Based on seven lecture series given by leading experts at a summer school at Peking University, in Beijing, in 1984. thi

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*Table of contents : TABLE OF CONTENTSPREFACENON-LINEAR HARMONIC ANALYSIS, OPERATOR THEORY AND P.D.D.MULTIPARAMETER FOURIER ANALYSISELLIPTIC BOUNDARY VALUE PROBLEMS ON LIPSCHITZ DOMAINSINTEGRAL FORMULAS IN COMPLEX ANALYSISVECTOR FIELDS AND NONISOTROPIC METRICSOSCILLATORY INTEGRALS IN FOURIER ANALYSISAVERAGES AND SINGULAR INTEGRALS OVER LOWER DIMENSIONAL SETSINDEX*

Annals o f Mathematics Studies Number 112

BEIJING LECTURES IN HARMONIC ANALYSIS

EDITED B Y

E. M. STEIN

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY 1986

Copyright © 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

T AB LE OF CONTENTS P R E FA C E NON-LINEAR HARMONIC AN ALYSIS, OPERATOR THEORY AND P.D .D . by R . R . C oif man and Y v e s Meyer MULTIPARAMETER FOURIER AN ALYSIS by Robert F efferm an

v ii

3

47

ELLIPTIC BOUNDARY VALUE PROBLEMS ON LIPSCHITZ DOMAINS by C a rlo s E. Kenig

131

INTEGRAL FORMULAS IN COMPLEX AN ALYSIS by Steven G. Krantz

185

VECTOR FIELDS AND NONISOTROPIC METRICS by A lexan d er Nagel

2 41

O SCILLATORY INTEGRALS IN FOURIER AN ALYSIS by E. M. Stein

3 07

A VER AG E S AND SINGULAR INTEGRALS OVER LOWER DIMENSIONAL SETS by Stephen Wainger

357

INDEX

4 23

P R E FA C E In Septem ber 19 8 4 a summer sch o o l in a n a ly s is was held at P ekin g U n iv e rsity. The su b je c ts d e a lt w ith w ere to p ics of current in te re st in the c lo s e ly in terrelated are as of F ou rier a n a ly s is , p seu d o -d ifferen tial and sin g u lar in tegral o p erato rs, p a rtia l d iffe re n tia l eq u ation s, re a l-v a ria b le th eory, and s e v e r a l com plex v a ria b le s .

E n titled the “ Summer Symposium

of A n a ly s is in C h in a ,” the con feren ce w as organized around s e v e n s e rie s of exp o sito ry lectu res w hose purpose w as to give both an introduction of the b asic m aterial a s w e ll as a d e sc rip tio n of the most recent re s u lts in th e se a re a s.

Our o b je c tiv e w as to fa c ilita te further s c ie n tific exchanges

b etw een the m athem aticians of our two cou n tries and to bring the stu d en ts of the summer sch o o l to the le v e l of current re se arc h in th ose important fie ld s . On b eh alf of a ll the v is itin g le c tu re rs I would lik e to acknow ledge our great ap p reciatio n to the organizing com m ittee of the con feren ce:

P ro

fe s s o rs M. T. Cheng and D. G. Deng of P eking U n iv e rsity, S. Rung of the U n ive rsity of S cie n c e and T echnology of C hina, S. L. Wang of Hangzhou U n iv e rsity , and R. Long of the In stitu te of M athematics of the A cadem ia S in ic a. T heir effo rts helped to make th is a most fru itfu l and e n jo ya b le meeting. E . M. STEIN

vii

Beijing Lectures in Harmonic Analysis

NON-LINEAR HARMONIC ANALYSIS, OPERATOR THEORY AND P.D .E . R. R. Coifman and Y v e s Meyer Our purpose is to describe a certain number of re su lts involving the study of non-linear a n a ly tic dependence of some fu n ction als arisin g naturally in P .D .E. or operator theory. To be more s p e c ific w e w i l l consider functionals i.e ., functions defined on a Banach sp ace of functions (u sually on Rn ) with v a lu e s in another Banach s p ac e of functions or operators. Such a functional F : B 1 -> B 2 is said to be re a l an alytic around 0 in B 1 if we can expand it in a power s e r i e s around

0 i.e.

oo F (f) = 2 A k(f) k=0 where A ^ (f) is a “ homogeneous polynom ial” of degree

k in f . This

means that there is a k multilinear function ^ k ^ l’" ^ : B 1 x B 1

x B1 ^

(linear in e ach argument) such that A^(f ) = A^(f, f, ---f ) and k a)

n v f i - f k)«B

< c k n i i fj«B . 2

j = i

1

for some con stant C .

(This la st estim ate guarantees the convergence of

the s e r i e s in the b a ll

||f||B

< g •)

3

4

R. R. COIF MAN AND Y V E S MEYER

Certain facts can be e a s i ly verified.

In particular if F

is analytic

it can be extended to a ball in B^ (the com plexification of B ^ ) and the extension is holomorphic from B^ to B^ i .e .,

F(f + zg) is a holomor-

phic (vector valued) function of z e C , |z| < 1 , Vf, g s u ffic ie n tly small. The con verse is a l s o true. Any such holomorphic function can be e x panded in a power s e r i e s , (where

is ~ x the

F rech et differen

tia l at 0 ). We w ill concentrate our attention on very concrete functionals arisin g in connection with d ifferen tia l equations or complex a n a ly s i s , and would like to prove that they depend a n a ly tic a lly on certain functional parameters. A s you know there are two w ays to proceed. 1. Expand in a power s e rie s and show that one has e stim ates (1). 2.

Extend the functional to the complexification a s “ form ally holo-

morphic” and prove some boundedness estim ates. Let L denote a d ifferen tia l operator like

S

j ) for some map n ( i,j) of

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

V & V -------------- ►V ® V

A : V xV ■

V

where K e i , e j) = e j ® e j r r C e ^ e j) = 6 ^

This is c le a r l y the situation for periodic functions.

It would be interesting

to understand the b ilinear operators admitting such a diagonizable lift to the tensor product. This observation indicates that the hypothesis concerning commuta tion with tran slatio n s, imposes s e v e re restrictions on the nature of a multi linear operation. We now return to the line on which we re a lize multilinear operations as

A k ( f i - f k)(K) = J

and

J

+

21

NON-LINEAR HARMONIC AN ALYSIS

a , ^ lx

A k(e

, •••, e

^ k xw \ ix(^ l+ " ' ^ k \ , t \ s ) (x) = e ^k^V"^-

(Note that this re a liza tion is only valid for multilinear operations v e rify ing some mild continuity conditions.) This re a liza tion permits us for example to show that the study of

^

* b ^

* f ^

can e a s i ly be con-

verted to the study of oo

J

xf/\ * (^rt * b - 0 t * f )

Y

0

which we considered p re v iou sly in the proof of Theorem [II]. In fa c t oo J*

/

oo

^ t *b 0 , N > 0 such that fo r each in te rv a l I there is a c on stan t C(I) (depending co n tin u ou sly on I ) for w hich |x e l : |b(x)-C(I)| | |I| . Then b e BMO and ||b||BMO - c Na ' The main idea to estim ate the BMO norm of T ( l ) is to replace inside e ach in terval I , TA by an operator TA fraction of I and A ' Ta (1) to Ta (1 ) .

where A j = A on a large

has a sm aller L ip sc h itz norm, and then compare

This is ac h ie ve d via the following lemma, the first of

which is the rising sun lemma (or the one-dimensional version of the Calderon-Zygmund decomposition). LEM M A 1.

Let A

be su ch that C-M < A'(x) < C+M

then for each I there e x is ts a function A j and a con stant C j such that Aj = A

on a s e t E

|E| >

|l|

and Cj - |

m

< A '(x) < C j + |

P roof. We can assum e C = M i.e ., a)

m j( A ') > M .

m

.

0 < A ' < M2 . There are two c a s e s :

36

R. R. COIFMAN AND YVE S MEYER

In that c a s e consider the sm a lle s t function A j > A with A '(y) >

then have Aj(y) = A(y) except for d isjoin t in tervals 1^ on which A j(y) = ^ ? .

But

J * i A i(y)dy = J *

+ J*

l ~u h = 2M|I| -

< 2M |I-UIk | +

|UIk |

U lk j

M |UIk |

j M |UIk | < (2M - mj(Aj)) |I| < (2M-M)|l| = |l|M

i.e ., M k l < f HISince

< AJ(y) < M2 , we have 4 M _ 2 M < A .( y ) < 4 M + 2 M

b)

mj(A') < M.

. We

NON-LINEAR HARMONIC AN ALYSIS

37

We con sid er the function 2 M - A /(x) = A '

|A'| < M2 .

Then we have mj(A^) > M and we construct A n

as above.

A j j = 2 M ( x -a )- A (x ) except on a s et of meas < ^ |I| A(x) = 2M(x-a) - A lx = A j A '(x)= 2M - A j = A j

|-M - | m < A'n < ^

+ |m

The main result is the following. T H E O R E M (G . D a v id ) .

for each I there e x is ts

A ssum ing that there e x is t 8 > 0 , c > 0 such that K j(x,y) s a tis fy in g standard e stim ate s uniformly

in I with T .(f) = I K T(x ,y ) f ( y ) d y

/ s a tis fy in g HT lH

2 2/ L^CD.L^d)

and there is a su b se t E C I w ith VxeE,

VyeE ,

—C 0

u

|E| > S|I| such that K ^ x .y) = K(x,y) .

38

R. R. COIFMAN AND Y V E S MEYER

Then T maps L°° to BMO with

llTllL

,BM O

C such that

d

= a (z) + ^ •

N O N -LIN E A R HARMONIC A N A L Y SIS

45

It is c le a r from these and other exam ples that the identification of the space of holomorphy, or a detailed study of the first bilinear operation A j ( a ) f is basic to the understanding of these functionals. R. R. COIF MAN DEPARTMENT OF MATHEMATICS Y A L E UNIVERSITY NEW HAVEN, CONN.

Y V E S MEYER CENTRE de MATHEMATIQUES E CO LE POLYTECHNIQUE P A L A ISE A U , FR AN CE

REFERENCES [1]

A. P. Calderon, Cauchy Integrals on L ip sc h itz curves and related operators. Proc. Nat. Acad. S c i., U.S .A. 7 5 ( 1 9 7 7 , 1 3 2 4 - 1 3 2 7 .

[2]

R . R . Coifman, D. G. Deng and Y . Meyer. Domaine de la racine caree de certain s operateurs d iffe ren tie ls a c c re tifs . Ann. Inst. F ourier 3 3, 2 (1983), 1 2 3 - 1 3 4 .

[3]

R . R . Coifman, Y. Meyer, L a v r e n t ie v ’s curves and conformal map pings, Rep 5. 1 9 8 3 , Mittag L e ffler Inst., Sweden.

[4]

R . R . Coifman, A. McIntosh and Y. Meyer. L ’integrale de Cauchy delfinit un operateur borne" sur L 2 pour les courbes lip sch itz ien n e s. A nn als of Math. 1 1 6 ( 1 9 8 2 ) , 3 6 1 - 3 8 7 .

[5]

R . R . Coifman and Y. Meyer, Au dela des operateurs pseudodiffere n tie ls . A ste risq u e 57. S ociete Mathematique de F rance (1978).

[6 ]

G. David, Operateurs integraux sin guliers sur c ertain e s courbes du plan complexe. Ann. S cient. Ec. Norm. Sup. 4 ° s e r i e , 1 7 ( 1 9 8 4 ) , 157-189.

[7]

G. David, J. L. Journe, “ A boundedness Criterion for Calderon Zygmund o p e ra to rs ,” A nnals of Math. 1 2 0 ( 1 9 8 4 ) , 3 7 1 -3 9 7 .

[8 ]

E. F a b e s, D. Jeriso n and C. Kenig, Multilinear L ittle w o o d -P ale y estim ates with app lication s to p artial d iffe ren tia l equations. Proc. Nat. Acad. Sci. U.S.A . 7 9 ( 1 9 8 2 ) , 5 74 6 -5 7 5 0 .

[9]

R. R o s a le s , Exact Solutions of Some Nonlinear Evolution Equations, Studies in Applied Math. 59, 1 1 7 - 1 5 1 .

[10] E. M. Stein, Singular Integrals and D ifferen tiab ility P roperties of Function, Princeton U niversity P re ss (1970).

MULTIPARAMETER FOURIER ANALYSIS Robert Fefferman Introduction The a r tic le which fo llo w s is an attempt to give an exposition of some of the recent progress in that part of Fourier A n a ly s is which deals with c l a s s e s of operators commuting with multiparameter fa m ilies of d ilations. In some s e n s e , this fie ld is not that new, s in ce already in the early 1 9 3 0 ’s the properties of the strong maximal function were being in v estigated by Saks, Zygmund, and others.

However, for many of the problems in this

area which seem quite c l a s s i c a l , answ ers have either not been found at al l , or only quite a short time ago, s o that our knowledge of the area is s t i l l fragmentary at this time. The a rticle is divided into s ix s e c tio n s .

The fir s t treats some basic

is s u e s in the c l a s s i c a l one-parameter theory whose multiparameter theory is then d is c u s s e d in the remaining s ec tio n s.

Since the reader is no doubt

quite familiar with the main elements of the c l a s s i c a l theory, we have omitted referen c e s to the materials in s ectio n one. The book “ Singular Integrals and D ifferen tiab ility P roperties of F u n c tio n s ” by E. M. Stein is an e x c e lle n t reference for v irtu a lly a l l of the material there. F in a lly , it is a p leasu re to thank P ro fe ss o rs M. T. Cheng and E. M. Stein for a l l of their hard work in organizing the Summer Symposium in A n a ly s is in China, as w e ll as many others whose generous h osp itality made the v i s i t to China such a very en joyable one.

47

48

1.

RO BE RT FE FFERM AN

The maximal function, Calderon-Zygm und decom position, and L ittle w o o d -P a le y S te in theory We hope here to review briefly some as p e c ts of the c la s s i c a l

1-parameter theory of these topics.

The three are in separable and we

hope to stre s s this. We begin with the fundamental C a l d e r o n - Z y g m u n d L em m a.

L e t f(x) > 0, f e L ^ R 11) , and « > 0 .

Then there e x is t d isjo in t cubes

such that

(1)a al| < | U $ k | + 1 ||f||! < ^ l l f l l j . From this weak (1 ,1 ) estim ate, interpolate to get the

result.

We now quote some important examples: 1.

C la s s ic a l Calderon-Zygm und C onvolution O perators. Here T f = f * K

where K(x) is a complex-valued function s a tis fy in g

M U L T IP A R A M E T E R F O U R IE R ANA L Y S IS

(a)

53

|K(x)| < C/|x|n ;

(P)

I

K(x)dx = 0 for a l l 0 < p 1 < p 2 J

P i< ix l c jf| |

R O BE RT FEFFERM AN

54

Now take S ( f ) .

We want to point out here that S is a singular

integral. In fact define K :R n

L 2( r (0 ) ; dydt) by

K(x)(y,t) =

- y) .

Then

and K s a t i s fi e s |K(x + h)-K(x)| < C - M |x|n+1 A ls o by a Fourier transform argument

| h | < J-| x | . 2

||Sf|| 0 L

< c l|f|| o (R

)

L

„

s o the

(R )

Calderon-Zygmund theorem a p p lie s. In fa c t, the adjoint operator a ls o maps L P (L 2( 0 ) -> LP(Rn) , 1 < p < 2 , becau se it is a l s o C-Z, s o again this explains why we get boundedness of S on the ful l range 1 < p < oo. 3.

The H ardy-L ittlew ood Maximal Operator a s a Singu lar Integral. Let

cp(x) e C°°(Rn) and suppose for

|x| < 1 , (x) = 1 and for

|x| > 2 ,

c£(x) = 0 . Then define K : R n 4-> L°°((0, « ); dt) by K(x)(t) = (x/t). Then |Vx K(x)(t)| = |t_(n+1)v ^

S-

< CHV^H

E CK ; |x|

(since if t > |x|/2 , V 0(x/ t) = 0 ). Again |K(x+h)-K(x)| L and we a l s o have

||f*K|| ^ L

Then Mf(x)

M < C J M - if |x|n+1

|f*K(x)|

|h| < I |x| , 2

^ e^x ^ ^or 6 > ^e x iste n c e a .e . of

lim f * K c(x) for f 0 estim ate

|{T*f(x) > a\\ < ^ a f R n |f| . It turns out that by using the Hardy

Littlewood maximal operator it is not d ifficu lt to p ro v e T*f(x) < ClM(Tf)(x) + Mf(x)! which immediately gives the boundedness of T* on LP(Rn) for p > 1 . However, it fa ils to give the weak type inequality for functions on L 1 (Rn) .

This inequality fo llo w s e a s ily from the observation that T* is

a singular integral. Let

if

|x| < 1

( 0 if

|x | > 2

(1

L 2(L°°) , s o H is w eak 1 -1 . 5.

The Maximal Function a s a L ittle w o o d -P a le y S te in Function. Let

f e L 2(Rn) , f(x) > 0 for a l l with a =

x . Use the Calderon-Zygmund decomposition

, j 0

cubes Q] where k

d eco m p o sitio n ,

—h r f IQU Qj k

k

f ^ C K Define f. a s in the Calderon-Zygmund J

[ - L

I

J

Q"

)m J f

su ffic ien tly large, to get (dyadic)

f

f(x)

'

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

k

if

XfQl

if

x/UQj k

k

56

RO BE RT FEFFERM AN

(1)

A -f liv e s on U Q] J

k

k

and has mean value

(2 ) A|f is constant on every

for

0 on each Q] . k

i 0 as

j -» - oo and f •-»f as J

j -» + oo s o f =

2

j=-oo

A -f. J

From (1) and (2) it is c le ar that the A jf are orthogonal so that

F in a lly , observe that the square function (S|A-f(x)|2) j

1 /2

e s s e n ti a lly ju st the dyadic maximal function.

In fact, if

then x (

It follo w s that

for some k and A j(f) (x )" v ,C^.

is

J

«

Mg(f )(x)

1/2 > cMgf(x) .

Before finishing this s ectio n , we s h a ll need estim ates near L 1 for the maximal function. If Q denotes the unit cube in R n then for k a positive integer

J * Mf(log+ Mf )k 1dx < oo if and only if

J * |f|(log+ |f|)kdx < oo .

Q

Q

The proof runs as f o l l o w s : If f e L (lo g + L)^ then

|ix|Mf(x) > a||

J | f(x )| > a / 2

and s o

f(x)d x

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

57

l Mf|1T „ k -!< L (,,l o g L)

00

^2I1ff(x)| ool

n

J

J ° ( l o g a ) ^ 1 ^-

J

|f(x)|dxda

cr/2

|f(x)|J^

Qq

< f

(log a )k - 1 dadx

1

k-

L(log L ) k

C o n v e rse ly , (Stein) Calderon-Zygmund decompose R n at height a > 0 . We have

J*

f(x)dx

C a

f(x)dx

cai| . k

This yield s 00

J

|f(x)|(log' Mgf(x))kdx

C na

OO

J*

h

J'

|f(x)|dx • (log a ) k_1da

M g f ( x ) > C na

< J

Mf(x) [log+Mf(x)]k~’1 dx . Qk

2.

Multi-parameter differentiation theory During the fir st lecture we d iscu ssed some fundamentally important

operators of c l a s s i c a l (and som etim es, not s o c la s s i c a l) harmonic a n a ly s is :

the maximal operator, singular in tegrals, and L ittle w o o d -P ale y -

58

RO BE RT FEFFERM AN

Stein operator. These operators a l l had one thing in common. They a l l commute in some se n s e with the one-parameter family of dilations on R n , x ^ dx , § > 0 .

The nature of the re a l variable theory involved does not

seem to depend at a l l on the dimension n . In marked contrast, it turns out that a study of the analogous operators commuting with a multi parameter family of dilatio n s re v e a ls that the number of parameters is enormously important, and changes in the number of parameters d ra s tic a lly change the re s u lts . Let us begin by giving the most b asic example, which dates back to J e s s e n , Marcinkiewicz, and Zygmund. We are referring to a maximal opera tor on R n which commutes with the fu ll n-parameter group of d ilations (x 1 , x 2 , ••*, xn) -» (5 1 x 1 , ^ 2x 2 , •••,^nxn) , where S- > 0 is arbitrary. This is the “ strong maximal operator,” M ^ , defined by

R

where R is a rectangle in R n whose sid e s are p arallel to the axes. like the c a s e of the Hardy-Littlewood operator, | i x | M(n)( f ) ( x ) > d | < g

||f||

Un

does not s a tis fy ,

.

L (R )

For i n s t a n c e whe n then for

|xj|,

n=2

and when

fg = s ' 2 ^(|xj X (| X l || 2 5 ,

M(2>(fg)(x1;x 2) =M (1 >(x|X i | |jx| |xj| |x2 | < L if a = 1 /8 .

and

8

< jxjl < 1 5| ~

log

,

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

59

If we have a weak type inequality, we must have ||M a l| < L ||fg|| so that ||fg|| > c log g- and the smallest Orlicz norm for which this holds is the L(log L) norm. A similar computation in R n reveals that for to map

boundedly to Weak L 1 we must have

The next theorem shows that indeed

C L(log L)n_1

does indeed map L(log L)n_1

boundedly into Weak L 1 . T heorem

o f J e s s e n - M a r c i n k i e w i c z - Z y g m u n d ( 1 9 3 5 ) [1].

For

functions f(x) in the unit cube of R n we have |!xf Q 0 ,M(n>(£)(x)>a!|g||f|| L ( l o g L)

(Q0 )

The proof is strikingly simple. Define M maximal function in the i

to be the 1 -dimensional i coordinate direction. Consider the case

n = 2 , which is already entirely typical. Let R be a rectangle contain ing the point ( x 1 , x 2), say R = I x J .

Then

g J J |f(x1 ,x2)|dx1dx2 = L J / j - J f C x j . x ^ d x J d X j R

(2 .1)

I '

J

j ' |f(xr x 2)|dx2 < Ul

J so (2.1) is

~ Fi f

M2f(Xl ’X2) d x l - Mx / Mx 2f ) ( Xl ’X2) I

'

60

R O BE RT FEFFERM AN

Thus, for a l l ( X j, x 2) f Q 0 ,

M(2 )f( x 1,x 2) < Mx ^ o Mx ^(f )(x1, x 2) .

We have seen that Mx

maps L(log L)(Qg) boundedly into L 1(Q0) so

that l,Mx2 f|lL 1 ( Q 0 ) -

C||f|lL d o g L ) ( Q 0 ) ’

and finally | t x « Q 0 |MX i M X 2 f ( x l > c l |

< % ' l “ x 2f l L l (Q o) £ ¥

I 'I lo o ,

L , cn |URk | 1_ n-1

W

l|exp( 2

% k)

l L i (B) < c •

Before we prove this theorem, let us show that it implies the J e s s e n Marcinkiewicz-Zygmund result.

L et a > 0 , and for each point

X f lM(n)f(x) > a ! there is a rectangle Rv containing x with

(2.2,

J 'R

x

Without lo ss of generality we assume

URX = URk where R k are certain

if the R x ’s . Apply the covering lemma to get R k with properties (1) and (2) above.

Then by virtue of (1) we need only show that

M * k l < £ M lL, (, l, o g L) B y (2.2),

i^ k i 4

|R^|

(Q0 )

|f I an d summing we have

f

< i - a

L ( l o g L)

n—1

e x p (nL l / ( n - l k' )

Now, let us prove the covering theorem. We s h a ll proceed by induction on n .

Assume the c a s e

n -1.

L et R 1,R 2 , •• *,R k , •• • be ordered such

that the x n side length d e c r e a s e s . For a rectangle R , let R^ denote the rectangle whose center and x^ side lengths,

i < n , are the same as

those of R , but whose xn s id e length is multiplied by 5 . Then we '"V.

'X,

d escrib e the procedure for s e le c tin g the R k from the R k : Let R 1 = Suppose R 1, R 2 ,***,Rk have a lre ad y been chosen.

.

We continue along

the lis t, and each time we consider the rectangle R we a s k whether or not

62

R O BE RT FE FFERM AN

|r n[u(Rj)d]|

h IRI •

U (»i>d St d r ^ b e f o re R

L e t us s l i c e a l l rectan gles with a hyperplane perpendicular to the x n a x is . Then if s li c e s are indicated by using S ’s

|sn[u(Sj)d]| > s o that

\ ( j)d

x i , x 2 , **’ , x n - l

1

coordinates.

instead of R ’s ,

\ |s|

on U R :, where J

is acting in the

By the boundedness of

on, s a y ,

(by induction) we have IUR j j < C|U(Rk)d | < C' ^ To obtain (2), notice that the R j ’s

Rk n

U

l&kl ^ C 'lu ^ k l • s a tis fy

(Rj)d

0 where

is a function increasing in each variab le se p a ra tely, fixing the

other variab le.

In other words, Zygmund next conjectured that since 31

is a 2-parameter family of rectangles in R 3 , the corresponding maximal operator, which we s h a ll c a l l Mz operator

should behave like the model 2 -parameter

in R 2 : |iMz f(x) > a, |x| < 1 1| < g ||f||L(log D d x l d ) •

Not long ago, using the methods we have ju st d is cu s s ed , Cordoba was able to prove this [7].

L e t us give the proof. Suppose

sequence of rectan gles with sid e lengths s ,t , must show there e x is ts a s u b co llection (1)

|URk |>c|U R k |

( 2)

II 2

^

Hexp(L) — C

and

is a s,t) in R 3 . We

such that

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

To prove th is, order the

69

so that the z side lengths are decreasing.

With no lo ss of generality, we may assume that that there are fin ite ly

|R k ^ [ U Rj]| < j- lR jJ > j^k many R^ and that the R^ are dyadic. (In fa c t, we

may assume this because if

/ |f| > a for some R e 5ft containing x , lR l R then there e x ists a dyadic R^^ whose R j (double) contains x such that —L f |f| >% .) Now let R , = R and, given |R,| R[ c 1

I

s e l e c t I?k ,

K

as fo llo w s: Let R ^+1 be the f i r s t R on the li s t of R^ so that

iSl J “ P( | ^ ) d x £ C ' We claim that the R k s a t i s fy $ k be R j , •• •,

/ “ ( I

f e x p ( 2 X-& ) 1 ^ • |x|j

* * ■ )= / " ■ ( ? , XEr- ) dx +

U» j

To s e e this let the

V

f

x*■) +" '+« f e l

*1

and

i M £

I ' x X

) £ c / exp( i ^ 7 ft.

) scii?i1'

s o we have

exp( S

Now let us show that rectangle. Then

\ ) ^

c S

^

•

|URj| > c|URj| . Let R be an u n selected

ROBERT FEFFERMAN

70

R

where the sum extends only over those chosen L e t us s li c e

which precede R .

R with a hyperplane in the x 1?x 2 direction. C a ll S,Sj the

s l i c e s of R and Rj . Then

jsj

f exp(XX^)dXldX ^C 2

(Again we sum only over those Sj which appear before S . ) Now, each Rj appearing before R has the property that its x 3 (or z ) sid e length e xce e d s that of R .

It fo llo w s that each corresponding Sj has either its ''V/

Xj

or x 2 side length longer than that of S .

C a ll those Sj having

longer x 1 sid e length than x 1 length of S of type I, and the other of type II.

Put S = I x J . Then

c * r a / / exp( 2 ^ J) ■ iifiJi / / “ p( 2 v | IxJ

1

x ) d,i,dX2

J

so it follows that > C X1 on URj ; hence

URj C V C '

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

71

i < C 1 UR j-|

So far what has been done su g g ests the follow ing general conjecture of Zygmund which s a y s :

L et { ^ ( t p t ^ - * * , ^ ) } = O , i = l,2,--*,n be fu n c

tions which are increasing in each of the v a r ia b le s t- > 0 sep a ra tely. Define a k-parameter family of rectan gles

Rf f f by 1 2 ’ ’ *' k

and a maximal operator on R n by

M0

I

|f(x+y)|dy.

l ^ t . . ‘“ .ti i J

T’ Then |{x||x| < l . M $ (f)(x) >«}|

||fII

k_ 1 . L ( l o g L)

Quite recently a b eautifully simple counterexample to the general c o n jecture was given by Fernando Soria of the U niversity of Chicago [8 ]. S o r i a ’s counterexample w as:

in R 3 , consider a l l rectangles @ of the

form s x t ^ ( s ) x t/s and 0 2(s) = ( ^ ( s ) on and

^>2(s ) - ^ i ( | ' 2 “ k_1)

3.

for a l l s f [ 2 _k'"1 , ~ ■ 2 _kJ

.

M ultiparam eter weight-norm in eq u a lities and ap p lic atio n s to m u ltipliers In th is lecture we want to describe further applications of the ideas

centering around the covering lemma for rectan gles previously described. We s h a ll begin with more about maximal operators, and then move on to multiparameter multiplier operators, and the connection they h ave with our maximal functions. The first topic we take up is that of c l a s s i c a l weight norm in eq u a lities, which have proven of enormous importance throughout Fourier a n a ly s is . Here, we want to know which lo c ally integrable non-negative weight func tions

w(x) on R n have the property that some operator T is bounded

on L^w(x)dx . The most b asic exam ples are the Hardy-Lit tie wood maximal operator, and Calderon-Zygmund singular integrals Tf = f * K . The theory was developed in R 1 by Muckenhoupt [9] and Hunt, Muckenhoupt and Wheeden [10], and in R n by Coifman and C. Fefferman [11].

We w ill

present only a sm all segment of that theory now and list some relevant fa c ts for which the interested reader should s e e the Studia article of Coifman-C. Fefferm an [11]. It is no coincidence that the c la s s of weights w for which the HardyLittlewood maximal operator is bounded on L^(w) is e xactly the same as the c la s s of w for which a l l Calderon-Zygmund operators are bounded on LP(w).

This is the so-called

c la s s of Muckenhoupt.

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

73

A nonnegative lo c a lly integrable function w(x) on R n is said to belong to AP if and only if for each cube Q C Rn

The sm a lle s t such C is called the AP norm. We s a y that w e A°° if and only if, whenever Q is a cube and E C Q ,

if

|E|/|Q| > 1/2 then

w(E)/w(Q) > 77 for some 77 > 0 . Let us list some properties of AP c l a s s e s : (a)

If p > 0 and w e PP then pw e AP with the same norm a s w .

(/3) If w e AP and

8

> 0 then w( 8 x) e AP with the same norm a s w .

(v)

If w f AP then w _1 /(P"- 1 ) e AP where —' 4- ~ = 1 .

(8 )

If w e AP then w e A°°. In fa c t, if w 1/2

implies

w(E) > 77. (For, in general if Q is arbitrary of sid e w( 1/2 |Q| , consider

and multiply w( 8 x) by the right constant p to have

f ^ / 8 pw( 0.

The constant Cg may be taken arbitrarily

>0.

(£) From (e) it is immediate (see a l s o (y)) that w e AP implies w e for some q < p . (77)

If f is a lo c ally integrable function in some

0 < a < 1 then (Mf)a 6 A 1 , i.e.,

s p ac e and

M((Mf )a)(x) < C(Mf )a(x) (for w e A 1

implies w e A^ for a l l p > 1 ). To prove this let f e LP(Rn) be given and a e ( 0 , 1 ) . L e t Q be a cube centered at x , and Q its double.

Then write f =

f j + f Q. We must show that

Tj

J

M(fj)a dx < CM(f)a ( x )

Q and ^

J

M(f0)a dx < C M ( f ) « ( x ) . Q

A s for the first inequality,

=

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

75

(This is an immediate consequence of the w eak type estim ate for M on L 1 .) This shows that

< M(f)a ( x ) .

JQ

'

VX' ^

7

A s for the second inequality, choose a cube C centered at x such that

± J igdx > \ M(f0,:)(x)

.

Now, if x eQ then any cube C' centered at x which in tersects

cv

is

contained in sid e a cube of comparable volume centered at x s o that

c'

c

We have proven that for a l l x e Q , M(f0)(x) < A •— f

|fQ|dx so that

|C| c

v a

( L

J

|f0 |dxj

c

< AaM(f )a* 0 0 •

/

L et us begin to d isc u s s the weight theory by showing that the HardyLittlewood maximal operator is bounded on L^(w) if and only if w r A P , 1 < p < oo [12].

In the first place if f = w _1 /(P- 1 )

and if M is

bounded on LP(w) one s e e s right aw ay that

w P ^P *^wdx Q or

76

RO BE RT FEFFERM AN

w i2 )(_ L i q i I iqi ;

('

w-i/(p-i)\P

< c

.

C o n v e rse ly , assume w e AP . Calderon-Zygmund decompose f 6 LP(w) at heights C^ where k e Z , and get Calderon-Zygmund cubes

-X -

f

C is large (to be described later) and

SQ^.

so

that

f - C k and l M f > y C k ! C U § k

Wj \ Jo k

- J

J

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

f f\ .

—-

Qk /

Now w e AP

w e A°° therefore w(Q^) < C^w(Q^) and so the above

e xp ressio n is

0 .

It

fo llo w s that Mf(x) < CMw (fP“ £) 1 /(P - 6) ,

and it just remains to show that Mw is bounded on LP(w). A quick review of the proof that

, n = 3 , is bounded on L,P(w)

re v e a ls that a l l we re a lly used was that w s a ti s fy an A°° condition in the Xj and x 2 v a ria b le s a s w e ll as a doubling condition in the x 3 variab le: w((R)cj) < C w ( R ) . A l l of these are s a tis fie d by our w here, and this concludes the proof sin ce Mw (f ) < M ^ ( f ) . Now we w ish to rela te some of our re su lts on multi-parameter maximal functions to the theory of multiplier operators. We s h a ll work in R 2 , and consider the following basic question: For which s e t s

S C R 2 is

For X s

be a multiplier of course means that, if for f e C^°(R2) we s et

Tf ( f ) = X s ^ ) ^ )

X s^)

a multiplier on LP(R2) for some p 4 2 ?

t^ien we bave the a priori estim ate

llTfHL PP/(R 2)

p 2*

L P(R )

In his celebrated theorem, C h arles Fefferman showed that if S is a nice open s e t in R 2 whose boundary has some curvature then X s ^ ) only be an L 2 multiplier [15]. The other nice regions left are those w h o se boundaries are comprised of polygonal segments. polygon then

If S is a convex

w ill obviously be an LP multiplier for a l l p ,

1 < p < oo,

ju s t becau se of the boundedness of the Hilbert transform on the LP s p a c e s in R 1 . The c as e that remains is the one we consider here. >

02

>

63

••• > 0 n > @n__i

Let

0 be a given sequence of angles

0

and let

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

85

Figure 2 be the polygonal region pictured above. Then we s h a ll define V 'te ) =

■

Consider as w e ll the maximal operator

M^f(x) = sup

by

-i-

defined by

f

|f |dt

xeReBn lK l J U

where

R

is the fa m ily of a l l rectan gles in R 2 which are oriented in

one of the d irectio n s

0 ^ , but whose side lengths are arbitrary.

We claim

86

RO BERT FEFFERM AN

that the behavior of Tq on LP(R2) for p > 2 is linked with the b e havior of

on L^P//2 ^((p/2) y is the exponent dual to p/2 ). More pre

c i s e ly , suppose that

is bounded on LP(R2) and we assume the

w e a ke st p o s s ib le estimate on M^, namely is of w eak type on L^P7^ l

(P/2) (R2) then

||m^Xe >

. C o n v erse ly , if

- ^|E| • Then

is bounded on

is bounded on L^(R 2) , for p'< q < p

[16].

To prove this assume first that Tq is bounded on LP(R2) .

Then

the fir s t ste p is to notice that this implies that 1/2

1/2

|IjJ the

RO BERT FEFFERM AN

88

segments pictured contain at least 1 / 1 0 0 of their measure in E^. duplicate the rectangle Applying

as shown, on th e se segments T^f^ > 1 /1 0 0 .

= Hilbert transform in the direction perpendicular to

we s e e

If we

^k^k^k^ > 1/^00 on

°f

6

^ to

Repeating tw ice more

we get

| ( S x Srk)

||l p - c j j ( X

This shows that

is of weak type (p/2) ' .

C o n v erse ly , assume that Sk =

is bounded on L^p//2^ . Define

= (£v f 2)|2^ < f i < 2 ^+1 i . Then if

multiplier operator corresponding to

W q To estim ate

~|(SWI2)

, we s e e that

/2||q

||(2 |SkT kf |2) 1 / 2 ||2 , let

J S

a l s o stan d s for the

!T ks kf i2 ^ < 2

=j

(2

'S kT kf |2 ) 1 / 2 ||q

•

ll' = 1 and let us estimate

/ 'T ki2 ^ •

But in R 1 we have the c l a s s i c a l weight norm inequality for the Hilbert transform: 0 .

It fo llo w s that (3.3) is

(3.4)

is bounded on that (3.4) is

< X

^

2m /'.-A j |Sk(f)|2M^ 1+£)

)

^ 1+e) if e is su ffic ien tly small.

It fo llo w s

89

M U L T IP A R A M E T E R F O U R IE R A N A L Y S IS

< c | | ( 2 l s kf l2) | L(1/ 2 < c 'llf llLq proving that

4.

is bounded on

.

HP s p a c e s — one and s e v e r a l param eters In this lecture we wish to d is c u s s another chapter of harmonic

a n a ly s is relatin g to differen tiation theory and singular in tegrals, namely Hardy Space theory.

In this lectu re, we s h a ll d is c u s s the one-parameter

theory, and, in the next, the theory in s e v e r a l parameters. ning when

In the begin

s p a c e s were fir st considered, they were s p ac es of complex

analytic functions in

= iz =x + iy |x 0 } which s a t i s f i e s

the s iz e restriction +oc

v l /p |F(x+iy)|Pdx )

0 , we have -f-oo

|IF H , d e fs u p

H

t>0

J |

|F(x+it)|dx ~ ||u|| J

!

L (R )

+ \\v\\ l

x .

L (R )

—OO

So we may view the sp ace

H1 through its boundary v alu es a s the space

of a l l real valued functions f e L 1(R 1 ) whose Hilbert transforms are as well.

L1

RO BE RT FE FFERM A N

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 = {(x,t)|x eRn, t > 0 ! whose

v a lu e s lie in R n+1 :F ( x ,t) = (u 0(x,t),« •-,un(x,t)) where the u-(x,t) s a t i s f y the “ G eneralized Cauchy-Riemann e q u a tio n s ,” “

Ai.

2

S T (x,t) s 0 ( t = x o} i=0 1

and (9u-

^uj

(9xj

dxi

for a l l i,j .

T h ese Stein-Weiss analytic functions are then said to be HP(R^+1) if and only if sup/ f

|F(x,t)|Pdx\

»■ [ J .

/

= 1|F ||

[17].

hP(r)

Again, these functions have an interpretation in terms of singular in tegrals, sin ce if a Stein-Weiss an a ly tic function F (x,t) is su ffic ien tly “ n i c e ” on R ^+1 , then the boundary v a lu e s u-(x) s a tis fy R-[u0 ](x) where R-

is the i ^

u-(x) =

R ie s z transform given by R^(f)(x) =

C„X1 tn i i f * — — . In particular we may consider an H (R . ) function (by

|x|n+1

+

identifying functions in R ^+1 with their boundary v a lu e s ) as a function f with re a l v a lu e s in L 1(Rn) each of whose R ie s z transforms R-f a ls o belong to L ^ R 11) .

An interesting feature of

s p a c e s is that they are

intimately connected to differentiation theory a s w e ll as singular integrals. To d is c u s s this, let us make some well-known observations.

For a

harmonic function u(x,t) which is continuous on R ^ +1 and bounded there,

u is given a s an average of its boundary v a lu e s according to the

P oisson integral:

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

91

u(X’t) = f * P t « ; f(x) = u(x,0) and Pt(x) = — L et T(x) = S(y,t)| |x-y| < t l .

Then sin ce con volvin g with

.

at a point

x can be dominated by an appropriate linear combination of averages of f over b a lls centered at x of different radii, it fo llo w s that if u*(x) =

sup (y .O fH x )

|u(y ,t)| , then u*(x) < cMf(x) .

Unfortunately, if u(x,t) is harmonic, for J' n |u(x,t)|Pdx < C then the domination

p < 1 , u =P [ f] , and u* < CMf is not useful, sin ce

M is not bounded on LP , and it is not true in general that u*(x) < °o for a.e. sa y

x e R n . On the other hand, suppose F

F f H 1^

is Stein-Weiss an a ly tic ,

1).

Then a beautiful computation sh ow s that if 1 > a > 0 is c lo s e enough to 1

(a >

then A(|F|a ) > 0 so that

|F|a is subharmonic.

If

s(x,t) is subharmonic and has boundary v a lu e s h(x) then s is dominated by the ave rag e s of h , i.e ., s ( x ,t) < P [h](x,t) . A pplying this to G = |F|a (which has

/ G 1 //a(x,t)dx < C for a ll t > 0 )

we s e e that G* < M(h) for some h e L 1 s o that M(h) e L 1

and s o G* e L 1

. Now Mis bounded on . It fo llo w s that F * c L 1 .

J u s t as for a random f e L 1 ^ 11) we do not n e c e s s a rily have R-f e L ^ R 11) (singular integrals do not p reserve L 1 ) it is a ls o not true that for an arbitrary

L 1 function f that for u = P [ f ] , u* f L 1 . But if f f H ^ R ^ 1 )

then u* f L ^ R 11) .

Thus the nontangential maximal function F*(x) =

sup IF(y,t)j e L 1(R n) ( y .O f H x )

if and only if the a n a ly tic function f e H1(R^l+1).

RO BERT FEFFERM AN

92

We know so far that we can characterize HP functions in terms of singular integrals and maximal functions.

There is another c h a ra c te riz a

tion which is of great importance. To d iscu ss it, let us return to HP functions in

as complex an alytic functions,

F = u+i v.

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 * e LP if and only if u* e LP ? In fact, this is true, and the best way to s ee this is by introducing a s p e c ia l singular integral, the LusinLittle w o o d -P ale y -S te in area integral,

re x ) which we already considered in the fir s t lecture. for a harmonic function u ( x ,t) ,

||S(u)||

A s we s h a ll s ee later,

£ ||u*|| L

for a l l p > 0 [18]. L

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

|Vv| = |Vu| .

When we combine the la st two re s u lts , we immediately s ee that

It is interesting to note that the first proof of

||S(u)||

l P 1 > p > 0 was obtained by Burkholder, Gundy, and S ilv e rs te in [19] by

using probabilistic arguments involving Brownian motion. Nowadays direct re a l variab le proofs of this e x is t a s we s h a ll s e e later on. To summarize, we can view functions f in HP sp a c es by looking at their harmonic e x tensio ns belong to LP(Rn).

u to R^ +1 and requiring that u* or S(u)

93

M U L T IP A R A M E T E R F O U R IE R A N A L Y SIS

It turns out that there is another important idea which is very useful concerning HP s p a c e s and their re a l v ariab le theory. spoken of equations.

So far, we have

functions only in connection with certain d iffe ren tia l Thus, if we wanted to know whether or not f e

we could

take u = P[f] which of c ou rse s a t i s f i e s Au = 0 . T his is not n e c e s s a ry . f

If f is a function and

0 n = l , then we may form f * ( x ) =

sup

e C£°(Rn) with

|f * 0 t (y)| , 0 t (x) =

( t , y ) e r (x)

t~n 0(x/ t) and if if/ e C£°(Rn) is su itab ly non-trivial (say radial, non-zero) and

[ifj = 0 we may form

s^(f)(x) =J J |f * 'At(y)|2 ^

t183

'

for

0 < p < oo .

T(x)

Then C. Fefferm an and E. M. Stein have shown that ll£ll p

„ ~ ll£*II p

HP(R )

n

L P(R )

!ls ,/,(0 || ^

n

L P(R )

T hus, it is p o ss ib le to think of HP s p a c e s without any referen ce to particular approximate identities like

P^(x) which re la te to d iffe ren tia l

equations. In addition to understanding the various c h aracterization s of s p a c e s , another important a s p e c t is that of duality of H1 with BMO, which we s h a ll now d is c u s s . A function 0 ( x ) , lo c a lly integrable on R n is said to belong to the c la s s

BMO of functions of bounded mean o sc illa tio n provided

j~-j

J* , * ) - * Q,dx. The BMO functions are re a lly functions defined IQI Q ^

modulo constants and

|| ||BMO *s defined to be sup

| 0

Rn

R+

for a l l functions

J * u*(x)pdx

u on R^J+1 . In connection with this type of measure

there is the characterization of functions in BMO(Rn) in terms of their P oisson integrals.

A function ai| < — Ill'll? a2

-L

J* S( u) a!| .

98

RO BE RT FEFFERM AN

The proof that

||u*|| < C l|S(u)|| r

r

which we ju st gave has been lifted

P

from Charles Fefferman and E. M. S te in ’s A cta paper [18].

To prove

the

re v e rs e inequality we want to go via a different route, and we s h a ll follow Merryfield here [20].

We prove the fo llowing lemma. In the next lecture we

show how this lemma proves

||u*||

> Cp||S(u)|| .

L et f(x) and g(x) e L 2(Rn) , and suppose 0 €C^°(Rn) ra d ia l

LE MMA.

and u = P [ f ] .

Then

J * J ' |Vu|2(x,t)|g *

1^( 012

=1

0 we have

rr

II

d t ic dt

f * * t i ,t 2 0 , l , , 2( * 1 - y 1 . * j - y 2) ' i y 1< i y 2 - ^

2

t2

In fact, taking Fourier transforms of both s i d e s , for the right-hand side we have /*

oo

JJ f c o i^ v ^ 2 R 2x R 2

=f(f) J 2

0

oo

J

|«A(ti ei ,t / 2)|2 ^ l^ = { (^ ) 0

We can use this representation to decompose the function f as fo llo w s :

R e 9 f d . S et ?1(R) = i(y,t) e R 2 x R 2 |y c R , £J < t 1 < 2 £ i where

, i = 1,2 R2 xR2 =

is the s id e length of R in the U

d ire c tio n !.

Since

?r(R), if we define

R ^ d

f R( xx, x 2) =

J J

f(y-t) (2).

It is a l s o

triv ia l that (2 ) = > (1 ), since if f e H1(R ^ x R ^ ),

?xR?>

J '

= J * Hx ^Hx ^(f Xx) g(x)dx

and sin c e f « H1 , Hv Hv (f) 2 ^S, k e Z . Set

a k( x i , x 2) =

2 R£9td

f R(x) •

|Rnfik |>l/2|R| |Rnnk+1| 1/1 OS.

118

R O B E RT FEFFERM AN

Then f = 2

where A.k = 2 ^ 1 ^ ! , and by the strong maximal theorem

|ftkl