(r , 0)

offers no cancellation, we find there is only one bad 0, 0 = 0. But still

AVERAGES AND SINGULAR INTEGRALS

383

n/4

f

JK0(r, 6)- K0(r', O')j dO = 1

.

J0

Let us consider now what happens with

We should expect no

help from the difference K(x,y) - K(x,y-1) when y > 0, if y-1 < 0. But this can only happen if y < 1 or 0 < 1/r . But over this set KE(x,y) is integrable at infinity 00

1 /r

KE(r, 0)d0rdr

1 50

5

1/r

00

Irfderdr

el-F

2

0

5

f1 00

<

,J

rl+e

dr .

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,t2). 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 of 48) for the curve (t,t2,t3). Essentially one needed a way to define a suitable analytic family Tz without using the asymptotic formula 48). Recall that Hyf = DP * f where

56)

Dp(e,rl) =

reieteint2 dt

384

STEPHEN WAINGER

So one might be tempted to define Hy,f = DP * f

57) w here

e

a irlt2 dt (1+t2)z/2

Dz(e,rl) p

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(Ac, A271) = Dp(c,rl) , for

any A > 0. Note that also the function mz(e,r/) defined in 52) also has this type of homogeneity, namely mz(Ac,A2r7) = mz((,77), for A > 0.

Now experience has shown that homobeneity is a powerful friend not

to be tossed away lightly. However Dp does not have this homogeneity. This situation can be remedied by defining

Hyf=Dzxf

59)

where 00

60)

Dp(c,rl) =

I

(1

+r72t4)-z /4 Xteir7t2 dt

-00

Note that for A > 0 61)

Dp(Ac, A2rl) = DP(c ,77) .

Let us see how formula 61) can help us. We would like to show 62)

C(z)

if Re z > -2 . By formula 61) we may assume r/ = ±1 , let us say Then by Van Der Carput's lemma with } = 2 , we see that

77

=1.

AVERAGES AND SINGULAR INTEGRALS

t

eXsel1

2

ds

1

ft d ese's2 dsI


So an integration by parts shows 00

J

(1+t4)-z/4eieteirlt2 dt < C(z) t

1

Now 1

j

(1+t4)-z /4 Xt eirlt2 dt

t

-1

f

1

e'et

t2 dt t

-1

1


r t2dtt
J-1

But we already know that

r J

-1

1

dt T
385

STEPHEN WAINGER

386

One may finally deal with -00 < t < -1 in the same manner that one treated 1 < t < 00. Hence 62) is proved. Let us calculate the kernel, Kz , corresponding to Hy if z = E > 0. Then

eiCx e myDp(e,rl)dCdrl

KE(x,Y) =fv

00

f f Lt t

00 +,72t4)-E/4

e-177Y (1

eirlt2

- 00

f

00

00

e-'6(x-t)dx

r

=

J

dt

J -00

00

t

f

i?7(t2-Y)(1+772t4) -E/4 ei 7t2&(x -t)dt

00

00

-.1

ir7(x2-y)(1 + 772x4)-E/4 drl

X

_00

0o

x3 f e

Ir7(x2y) 1

(1+,72)1/4

dr7

-00

1p

x3

x2'-Y2 e/2

(

x2

where PE/2 is a modified Poisson kernel. PE/2 decays exponentially

fast at oo and PE/2(u) ti JuJICE/2 as u -, 0. See

[SWE].

Thus KE(x,y) has a singularity near the curve (t,t2) of the form which is just the improvement over the 8-function that we 1

1e_eo1i-E/2

seek. A modification of these ideas worked for curves

AVERAGES AND SINGULAR INTEGRALS

a' Y(t) = (t

,ta2,...

,

387

tan)

an. See [NRW]. al < However, there is a natural generalization of these curves. All of

these curves satisfy an equation of the form Ye(t) = A Y(t)

63)

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,t2)

A curve satisfying 63), where all the eigenvalues of A have positive real part is called a homogeneous curve. A will generate a group of transformations TX = exp (A loga) . Then

Hyf - Dy*f

64)

where 65)

Dy(Tx) = Dy(e )

Moreover there is a distance pA(x) defined on Rn such that P(TAX) _ AP(x)

In the case of the cure (t,t2) we may, as we said before take

38$

STEPHEN WAINGER 0

A =C1 0

2

)

Then 0 ),

T-

,

0

and p(x,y) = (x4+y2)1 /4

It turns out that in the case of a general homogeneous cure, we can obtain a satisfactory analytic family of operators by defining 66)

Hzf=Dz*f

where 67)

z(e) =pA*(() j ItI-zexp(ie'y(t))dt

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

p * A

y(t) = (tacos (j3logt), tasin(/3logt))

is an example. So one could believe it might be rather messy to prove integrals involving exp to be bounded. It might be difficult to show that at each t some derivative of , y(t) would be non-zero. However, if one makes a change of variables t = eu , we would be led to consideration of integrals involving n(u) where n(u) = y(eu). If A y'(t) = y(t), n(u) satisfies 68)

n'(u) = Ai(u)

.

AVERAGES AND SINGULAR INTEGRALS

389

We shall show that if rj(u) is a curve in Rn satisfying 68) where the

eigenvalues of A have positive real part, then either r)(u) lies in a 0 and u there is a j proper subspace of Rn or for every

1<j
0.

rl(u)

du)

From 68) we see that dI+17

Ajrl

(u)

(u)

du1+1

By the Cayley-Hamilton theorem, we can find numbers aj , 0 < j < n, such that

jAj=0. j=0

So n

Yd j+t

aJ duI+1 '7(u) = 0 j=O and n

j=0

a j

aj

.

rr'(u)

=0.

duJ

In other words rJ,(u) e satisfies an nth order constant coefficient differential equation. So if for some u and d, n'(u) .

=0

j = 1 ,2,...,n-1

du3

rj'(u)

6 = 0 for all u. Thus r)(u) 6 is a constant. But r!(u) 6 , 0

STEPHEN WAINGER

390

as u -, -oo since the eigenvalues of A have positive real part. Hence rt(u)

is in the subspace of Rn orthogonal to e.

We shall conclude this section with the statement of some theorems that follow from the reasoning discussed above.

THEOREM 1. Let y(t) satisfy A 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 IIHyf1I

<

LP

Cp/Y

1
IIf1ILp

We say that a curve y(t) in Rn is well curved if dJy(t)

dyj(t)

j = 1,2,... t= O

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,

f f(x-y(t))

dt

<ApyIIfIILp,

1
A general theorem in L2 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,t2). Thus we wish to consider averages h

70)

r f(x-t,y-t2)dt

Mhf(x,y) --

F

.

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 Mhf(e,r1) = mh(e,r))f(e,rl)

where h eit e eit2 'Idt

mh(e,r1) = h 0

Now mh(e,rl) cannot be evaluated explicitly. If one 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 [FHI. 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

f exp Q(x)dx I. J
where Q(x) was a quadratic function of x .

(' a

Instead they considered

Ixl2

e J e R

F

+Q(x) dx

STEPHEN WAINGER

392

which could be calculated explicitly. This suggests to consider instead of 70) vh * f(x,y)

71)

f

exp( t2)f(x-t,y-t2)dt h/

.

Then

vh (e,q) = v(he,h2rl) f (C,rl)

72)

where v is the measure considered in 23). In particular 00

v(he,h277) = f e t2eiheteih2r)t2dt

,

and this integral can be computed explicitly by completing the square. We find

73)

v(h,h2) = (nice smoothly decaying function) exp i

2 h477 1 +h 4,772

One might guess that the appearance of the oscillatory factor exp i h42 't is a reflection of the fact that

is a singular measure.

1+h477 2

On the other hand we see that if h2r) is large

expii h27,expie77 which is independent of h. Thus one might try to write (from 72) vh* f (e,n) _ (i(h,h27l)exp(_ l

2

rl //

((exp

i

)f

One might now hope that if one defines a measure vh by the formula

AVERAGES AND SINGULAR INTEGRALS

h(6,r!) = v(he,h2rl) exp - i

393

r7

vh could be dealt with by classical arguments, while

{exp g

(

'I ) f( ,14 =

is another L2 function having the same norm as f. So one

could hope that lsup vh * fllL2 = IISUp vh * gllL2

74)

< CIIgIIL2 = CIIfIIL2

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 vh and not so much the explicit expression of vh as was used above. Stein [Ssp] and [SH] succeeded in doing this by introducting appropriate g-functions. Stein's first argument with g-functions dealt with the averages MB f of equation 4). Recall r

75)

MB r

f(x) = J f(x-y)dgr(y) Br

where Br is the ball of radius r centered at the origin, and dur is the unit rotationally invariant measure on Br. We set, as before, 76)

911f(x) = sup IMB f(x)I r>O

r

STEPHEN WAINGER

394

Stein used g-functions to prove THEOREM 3: Cp,n 110

77)

LP

if p>nn-1 and n>3. Simple examples of the form n-1

IxI O log' IXI where

1

near the origin and has compact support show that p > nn1

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

g(f) (x) _

78)

f

1 /2

00

tI dt Mtf(x)I2 dt

Assume that we could prove IIg(f )IIL2 < C(n)IIf1IL2 '

79)

and let us see how 77) would follow. Now

rnMrf(x)

CT-

snMsf(x)ds

0

=n

r sn-1 Msf(x)ds +

r sn ds Msf(x)ds

AVERAGES AND SINGULAR INTEGRALS

395

Thus r

Mrf(x) < n

r

r sn'1 Msf(x) ds + 0

n

r

sn d Msf(x)ds

0

= I(r) + 11(r)

.

Now l(r) is dominated by the Hardy-Littlewood Maximal function and

II(r) < n

f

sn-1 /2 s1 /2 Msf(x)ds

rJ

0

r

fr -rn J <

1

1 /2

s2n-1 ds

g(f) (x)

0


We turn now to the proof of 79). W

JRn

Ig(f)(x)I2dx= J t J

IdtMtf(x)I2dxdt

Rn

0

0 =

r t Jr

J0

Mtf(6)12d6dt

Rn

00

tJ

_ 0

Rn

Imtr(6)I2dedt

.

STEPHEN WAINGER

396

But Mtf(e) = f(()m(tlel), where m(r) = n12 Jn-2(r) r 2

2

Here Jn_2 is the usual Bessel function. We shall need to know 2

dmr I < C r

and

dorm

=C

1

1

n-1

n-2

dr

2 r1/2

r 2

is bounded. See [SWE]. Thus

f f Jtm(t()Ide 00

f lg(f)(x)I2dA <

t

Rn

0

00

< f If( )12 5 t at m(t 1j1)2 dt Rn

0

Thus to obtain 79) we need to show 00

r tIdm(tIibI2dt
First since

is bounded,

f

1 /IeI

1/161

dt

0

0

Next, since Im'(t)I <

<1e12f

C

n-1 t 2

,

tdt
AVERAGES AND SINGULAR INTEGRALS

397

00

t

dt m(tIeI)I2 dt

/IeI 00

Ifi2 f t

dt < C , (t lel

)n-1

1 /ICI

if n>4. Let us be more precise about the counterexample in 2-dimensions. We take 1

x very near 0

IxIlogI1

in Co

away from 0 .

We can disprove 77) for p = 2, n = 2 by showing 80)

MBIxIf(x) _ 00

for all small x. Because of rotational symmetry it suffices to prove 80) for points (a,0) with a small. In that case

dO

MIxlf(x) _

(a2(1-cos)2+a2sin2ej1 /2 log -n

a2[(l-cos)2+sin2O] E

dO

ti

-E

1

IeI In IeI

= 00 .

398

STEPHEN WAINGER

A similar argument shows sup IMB f(x)I = 00

81)

1
r

a.e. for an appropriate f . We wish now to prove a theorem indicating how bad 81) fails in L2(R2). We set 82)

)Rkf(x,Y) =

sup

f(x)YI

IMB

2k<j<2k+1 2k

where f is in L2(R2). We shall show 83)

Il)KkfpL2 < CkIIfIIL2

To prove 83), it suffices to show that for each function j(x) taking the values 84)

IIMBI+j(x)2-k f(x)II < CkpfIIL2 ,

with C independent of the function j(x). We show 85) by induction on k. That is given a j(x) taking the values we shall define

a function j*(x) so that j*(x) takes values in the set and 85)

IIMBI+j(x)2-kf

_MB1+j*(x)2-k+l

fIIL2 < CIIfIIL2

85) provides the inductive step to prove 86). If j(x) is given define if j(x) is even

if j(x) is odd .

AVERAGES AND SINGULAR INTEGRALS

Then, if we set 2k

gf(x) _ I IMB

86)

J=0

f(x)-MB

k

1+

f(x)I2 1+11 2k

2

we see IMBfI < gf(x)

Thus to prove 86) and hence 85) it suffices to prove 119(fAL2 < CIIfIIL2

87)

We shall prove 87) by using the Fourier Transform. 2k

('

J Igf(x)I2 dx = I 1

fMf(x)-MB

l+2k

0

l+-

2k

=I

IMB

W) - MB

j=0

f where m(r) = J0(r). So to prove 87) it suffices to show

88)

2

399

400

STEPHEN WAINGER

But 88) follows because for s positive and

r

positive

IJ0(r)) < C/'

89)

and

IJ0(r+s)-Jo(r)I < Ct.

90)

To prove 88) we use 89) if ICI > 2k and 90) if 161 < 2k Finally we will show how Stein [SH] proved the maximal function along the parabola (t,t2) is bounded by using g-functions. We start with the measure dµ defined in 24) 2

qS(t,t2)dt

dµ(O) = J

.

1

We set 2

91)

dµh(qS) =

f

-O(ht,ht2) dt

.

I

Then 2

dµh * f(x,y) =

r

f(x-ht,y-ht2)dt

,

1

or

92)

dµh * f(x,y) _

f

2h

f(x-t,y-t2)dt

.

h

We choose a function &i(x,y) E C0m(R2) with %&(0) = 1. We set

AVERAGES AND SINGULAR INTEGRALS

93)

'Ph(x,Y) = h3 0 x , h)

and

94)

g(f)(x,Y) _

Idµh *f(x,Y)-Oh *f(x,Y)I

I

Let us first assume 95)

119(f)IIL2 5 C IIf1IL2

Note that E

sup

f Idµh*f(x,Y)-h*f(x,Y)Idh

E>0

0

sup

1f

E

Idµh *f(x,Y)-oh *f(x,Y)I2dh

E>0 E1/2

0

<

r

1 /2

*f(x,Y)I2 dh

Idµh

0

< g(f) (x,Y) So E

96)

sup I E f dµh *f(x,Y)dh1 t>o 0

<- Of )(x,Y) + sup 10h *f(x,Y)I h>0

401

STEPHEN WAINGER

402

A classical argument (see (RI) shows sup Oh *fp < C tIfIILp Lp h>0

Thus by 95), we see E

sup I e fd1zh*f(xY)dhl

< CIIf11L2

E>0

L2

0

If f>0, E

(' J dµh*f(x,Y)dh 0

2h

E

f

f(x-t,y-t2)dtdh h

0

E

E

>

f r f(x-t,y-t2)

f

dh

/2

0

('E

E >

f

f(x-t,y-t2)dt

J

.

0

So from 96) we infer E

sup If

E>0

eJ

0

It remains to prove 96).


f(x-y,y-t2)dt

L2

AVERAGES AND SINGULAR INTEGRALS

fI(gf)(x,y)I2dxdY

403

r T J Idµh*f-ch*fl2dxdy

=

0

ff 00

rd h(e,-0)h(e,7)I2 If( ,77)I2 d dii

0

f

a* .77)12

f0,1f(

dry.

0

0

So to prove 96) it suffices to prove 00

I h(

J

97)

,rl)I2 < C .

0

The integral on the left side of 97) is

JT

98)

0

Thus by replacing h by Ah A > 0 we may write the expression in 98) as

0

J0

µ(Xhe,)12h271)-0(Ahe,42h277)I2

By choosing A so that \2e2+X4772 = 1 , we see that it suffices to estimate 99) when e 2 +772 = 1 . In this case we see 1

1

(' 99)

JJ

dh dµ(he,h27)-cb(he,h27)I2 < C

0

since dµ(0) = 0(0) = 1 .

r h2 dh < C 0

404

STEPHEN WAINGER

Then from 27) Idµ(he,h27l)I < Ch-3

for some 6>0. So ('00

100)

('00

ld26 < C

T Idµ(he,h2rl)I < J

J

J

h

1

1

CN

Also k1;'"?) <

for any N, so

(1+e2+772)IJ 00

('

101)

1

dh Ii&(he,h277)I < C

.

1

Now we obtain 98) and hence 96) by combining 100), 101) and 102).

In this section we have emphasized L2 methods. LP results for p > 1, can be obtained by combining the L2 estimates presented here with the techniques of section 2. Altogether one can prove the following theorems :

THEOREM 4.

If y(t) satisfies y'= A y(t) where all the eigenvalues of

A have positive real part, h

IIo
THEOREM 5. Let y(t) be a curve in Rn. If the vectors y'(0), y"(0), y(3)(0) ... span R° ,

II

sup

0
f 0

h

If(x-y(t))IdtII

< CpIIfIILP LP

1
AVERAGES AND SINGULAR INTEGRALS

405

IV. Vector field problems We turn now to problems III and 1II'. 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,0) = 0, let !'1

102)

Hyf(x) =

f(x-y(t,x)) dt

f -1

let !'h

103)

f(x-y(t,x))dt ,

Myf(x) _ 0

and 104)

)11 f(x) = Y

sup Ahf(x)l 1>h>o

y

.

We are now ready for the statement of problem IV and W. Problem IV: When do we have UHyf(x)11Lp <- Cpilf11Lp ?

Problem IV': When do we have f(myf11Lp <- CpIIfhILp ?

The change of variables described above preserves tangency and

curvature conditions. So for example N It=o will be parallel to the x

406

STEPHEN WAINGER

axis, and if the curvature of the integral curves of v(x) never vanishes a2y

will not be zero. It turns out that one can prove the following

at2 t=0 theorems:

THEOREM 6. Let y(t,x) = (t,r(t,x)) be a smooth curve in R2 satisfying

ar

x

It=o = 0

and

a2rlt=ono. at

Then 105)

IIHyf IIL2(R2) <- Of I!L2(R2)

and

106)

IItyfIIL2(R2) <- CIIfllL2(R2)

A consequence of Theorem 6 for vector fields will then be THEOREM 7. If v(x) is a smooth vector field in R2 such that the

integral curves of v have nowhere vanishing curvature, IIHvf1IL2(R2) <_ IIfIIL2(R2)

and h

lim h1 h-00

f(x-tv(x))dt = f(x)

,

a.e.

1 0

for f in L2(R2). 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 59 y. 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 denotes composition to consider Hy H* and Mh(x) . (Mh(x))* where

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,t2) 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 t(u,v)Iu=t-s,v=t2-s2J. Hence one might hope that K would have a much smoother kernel than Hy H. (Of course if K were bounded on L2 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

H*f(x,y) = J f(x+t,y+y(t)) dt

and hence that

HYH*f(x,y)

f(x+t-s,y+t2-s2) dt

tss

=ff

Let u=s -t, v =s2-t2, and we find HYH*f(x,y)

=ff f(x-u,y-v)k(u,v)dudv

where C

k(u,v) = IuI

u-u u+u

.

Now k(u,v) does have its support spread out. However the singularities of k across the curves v = ±u2 are not locally integrable away from the

408

STEPHEN WAINGER

origin. Hence HYHY cannot be a Calderon-Zygmund operator. Let us try to see what goes wrong on the level of the Fourier Transform. Recall

HYf=Qp*f where

p(e,r/) _ f

e(iet+ir7t2)dt

-00

We know from 48) that

Qp((,ri) = sgn + C

IjII/2 e

77

6

+ better terms.

The multiplier for HY HY would be essentially IQp12 . Notice that IQp12 doesn't look any nicer than Q. However IQp(e,r7)-sgneI2 is

much nicer than Qp(e,rj) or Qp(e,rj) - sgne. Now sgne corresponds to the operator Lf =

ff(x-t,y) dt

L is known to be bounded in L2. This suggests that we try to consider M = (L-HY)(L*-HY) .

If M is bounded in L2 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 107)

Ph(x) (Ph(x))*f(x) < CIPh(x)f(x)+P*(x)f(x)]

where Ph is the Poisson Kernel. It follows from 107) that

409

AVERAGES AND SINGULAR INTEGRALS

IIPh(x)f Il L2 < Clip h(x)f(X)ll

L2

and hence

IIPh(x)f1IL2
Since the function h(x) is arbitrary we have Ilsup Ph*f1IL2 <_ CIIf1IL2 h

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

h(x,y) Rh(x,Y)f

(x,Y) =

h(x,y) f

f(x-t,Y)dt

.

0

(We know

IIRh(x,y)f(x,Y)IIL2 <- Cllf(x,Y)lIL2 )

We might hope to prove 108)

Mh(x)(Mh(x))*f(x) < C(Rh(x)Mh(x)f(x)+Mh(x)Rh(x)f(x)+Bh(x)f(x))

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 0

for 0
t-1

for t>1

r(t) = Then

n

Hrf(6,71) = my(6,17)f(6,q)

where

r

my(4,i7) =

00

ei4teirir(t) dt

-00

=

r

+

00

1

sintt dt

0

at

1

In fact it is easy to see which is easily seen to be unbounded if the following: Suppose y(t) is odd and linear on a sequence of intervals ai < t < bi < 1 and assume that the linear extension of y on b [ai,bi] does not pass through the origin. Then if the ratios a are t

unbounded,

lim E--0

I

f(x-y(t)) dt

ItI<E

cannot exist in the L2 sense for every f in L2 . It is somewhat more difficult to produce counterexamples for the operator 1R y. For example in the case of the two dimensional curve

(t,r(t)) described above, it is easy to see that Vy is bounded in every LP, because it is dominated by

AVERAGES AND SINGULAR INTEGRALS h

!'h

sup

I

0

411

If(x-t,y)1 dt + sup h

f

f(x-t,y-t+I)dt t

.

0

Both of these operators are bounded by the classical theory. We refer to [SWI for an example to show that the maximal function along a Coo 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 r(t) convex for positive t. Let h(t) = tr'(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,P(t)) with r(t) convex and increasing for t > 0. If r(t) is even IIHyfIIL2 <- CIIflIL2

if and only if

Ir'(ct)I > 21r'(t)I

t>0

for some C>0. If y(t) is odd IIHyfIIL2 <- CNfIIL2

if and only if h(Ct) > 2h(t) ,

t>0

for some C>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

yfliL2 <- CIIfIIL2

if * holds for some C > 0, however the maximal functions can be

bounded on L2 (and in fact in LP for any p > 1) for some convex curves even if * 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 a and an a dimensional CalderonZygmund kernel K(P,Q). Then they consider

Tf(P) T (Q) K(P,Q)dm(Q)

where dm(Q) is a measure on Mp M. They show that if n > 3 and

k = n-1 IITfIILp <_ CIIfIILP

if Mp satisfies a kind of generalized curvature condition. See [PSI. 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], [STR1], 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 00

A-1)

1(I) = f e ax2 dx z

413

AVERAGES AND SINGULAR INTEGRALS

for large A. Of course we can make a change of variables

and observe

A-2)

I(A) _ V"X

where 00 e-t2

B=

A-3)

dt .

-00

The point we wish to make here is that if A is large most of the contribution to the integral I(A) comes from a small neighborhood of the origin. In fact

5e

At2 dt

<

e

f

('

At2

A__t2

dt + J e 2

2 dt

t>1

A21/5
It1,A21/5

<e-AI/5+e,\/2 -gyp very fast as A. o . 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(A), we

A2/5 - -A2/5

can expand the integrand in a power series in that little interval. For example we might consider 00

JA) =

J

e Xh(t)dt

STEPHEN WAINGER

414

where h(t) > t2 for large t, h(t) = t2 +0(t3) for small t and h(t) > 0 for t > 0. Then one can easily see that

f

dt

Itl>A2/5

is exponentially small as before. Now e Ah(t)dt

J

= f e-At2 . (1+O(at3) Itl«

ItIU 2/5

00

e-at2 dt

e-'fit2

dt + Ea I /5

ltl«2/5 =

B + O(e-AI

f

-00

/5)+0

(1

l

.

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, if we had

K=

r

eikt2+.k P(t)dt

-00

and P(t) were very negative at infinity and 0(t3) near t = 0 we would try to write t = a + it and integrate on the line a = r for It l < 1/, 2 /5 , and we would expand in a power series in this small interval.

AVERAGES AND SINGULAR INTEGRALS

415

More, generally, if we were concerned with the asymptotic behavior for large values of k of an integral of the form fg(z)eAh(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'(C) = 0. Thus the main contribution to our integral should come from a point where

h'(C) = 0 or an endpoint. This is also reflected in Van Der 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 phase and quantum mechanics

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 b

I=

r eiAf(t)dt a

with f(t) real gets most of its contributions for large A, 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 -oc to oo and f oscillated very rapidly for large t , we would expect the main contribution to the integral to come from small neighborhoods of a zero of V.

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 = 2 [x(t)]2 - V(x(t),t) . The action along a path x

is defined by

tb

S(x) =

r

L(x(t))dt .

ta

The path on which the classical particle moves will be a path x(t) such that z(ta) = xa , z(tb) = xb , and such that S(x +Sx)IS=o

=

0.

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 im-

portant 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

Fje"h

s(x)

Dx

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 contribution to the integral F to come from paths which are very close (on a

scale of h ) to the path x defined by *. There have been many papers in the mathematical and physical literature dealing with the problem of making sense out of the definition F. See for example [CS] 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 21 subintervals, Ij , of equal length [b_ta]k

Ik =

l

+ ta, [tb-ta]2

nl

+ to

.

We consider only paths which are linear on Ik . Such paths are determined by xk = X (ta

[tb-ta])

+ 2

k=

We then take

1 S(x1 ... x2j-1)

F = lim A. j-,°°

J

eh

dx 1dx2,...,dxk

where S(x1,...,x2j_1) is the action along the polygonal path determined by x1'...,x2j_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,

[CS]

[CSS]

[CW]

[C] [CS]

[CORI]

pp. 226-256. R. Cameron and D. Storvick, A simple definition of the Feynman integral with applications, Amer. Math. Soc., Providence, 1983.

H. Carlsson, P. Sjogren, and J. Stromberg, "Multiparameter maximal functions along dilation-invariant hypersurfaces" to appear in Trans. of the A.M.S. H. Carlsson and S. Wainger, "Maximal functions related to convex polygonal lines," to appear. M. Christ, preprint. 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. 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

[CF1] [CF2]

[CF3]

419

A. Cordoba and R. Fefferman, "A geometric proof of the strong maximal theorem," Annals of Math., Vol. 102, 1975, pp. 95-100. , "On differentiation of integrals," Proc. Nat. Acad. Sci., U.S.A., Vol. 74, 1977, pp. 2211-2213. "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]

[FH] [H] [KS]

[NRW] [NRWM]

R. Fefferman, "Covering lemmas, maximal functions, and multiplier operators in Fourier Analysis," Proc. Symp. in Pure Math., Vol. XXXV, Part 1, pp. 51-60. R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York. L. HOrmander, "Estimates for translation invariant operators in LP spaces," Acta Math., Vol. 104, 1960, pp. 93-139. R. Kunze and E. Stein, "Uniformly bounded representations and harmonic analysis of the 2x2 unimodular group," Amer. J. of Math., Vol. 82, 1960, pp. 1-62. A. Nagel, N. Riviere, and S. Wainger, "On Hilbert transforms along curves, It, "Amer. J. Math., Vol. 98, 1976, pp. 395-403. A. Nagel, N. Riviere and S. Wainger, "A maximal function

associated to the curve (t,t2)," 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 Mathematica.

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 1, 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, [NSW]

pp. 735-744.

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. J. of Math. [NVWW3] , "Maximal functions for convex curves," Preprint.

-

[NW]

[NW2]

[NE]

[P] [R]

[PS]

[S]

[SBC]

[SHI

[Ssp]

[SPL] [SI] [SWA]

A. Nagel and S. Wainger, "Hilbert transforms associated with plane curves," Trans. Amer. Math. Soc., Vol. 223, 1976, pp. 235-252. -, "L2 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. W. Nestlerode, "Singular integrals and maximal functions associated with highly monotone curves," Trans. Amer. Math. Soc., Vol. 267, 1981, pp. 435-444. R. Paley, "A proof of a theorem on averages," Proc. Lond. Math. Soc., Vol. 31, 1930, pp. 289-300. N. Riviere, "Singular integrals and multiplier operators," Ark. Mat., Vol. 9, 1971, pp. 243-278. 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. E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. , Boundary Behaviour of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972. , "Maximal functions: Homogeneous curves," Proc. Nat. Acad. Sci., U.S.A., Vol. 73, 1976, pp. 2176-2177. , "Maximal functions: Spherical means," Proc. Nat. Acad. Sci., U.S.A., Vol. 73, 1976, pp. 2174-2175. , Topics in Harmonic Analysis related to the LittlewoodPaley Theory, Princeton University Press, Princeton, 1970. , "Interpolation of linear operators," Trans. Amer. Math. Soc., Vol. 88, 1958, pp. 359-376. 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 [SWE]

[STR1] [STR]

[STRO]

[V]

421

E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton. R. Strichartz, "Singular integrals supported on submanifolds," Studia Math., Vol. 74, 1982, pp. 137-151. J. Stromberg, "Weak estimates on maximal functions with rectangles in certain directions," Ark. Mat., Vol. 15, 1977, pp. 229-240. J. Stromberg, "Maximal functions associated to rectangles with uniformly distributed directions," Ann. of Math., Vol. 107, 1978, pp. 399-402. J. Vance, "LP boundedness of the multiple Hilbert transform along a surface," Pacific J. of Math., Vol. 108, 1983, pp. 221-241.

[WE]

[Z]

D. Weinberg, "The Hilbert transform and maximal function for approximately homogeneous curves," Trans. Amer. Math. Soc., Vol. 267, 1981, pp. 295-306. A. Zygmund, Trigonometric Series, Vols. I & II, Cambridge University Press, London, 1959.

INDEX

approach regions admissible, 245 non-isotropic, 261 non-tangential, 244

Ap classes, 73, 353 area integral, 92 atomic decomposition, 114, 156, 159, 340

convex curves, 411 curvature, 321, 361

and Fourier transform, 321, 325, 268, 375 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+xR+), 101

electrostatics, 163

Bochner-Martinelli formula, 196

Bochner-Riesz summability, 344 Calder6n-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 equation, 201 convergenge of averages over spheres, 358, 361 along curves, 361 along vector fields, 361, 406 covering lemmas, 60

g-functions, 393

Hardy space, 144

Hp spaces, 89 HP(R+ x R+)

,

101

harmonic measure, 141

Hartogs extension phenomenon, 207

heat operator, 254 Henkin integral formula 219, 336

423

424

INDEX

Heisenoerg group, 257, 335 Hilbert transforms along curves, 361, 372 along vector fields, 362,406

hydrostatics, 177 hypoelliptic differential operators, 281 Kohn (canonical) solution, 209 Laplacian, 266

non-commuting vector fields, 267

oscillatory integrals (first kind),308 (second kind), 335, 344, 349 Poisson integral, 187, 243 bi-Poisson integral, 102 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 local singular function, 210 Lu Qi-Keng conjecture, 235 maximal functions, 48, 49, 245 strong, 60 spherical, 359, 393 on curves, 391, 400 on vector fields, 406 method of layer potentials, 133, 143 Mobius transformation, 186

multilinear Fourier analysis, 18 multiparameter differentiation theory, 57 multipliers, 72

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 surfaces (non-zero curvature), 321 systems of elliptic equations, 133, 163

Szego kernel, 193, 265, 296, 297 Transference theorem, 39 Van der Corput lemmas, 309, 370

weight norm inequalities, 72 Zygmund conjecture, 67

Library of Congress Cataloging-in-Publication Data Beijing lectures in harmonic analysis.

(Annals of mathematics studies ; no. 112) Bibliography: p. Includes index.

1. Harmonic analysis. 1931- . H. Series. QA403.B34 1986 ISBN 0-691-08418-1

I. Stein, Elias M.,

515'.2433

86-91452

ISBN 0-691-08419-X (pbk.)

Elias M. Stein is Professor of Mathematics at Princeton University