Beijing Lectures in Harmonic Analysis. (AM-112), Volume 112 9781400882090

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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