Euclidean Harmonic Analysis

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Departmentof Mathematics University of Maryland,College Park Adviser: R. Lipsman

779

Euclidean Harmonic Analysis Proceedings of Seminars Held at the University of Maryland, 1979

Edited by J. J. Benedetto

Springer-Verlag Berlin Heidelberg New York 1980

Editor John J. Benedetto Department of Mathematics University of Maryland College Park, 20742 USA

A M S Subject Classifications (1980): "31 Bxx, 42-06, 42A12, 42A18, 4 2 A 4 0 , 43-06, 4 3 A 4 5 , 4 4 A 2 5 , 4 6 E 3 5 , 8 2 A 2 5 ISBN 3-540-09748-1 ISBN 0-387-09748-1

Springer-Verlag Berlin Heidelberg NewYork Springer-Verlag NewYork Heidelberg Berlin

Library of Congress Cataloging in Publication Data Main entry under title: Euclidean harmonic analysis. (Lecture notes in mathematics; 779) Bibliography: p. Includes index. 1. Harmonic analysis--Addresses,essays, lectures. I. Benedetto, John. I1. Series: Lecture notes in mathematics (Berlin); 7?9. QA3.L28 no. 7?9 [QA403] 510s [515'.2433] 80-11359 ISBN 0-387-09748-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher~ the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

TABLE OF CONTENTS

INTRODUCTION

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1

L. CARLESON, Some analytic p r o b l e m s r e l a t e d to s t a t i s t i c a l mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . Y. DOMAR,

On spectral synthesis

in

~n,

n ~ 2 . . . . . . . .

46

L. HEDBERG, Spectral synthesis and stability in Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

R. C01FMAN and y. MEYER, Fourier analysis of m u l t i l i n e a r convolutions, Calder6n's theorem, and analysis on Lipschitz curves . . . . . . . . . . . . . . . . . . . . . . . . . . . .

lO4

R. COIFMAN, M. CWIKEL, R. ROCHBERG, Y. SAGHER and G. WEISS, The complex m e t h o d for i n t e r p o l a t i o n of operators acting on families of Banach spaces . . . . . . . . . . . . . . . . . .

123

A.

CORDOBA~

i.

Maximal functions:

2.

Multipliers

of

.

]54

F ( L p) . . . . . . . . . . . . . .

a problem

of

A.

Zygmund

162

INTRODUCTION During Euclidean lecture

the

series

molded

semester

analysis

comprising

Euclidean a vital

spring

harmonic

harmonic

relationship the subject

this

fundamental

and,

not only

theory

provides

correlation problems,

in turn,

In the

first

the two m a i n

lecture

problems

fication

of e x p e c t e d of the Gibbs

a include

function,

discusses

series

theory

as a r i g o r o u s

equation

ator

and

ensemb l e

systems.

results

progress

in applications.

The

first

that

Fourier

series

f

of problems

b. the

of phase first

to e q u i l i b r i u m

the Gibbs

of

free e n e r g y

that

of h a r m o n i c theory

analysis

it

oscill-

for an

is p e r v a s i v e

is to introduce

in

some

lead to further

in this

volume,

as well

into one or the other

as

of two

synthesis

is an element

such

series.

problem

famous of

Carleson

function

deals w i t h

f

synthesis

is to d e t e r m i n e

converges

question

L2[0,2~) answered

of pre-

gave

whether

or not

in some d e s i g n a t e d

in this area

and c o n v e r g e n c e

this

is the p o i n t w i s e

C. F E F F E R M A N

the

a given phenomenon.

of a function The most

everywhere.

every

and

The results

b Carleson

properties

contained

addressed

a. the veri-

of the existence

harmonic

fell

with

and these

CARLESON

of the

In part

lectures

visitors,

to d e s c r i b e

function.

in w h i c h almost

category

fundamental

Fourier

to the

by our other

spectral

associated

properties

w h i c h may e v e n t u a l l y

series

is used to this

of problems.

harmonics

The the

lecture

L.

and the a p p r o a c h

Classical

and p r o b l e m s

The r e m a i n i n g

scribed

models.

and the point of his

analytic

categories

verification

dynamical

of

Wiener's

problems,

systems.

properties

shows how one can verify

of such

the lectures

volume

thermodymamic

classical

He then considers

analysis

for

example

spaces.

for d y n a m i c a l

describes.

his approach;

Hp

light;

statistical meehanies:

of the basic

for c e r t a i n

systems

to

of this

equilibrium

a Boltzmann

Fourier

a neat

t h e o r e m but

such as white

have

applications

interplay.

number

six

and m a i n t a i n s

in fact,

provides

and p r e d i c t i o n

of c l a s s i c a l

proofs

as well

transition

for the

in f i l t e r i n g

lead n a t u r a l l y

validity part

the prime

theory

which,

mysterious

in

The

part of our program.

significant

theorem

for p h e n o m e n a

perspective

functions

a major

areas

it with

extent,

characterizes

properly

other

Tauberian

to some

were

a program

of Maryland.

has a rich basic

several

and e n l i v e n e d

Wiener's

spectra

volume

analysis

150 years.

define

this

with

over

theorem

of 1979 we p r e s e n t e d

at the U n i v e r s i t y

question almost

a conceptually

treats

way

the case

is p o i n t w i s e

in 1966

by proving

everywhere

sum of its

different

proof

of

Carleson's

theorem

as a c o m p a r i s o n lecture Math.,

between

series. 98

volume,

in 1973,

Since

(1973)

space

is

in nature

L2

operators theorem

Carleson's

first

and depends

formulated

classical

is the formula,

expresses

SNf

boundedness

into

the basic

IISN(.)f(')II I ~ CHfll2, where

that N

(i)

from the

For each

pieces

by making result

for the case

of

a proper

dyadic

In his N(x)

= ix, w h i c h

Carleson's the

analyzes

corresponding

function

both Y. D O M A R

let

of the

method

Synthesis

the

contain

category X

subset

was

of

the

space

simplicity

the given N, and subject

and L. HEDBERG.

and

~n

of d i s t r i b u t i o n s determine

depending

on

of

whether

SNf

and

x.

is

the corresof

T

f

and

method

of the

into

data on small f

or

N,

is o b l i v i o u s does the

lecture

to

opposite.

series

by

fall

into

formulation:

contained

or not a given

the

he e x p l a i n e d sums of

of

following

support

and

illustrated

they d i s c u s s e d the

T

and o r t h o g o n a l

N(x),

frequency

Fefferman's

with

f

TNf(x)

he verifies

function

The problems

approach.

of

to the estimate,

or c o m p l e x i t y

matter

nature its

for F e f f e r m a n ' s

for a r b i t r a r y

of spectral synthesis and have

be a class E

also

both

The

kernel,

in fact contains

and d e c o m p o s i t i o n

which

f.

including

lectures, F e f f e r m a n

procedure

Regardless

of

Dirichlet

decomposition

his

operators

series

independent

and then,

intervals.

estimate,

representation

N(x),

argument;

local

method

inequalities,

germ of the whole combinatorial

Carleson's

N(x), where

to the r e l a t i v e l y

of the d e c o m p o s i t i o n .

the m e t h o d

for

of linear

with,

analysis,

is a f u n ct i o n

follows

H(eiN(x)yf(y)).

Cotlar's

Fefferman's

property

is e q u i v a l e n t

essentially

inequality

comments.

theorem

function

is the

direction

functions

applying

in this

can even be used

hand,

Fourier

transform

IITNfllI ~ CllfIl2, for a r b i t r a r y

ponding

Carleson's

(Ann.

H, and the f u n d a m e n t a l

harmonic

the Hilbert

(i)

DN

transform

by noting

that

lectures

a few of his

To begin

sum of the

in E u c l i d e a n

he begins

of his

appeared

of proof

of the maximal

as well

[ I s u p l S N f ( - ) l l l m ~ CI/fll 2, N

on L 2, provides

Then he o b s e r v e s

method

SNf = DN*f , where

H

proved

proof

subject

his

On the other

partial

of s u b s t i t u t i n g

(i),

included

by Cotlar.

as a Hilbert

of the o p e r a t o r

Instead

N th

the

already

on an o r t h o g o n a l i t y

Vf(L2[0,2~),

SNf

of this

we m e n t i o n

in 1968 Hunt

is an easy c o n s e q u e n c e

(1)

where

not

L log L(loglogL).

was

paper has

omission

that

p > i, and that

for the

Carleson's

we have

of this

We begin by r e c a l l i n g LP[0,2~),

it and

Fefferman's

551-571)

and b e c a u s e

and an e x p l a n a t i o n

in a fixed

element

~ ~ X

is the in

limit

X.

in some d e s i g n a t e d

In Domar's

L ~ ( R n)

and

is a curve

classical

Tauberian in

R 2

of the c u r v a t u r e manifolds in terms

the Fourier

the t o p o l o g y

of Beurling's on W i e n e r ' s

case

E

is weak

spectral

theorem.

of b o u n d e d

transform

E.

~n,

synthesis Domar

He also

of

problem

properties

of

collection

of Sobolev

logy

norm

the

spectral

to the

synthesis

stability,

essentially in various criterion spectral

space

for r e g u l a r

second

of o p e r a t o r s

category

of

Lp

of Zygmund,

The o m n i p r e s e n t

are an e s s e n t i a l theory,

and

In o r d e r operators,

latest

~

cally uses

of problems

feature

X

X

in w h i c h

is e q u i v a l e n t

of closed this

sets

equivalence Wiener's

Sobolev

to v e r i f y

from

theorists~

maps

in h a r m o n i c

were

Commutators

proble m s

of

for elliptic

space

to extend

G. WEISS,

which

spaces

and

The basic functions.

are

associated

constructed

interpolation

result

An i n t e r e s t i n g

the c e l e b r a t e d

with

it p r o v i d e s Large

theory

parts

a means

estimate others,

spaces

is stated

naturally

for

systematifor

to calcu-

H

and

and b i l i n e a r value

when

one

to curves.

Next,

set forth a t h e o r y theorem He dealt points

and with

of the theory

of

Stein's a continuum

of a d o m a i n

for each point

in terms

t h e o r e m which,

ago to

of Coifman's

of c o m m u t a t o r s

the b o u n d a r y

corollary

Wiener-Masani

of real and com-

Boole's

of operators.

intermediate

functions,

and r e l a t e d

long been a staple

the R i e s z - T h o r i n

families

H

in the study of b o u n d a r y

several

includes

maximal

top£cs.

for

and they arise L2

of our

of over a century

theorem. and has

H.

used

the classical

for a n a l y t i c

of Banach

H

of

several

a range

c a l c u lu s

in the context

in joint w o r k with

interpolation

presented

analysis

from the

its g e n e r a l i z a t i o n s

and m u l t i p l i e r s ,

analysis

equations,

as

and

estimates

of Calder~n's

function

lectures

Lp

symbolic

preserving

H

emerged

some of its major

and Y. M E Y E R

Boole's

proofs

measure

various

the h a r m o n i c

have

as well

transform

are

given

D ~ ~n

case,

setting

to c h a r a c t e r i z e

deals w i t h

of the area,

interpolation

late the d i s t r i b u t i o n

theorem

verifies

for

results

and the topo-

and g e n e r a l i z e s

and Stein,

Hilbert

Meyer's

wishes

of

theory,

These p r o b l em s

Calder~n,

R. COIFMAN

plex methods,

maps.

Hedberg

in order

spaces.

guests.

ergodic

is the

for all elements of p o t e n t i a l

E.

problems

spaces

E

in terms

synthesis

In Hedberg's

This

and a n a l y z e s

points

research

the

to

spaces,

in w h i c h

synthesis.

The

Hp

sense

complementary Sobolev

topology.

property

in the

E.

of

setting

ultimately

synthesis

spectral

of the g e o m e t r i c

Sobolev

based

some analogous

n ~ 3, and obtains

can be any one of a large is the

is the

the case

spectral

contained

is a subset

This

considers

solves

measures

X

* convergence.

and he c h a r a c t e r i z e s

of

in

topology

of

D.

of s u b h a r m o n i c is an e x t e n s i o n

in turn,

provides

of

important

factorization

criteria

Finally,

A.

thorough

mix of many

problems

and concepts.

questi o n

CORDOBA

solved

The

first

on the d i f f e r e n t i a t i o n and estimates

maining

results

arising

from c l a s s i c a l

We wish

work;

Dorfman,

Ward

editorial

include

to thank

result

a rather

complete

Berta

problems

involving

second

settles

a basic real

theory

Besides

for m u l t i p l i e r s

Cindy

Edwards,

of our technical

Johnson,

and

Pat Rasternack,

typing

staff

to Alice

C. Robert

Warner

many

of the analysts in our p r o g r a m

for their

at the

University

of Maryland,

included:

L. Ehrenpreis

A.

Picardello

Baernstein

E. Fabes

H.

Pollard

M.

Benedicks

C. Fefferman

E.

Rrestini

G. Benke

R. Fefferman

F. Ricci

R. Blei

A.

G. Bohnk6

L. Hedberg

P. A.

H. Heinig

D. Sarason

R. Hunt

P. Sarnak

Boo

L. Car l e s o n

Fig~-Talamanca

L. Rubel C. Sadosky

P. Casazza

C. Kenig

P. Soardi

L. C a t t a b r i s a

T. K o o n w i n d e r

A.

R. Coifman

J.

J.-O.

C6rdoba

Lewis

Stray Stromberg

L. Lindahl

N. Th.

L. de Michele

L. Lipkin

G. Weiss

J. D i e u d o n n 6

Y. Meyer

G. W o o d w a r d

Domar

Robert

Benedetto Park, M a r y l a n d

A.

P. Duren

for their

Chang,

M. Ash

Y.

The re-

assistance.

the p a r t i c i p a n t s

of

variable

on a c o v e r i n g

function.

John J. College

A.

a

category

and depends

maximal

our a p p r e c i a t i o n

Raymond

problems.

methods.

Casanova,

Slack,

and p r e d i c t i o n

in this

of integrals

summability

and to express Evans,

specific

on the a p p r o p r i a t e

Schauer, and June

expert

several

filtering

of the real methods

theorem

Becky

for certain

Varopoulos

C. M o z z o c h i

R. Yamaguchi

D. Oberlin

M.

Zafran

SOME A N A L Y T I C PROBLEMS RELATED TO S T A T I S T I C A L MECHANICS Lennart Carleson Institut M i t t a g - L e f f l e r

Apology.

In the following lectures,

I shall give some analytic

results which derive from my interest in statistical mechanics. not claim any new results statistical mechanics

for applications,

It is my hope that

that i n t e r e s t i n g and difficult analytic

problems are suggested by this material; make c o n t r i b u t i o n s

and any serious student of

should consult other sources.

analysts will find, as I have,

I do

of real significance

and that they will e v e n t u a l l y in applications.

I.

Classical i.

We

Hamiltonian

Statistical

consider

Mechanics.

a system

N

particles

classical

=

equations

for the m o t i o n

are oH

qi are

the

It f o l l o w s preted

momenta that

as the

l 2 7 ~ Pi +

We

now

assume

~0-1N,

where

H

Denote the

basic

~(t)

the m o t i o n is the

during

assumption

S

at least ables

for s i m p l e

and

is more

belonging

natural

C~.

to a s s u m e

~

where

the Gibbs with

a bounded

responding late

this

T

limit.

avoids

in m o r e

a

A natural

number

number

I

of vol-

total

energy

surface

en-

particle. EN

in

~ = (p,q). mechanics

is now

that

the mo-

i.e.,

/~N ~(p'q)d~ =

lim N~

from

a(Z N)

on a f i n i t e

number

a physical

point

zero.

We

then

is also

that

of vari-

of view

it

~(c~(t))dt 0

set of

density

assumption

of d i f f e r e n t

of s y m m e t r i e s detail;

inter-

that

lim ~1 T~exists,

is

AN

The

of the

depending

Actually,

a box

per

energy-surface~

functions to

inside

particles. energy

of s t a t i s t i c a l

i ~T lim lim ~ J [ e(@N(t))dt N~ T~ 0

H

is

average

element

particles.

(q3i+l' q3i+2' q3i+3).

place

of points

on the

and

situation

=

of the

is the

surface

space

takes

for the

the m o t i o n

~i

density k

coordinates

A typical

~ %(qi-qj )'

the

is e r g o d i c

: -~qi

position

system.

so that

d~

Pi

is c o n s t a n t

~ i~j

p

6N-dimensional The

tion

by

the

of the

that

= E N ~, XN,

~H 0p i ' qi

H(p,q)

=

ergy

and

energy

H(p,q)

ume

to a

H(Pl,...,P3N,ql,...,q3N).

_

(i) Pi

movin F according

function H(p,q)

The

of

Backsround

here

particles,

in the

the m e a n i n g

and

function in c o n c r e t e

we

therefore H.

I shall

cases

speak are have

of

dealing a cor-

not

formu-

is q u i t e

clear.

Gibbs'

contribution

ing the density

here is that he has given a formula for comput-

do/~(Z)

= d~.

Let us observe

d~ = do dE Let

~

be a parameter F(~)

where

V(E 0)

in

that

~N"

and consider :

I

e-~E d~

is the volume

fEtE

=

I~ e-~E dv(E)

d~.

By partial

integration

0 F(p)

=

8 r~ e-SEV(E)dE. J0

The dependence E and

=

Ne,

v(e)

integral

on

N

is now such that

V(E)

is expected essentially

:

9(t)

where

vN(e)

to be a smooth function.

~ v(e),

We are dealing with an

of the form IN

where

vN(e)NCN ,

C N I~ e-N[~t-9(t)] dt

=

is an increasing

function

bounded

from above.

If we

define (2)

-~/*(~)

we realize

=

sup(~(t)-~t) t

that

IN(B)

E

e - N g * ( p ) . Const.

and I~ IN([) ¢ C N

Ng(t0) e

• e-N~tdt

=

Const.

e-N¢*(8) N

to Hence face

IN t

and so where

~*(~)

F(~)

get their essential

contribution

is the Legendre

transform of

~(t)

+ ~*(~)

~(t). ~

~t.

Hence

~**(t) and

4**

from the sur-

the supremum is taken.

is the smallest

~

~(t)

convex majorant

of

4.

Observe

that

0nly give

those

ambiguous

values values

of

P

of

-4*(@)

which

t :

in 4(t)

correspond

(2).

We have

- @t

and

:

if

to

linear

4'(t)

pieces

in

4**

@

:

so t h a t ~*'(@) If the

graph

Hence,

if

of

4"

Going

4 *~ is

back

contains

smooth,

to

9"

a straight

then

FN(8) ,

t

~ 0.

line

then

4** is s t r i c t l y

the

proper

9"

shows

a corner.

convex.

definition

is

log FN(@) - log C N f(8)

Unless have

the

energy

ambiguity

inition face.

of

:

surface

in

t

F(~)

we

lim

N

is one

of the

can c h o o s e

is c a r r i e d

out

@

exceotional so that

essentially

values

the

for w h i c h

integral

on the

right

in the

energy

we def-

sur-

If e-~Ed~ N

it

then

follows

that

[ ~(p,q)d~

=

]

and

this

results the

is Gibbs' if

formula In the

rule.

We also

first

I ~(p,q)d~ J

see

that

f(~)

has

a singularity--in

gives

the

correct

case

of the

integral

over

we

can e x p e c t

these cases

exceptional

it is not

clear

that

Hamiltonian,

i[o?+[+(-

-

~z

qi-qj ) '

p

, ,N

result.

simple

i

the

lim N~-

gives 3 cNB - 7 N

Classical an

inverse

thermodynamics

temperature.

The

tells

us

that

second

part

is

we

should

interpret

~

f. E(X).

Clearly,

e tNd

F(X) N

N(lt-~N(t))

=

[~

kN

e

dt.

Hence, N log F(k)

=>

e

kN e x p { N ( i n f ( k t - P N ( t ) )

0

}

t

e

Nktd t

NP-N ( k ) ~

J-~

and

e

N log F(X)

_-< e

N~N(X)

I N2

dt

=

N2

eN~{] (X)

0 Therefore,

Since

log F(X)

is

smooth

it f o l l o w s

l i m ~N(t)

=

In a s i m i l a r

way

one

that

sup(kt-log

N ~-

F(X)).

k

can

compute

high moments

aN E One

not-

~.

closely

of l a r @ e

following

related

finds

N

~

ebN'

E(X)

> O.

that b

=

a log a - a log k - a + log E(e XX) E ( X e XX ) E ( e XX )

-

a .

e

dis-

10

2.

In the

complicated. of the ify.

The

of the

classical

but

At

n.

We

each

to the

theory

at the

same

we h a v e

time

the

states

in this

way

introduced

time

of the

think

of

The

we

general

n

just

is e x t r e m e l y

in the

description

as d i f f i c u l t

shall

model

between

present

which

to verparticles

an e x t r e m e -

contains

some

theory.

particles,

state

(i)

collisions

Here

Boltzmann

by

element

but

elastic

very

of the

particles

(~,~).

a random

fashion.

a system

described

plausible

concerns

in a r a n d o m

should

the m o t i o n

is h i g h l y

characteristics

Suppose N
0

which

(C)

Proof.

on

H' ( t )

~

) pj +

where

f ~ 0

(0,to).

Since

g(@)

[

on

> @,

The g e n e r a l

-[ p (t)log Pv(t) i

: -

[ i,j~v,~

A piPj ,

(0,to).

Hence,

=

is a c o n t r a d i c t i o n .

H(t)

+ A

it f o l l o w s

case

follows

that

pv(to)

f r o m density.

is n o n - d e c r e a s i n g .

Al~piPj log p~

i,j,~,~ =

-7

[ A~PiPj(l°g

P~ + log p )

1

-~ [ A~PiPJ(!°Z

Pv + log p -

log P i - log pj)

PvP~ =

There

-~

is e q u a l i t y

(D)

Let

-

1 ~ A~(piPj

if and only if

A

be the l i n e a r

PvP~ ) l o g piPj

:

[

~:i of i n i t i a l

X ( i

values

N

X : {l }i

~ 0.

such that

k~Pv(O)

the t r i v i a l

interpret

pv(0).

theory

if and only if

We can t h e r e f o r e

A~

~:i

In c l a s s i c a l

Here we first h a v e

whenever

N

kvPv(t) for any c h o i c e

: p~p~

0 .

space of v e c t o r s

N

of the motion.

~

PiPJ

A

is c a l l e d

invariant

and the energy.

X = {i}.

A T ~ ~ 0 = X. + X.

A

the i n v a r i a n t s

they are the m o m e n t s

as an a d d i t i v e

=

~

invariant

+ X under possible

interactions. Proof.

Assume

A

satisfies

the c o n d i t i o n .

Then

N

' X k ~p~(t) i Assume,

:

[ A i j k pip j : ~ [

conversely,

N [i Pi --- i. quadratic

that

[ A~XvpiPj

We may also a s s u m e f o r m has

that

to be a c o n s t a n t

A

(kv + k b - X.l - X')PiPj3

e 0

for all

IX v = 0. multiple

pi > 0

It f o l l o w s of

(Zpi)2 ,

=

0.

for w h i c h that the i.e.,

12

A.".~(X

i]

+ X

v

b

)

:

C.

Consider [ A~(X i]

* X v

- X. - k.) 2 l 3

b

:

[ A~[(X

-

The

first

sum

vanishes.

The -c

+ k )2 , (k. * k.) 2] ~ i 3

~ 2

second

X i,j

[

AY~(X. l]

+ k.)(k

l

]

).

+ k

V

equals

(~. + x

)

:

0.

i

Hence,

k. + k. i 3 (E) ing

Let

sense.

us

now

Let

E

=

assume be

any

k

that set

+ X

v

the

of

E1

A~ ¢ 0}. T h e n t h e s y s t e m is m] = E' E2 = E l .... 'Ek = E k - l ' and

We

choose

-

so

(5)

tham

system

Let

E

that

be v,b

i.e.,

~i We

the

set

( E.

where

Hence,

# 0

=

for

all

called

if

it

E

indices

By

A~ i]

If

"ergodic"

ergodic

E k = all

~ ~ v ~

~. ~ 0. m : E and

E

is

~ O.

Let

pv(t n ) ~ ~ v .

~.~. i 3

A~ i]

indices.

with

tn

if

~

i,j

=

in

the

{vI3 ~

and

if

for

for

any k

log

~

( E

follows

=

and that

E

By

(5) ,

log

is

an

x

:

H(~)

[ p(O)~

invariant, :

Finally,

let

x

sup(-~

solve x

log

the

exp{-~

Lagrange

theory

c(X)X X

extremal

xv) ,

V

}. v

problem

~ x X

V

By t h e

.

i.e.,

v

=

~ ~ X

~

V

we h a v e x

: "o

exp{-[

d(k)X k

E,

large enough.

AY~ ~ 0 it f o l l o w s m] = all indices,

and ~

i,j ( E

~ 0.

have ~

set

(C),

i.

-~

follow-

}, v

and

X

( A.

13

and

x

is x

since

unique

log

by

~

:

log

~

and

0

:

[(~v

=

~(x

Jensen's [ ~

log log

log x

~

and

are

x

-~

log

-~

~

We

~ ~

~

x

+ x

have

log

invariants.

- x

)log

v

inequality.

x

:

~ x v log

x

Hence, log

~

- ~

log

~

)

0,

X V

\

which

gives

~

=

x

.

v

Let

us

Theorem.

summarize

Let

the

(A~)

be

result an

ergodic

lim t~exist

and

~

>

0.

{log

~

v

H

II.

The

Harmonic

i.

We

of

sional

for

all

but

particles

the

Hamiltonian

assume

a

Many

for

N ~ -,

The

~ibbs'

an

The

limits,

~,

invariant

and

{~

maximizes

}

the

v

with

make

small

: a

results

given

invariants.

--2 i P~

would

+

N ~

=

is

let

us

a e

i~x

we

this

m

0,

>_-

assume

case

P

true

assume and

U(q)

and

in

be

is

in

=

nlaced

the

that

the

a _~qvq~

that

~ A(x),

theory

a particle

oscillations

and

AN(X)

where

simplicity,

AN(X)

When

is

a model

=

We

]

:

matrix.

Oscillator

a lattice.

The

p(t)

distributions

consider

case

transition

M

entropy

point

in a t h e o r e m .

several

the

movement

llp[2

at

dimen-

lattice is

each

is

Z.

governed

by

+ U(q).

i.e.,

0.

a

~

trivial.

0

sufficiently

The

free

rapidly.

energy

is

r _jpl2 } NlogFN() : loglIe dple U q) q :- log +CN so

that

We

write

F(~)

= C6.

The

connection

between

energy

and

6

is

simple.

14

-

}P

I~

I e

dp

The m a i n

:

c

r

contribution

N-I-~ e

- r2+(N-l)l°sr dr

to the

=

c

integral

Jo

dr.

e

comes

from

I T

r ~, ( N / p )

so that

i

i.e., the

the k i n e t i c

potential

r2/N

=

energy/particle

is

energy

± 2@ ' i/2@.

1 U(a)

-

energy

tial

and To

is, t h e r e f o r e ,

kinetic study

the

same

comnutation

for

i 2@

N The

The

yields

in e q u i l i b r i u m ,

e@uallv

divided

between

poten-

energy. time

evolutions

we have

to c o n s i d e r

the

eauations

p~ writing :

b~.

and

Yv We

for

assume

qv"

Assume

Ibvl

~ i

for

simplicity

or s o m e w h a t

more

y (0)

= 0

generally

and

set

!

yv(0)

Jb v j < C Jr Jc

that

Z la~il~l ~ denotes

Yv(t;N)

the

solution

and we




e

i~x

:

formula

makes

sense

hand

side.

(i)

y (t)

using

integral

for y6(t)

0, we

see that

y(t)

buted

as the

stays

take

i i ~

: 0, u s i n g

sin a(x)t a(x)

= 2cos

b

the p o w e r

e-iVx

The

e~,

series

dx,

corresponding

i.e.,

sin a ( ~ ) t a(~)

b

trajectory

Observe

T , Y~(t)2dt 0

need

result

reason

:

special

on k i n e t i c

we t h e r e f o r e special

(t]..

measure.

formula

b = 6

C~

+ 6

-@

energy

to a s s u m e

B

energy

be a r a n d o m

~.

is not at all

distri-

that

P

I,

-7

holds.

some

on k i n e t i c

let

i

for all

and

however i R

,

To o b t a i n

symmetry

the

Gibbs

on the

initial

hold

more

generally.

variable

with

distribution

may

values

assume

I~

b : {b }"

be the

A(x)

Fi~ I ~-~ b ( x ) c o s ( a ( x ) t ) e -l~x dx.

=

on a v e r y

the r e s u l t

and

when

for d i s t r i b u t i o n s .

example,

GibBs

For this F(b)

is

~ 0,

I

the

y (t)e i~x

sin a ( x ) t a(x)

b(x)

I~ i ~-~ b(x) 2~

notations

a(~)

Hence

theory

= ~

Hence

y

so that

Y(x;t)

is

As a s i m p l e if

:

also

-

y~(t)

Then

WTEC~Y.

and

for the r i g h t

the

b

a(x)

Y(x;t)

Let

~

let

Assuming

but

:

be an

corresponding

introduction

bdF(b)

we

=

0

,

independent

solution. shall

I~ b 2 d F ( b )

sequence

=

1.

from

B

In a c c o r d a n c e

say that

Gibbs

with

theory

and the

holds

let

y(t;b)

discussion if, g i v e n

in

any

16

weak

s-neighborhood

T(g)

so that, for a n y

in t h e

distribution)

that

does

not

in t h i s

What

this means

fall

space

T > T(g),

the

of m e a s u r e s

the

distribution

R 2n+2 ,

(in the

there

initial

( y 0 , Y 0v , . . . y n , y ~ )

of

e-neighborhood

computationally

in

probability

of the G i b b s

is soon

clear.

is a value

on

(0,T)

distribution

The

following

is

<e.

theorem

holds.

Theorem.

For

(2) If

(2)

each

fixed

a'(x)

=

is v a l i d

(a)

for

z z

only

for either

all

z

if

consider

on a set

o f the

F #

i

the

condition of m e a s u r e

zero.

cases, e-b2/2

(b)

for

z : 0

t h e n the G i b b s t h e o r y h o l d s for the h a r m o n i c o s c i l l a t o r values

B.

~e_mma.

Assume

condition

sup

for all Proof.

t,

I.

3

By

Subdivide

(2), a(x)

(a) a n d

]_ A ( x ) e - l v x

6T _< t -< T,

(Lemma)

T ÷ ~.

(2)

if

let

A

if

A

F -

i

and the i n i t i a l

be a p o l y n o m i a l .

eos(a(x)t)dx

e-bL/2,

Then

< 6

T > T(6).

(0,2~)

is b o u n d e d

into below

K

intervals

except

lj,

in o ( K )

K

fixed,

of the

a n d let

intervals

•

In e a c h

interval

(3)

I.

consider

(v,t)

so t h a t

Iv - a ' ( x ) t I > K26T~ , t > 6T,

For e a c h

choice

of

(v,t)

the

inequality

x

( I.. 3

(3) h o l d s

for a l l

but

o(K)

intervals

I. - u n i f o r m l y in (v~t) - u n l e s s a'(x) = constant = e J a set o f p o s i t i v e m e a s u r e . If (3) h o l d s a p a r t i a l i n t e g r a t i o n s h o w s that II

e - i V x A (x) cos (a (x)t) dx


0

i [I b(x)b(y)w(T(a(x)+a(y)) )dxdy. 8~ 2 JJ

Since

a(x)

tegral

is easily proved to tend to zero using localization.

now also use localization study

x,y > 0,

(8)

Let

hT(U)

on the support of

- a(y)

Observe

=

derivative

and D2

llhllI E C. and

support

If

Ix-yl

the last inWe shall

It is then enough to

first that

a'(~)(x-y)

be a function with

lul < T -I+6

b(x)b(y),

in the first integral.

by symmetry.

a(x)

and

in

+

0((x-y)3).

lul < 2T -I+6,

> T -I+6

hT(U)

~ i

in

then for any second

~-6 e x~y e 8

ID2w(T(a(x)-a(y)))I

< CNT-N

for all

N.

Hence L

II((l-hT(X-y))w(T(a(x)-a(Y))))vll < CT -N. We may therefore hT(X-y).

If

restrict

Ix-yl

the first

< T -I+6

We may therefore from

also replace

- a'(X~Y)(x-y))I a(T(a(x)-a(y)))

(8) and we may drop

hT

of b(x).

inside

(0,~)

The result

I

(9)

-Z

where

h = hT0

by

and is

~ 1

< T< I+6.

by the similar ex-

by the same argument

Finally we may introduce a function strictly

by m u l t i p l i c a t i o n

we also have

ID2(a(T(a(x)-a(y)))

pression

integral

~(x)

( CO

which has support

in a n e i g h b o r h o o d

is that we should p r o v e c o n v e r g e n e e

~(~)h(~)

as above.

of the support as

T ÷ ~

of

e i~x+i~y w(Ta' ( ~ ) ( x - y ) ) d x d y

--JI

for some fixed

T O . We introduce the new notation,

x - y = 2~, ~ + ~ = n, m - ~ : m, and have to compute W(m,n;T)

= [I

~(~)h(~)ein~+im~(2Ta'( 0,

+ 0 (T-N) .

we obtain

(i0)

:

W(m,n;T)

I

~(~)ein~

m W(~a--~-~)

~ dE

+ O(T-N).

_co

Observe

that

besides

the e s t i m a t e

(i0) we have

(ii)

[W(m,n;T)l

~

C + O(T-N), Tn 2

(12)

[W(m,n;T)I

$

C

T4

We write

(n2+m2)2

"

(9) as co

[ bn+ m bn_ m W(n,m;T) m =-~ n 2 2 The

second

place

sum is e a s i l y

W(n,m;T)

by (i0)

=

estimated

by

! Iml T4 (12).

! Inl T4 + (Rest) In the

first

sum we re-

and can omit the r e m a i n d e r term, leaving

us with

~oo

Inl!T 4

~m bn+m b n-m 2 2

I %(~)e lq~ w( 2a' m(6J-T) 2a'dE(6)T j_~

Observe now that the inner sum only fore have the trivial m a j o r a n t

[ni!T 4 and can t h e r e f o r e By a s s u m p t i o n ,

compute

the

extends

over

Iml < CT.

We there-

C Im] M -I/2

has b e e n

function

to the

same

< M-I/4+P

interval


i, and a s s u m e

in front of

~

studied

r

2~(r2 )

by Kac,

dx 2A(x)

> -

i.

A

of s i n g u l a r i t i e s

depends

on w h e t h e r

or not

first

_~i ~ 2~ ]_~

6

types

discussion

i.e. , w h e t h e r i I~ dx 2-Y -~ 2A(x)

(24)

(25)

of a p a r a m e t e r

• @(r)

= 0,

r ~ I,

can can (22)

31

Differentiating

(23)

taken

for

A(x)

is r e p l a c e d

means

I = 0

that

if

does

not

this

region. If,

~A(x).

holds with

back

~

to our

whole

we

then

not

C > 0,

that

expression

the

it m u s t

original

see

In terms

does

limit

2 x. 3

0

(22)

that

hold, the

as

=

time

it does

limiting

in

C log

(23)

~

is

if

e v o l u t i o n this 2 for [ qv < 2N + i

U = 0

because

hold

the m a x i m u m

becomes

of our

fact

the m o t i o n ,

(25)

In the 6N

Going

the

by

(25)

however,

using

and

interfere

changes.

and

not

take

procedure

place

of

in

(21)

that

first

variables

N ÷ ~, t h e n

this

means

~ ÷ 0.

that

for all

m

2

where The

E

free

is e x p e c t a t i o n energy

In t e r m s to the

can

easily time

[q~=

2N + i

those

parts

III.

One-dimensional

the

of the

shall

interaction

more

the p o t e n t i a l compensated Let

is a s s u m e d

of l e n g t h

(x) Hence,

of

that

:

$(x)

cos

v,~:l We are

(i)

models

interested

in the

e -NfN(6)

:

choose

sets

real

called long

line

where

more

van Hove's.

range

function

which

i

N

)

2~inx v 2

dx

on the

and has m e a n

2~nx.

behavior

so that

in

trapped

is i n c r e a s i n g l y

is e v e n

asymptotic

E N c ~N

close

and Here

is

forces.

v:l

il "0" .II e - ~ ( x v - x

also.

place

is small.

on the

usually

n:l

0

We now

or get

we h a v e Cn

case

take

Chains.

differentiable

~ c n i

in this

U(q)

homogeneously small

distribution.

will

freely

particles

a model

to h a v e

We a s s u m e

move

potential

consider

be a c o n t i n u o u s

i.

explicitly

and M a r k o v

with

by an a s s u m p t i o n

$(x)

the

successive

We b e g i n

Gibbs

the m o v e m e n t

either

where

chapter

between

dependent.

and

Models

to the

computed

evolution,

sphere

in this

respect

be

of the

sphere

We

with

of

1 • . . dx

N •

torus

value

zero:

32

i N 2winx -- X e ~ N i

(2)

tl >_- 0, ~ d d : i, J 0

where

[i 2 w i n x ] e 0

+

dd(x)

and

-N6~Cn[@(n)12 (3) This

e is c l e a r l y

Lemma.

Divide

suppose

that

-NfN(6)+o(N) mE N

always the

possible.

interval

a.N] xv's

~(n)

:

=

We use the

(0,i)

belong

e

to

into

following

k

equal

intervals

I..3 This d e f i n e s

I. and ] E(a l , . . . , a k).

a set

T~en k m E ( a I ..... a k)

:

exp({-~

ajlog

aj-log

k]N+o(N)).

Proof. mE

N[ (alN)! "" "(akN)!

:

To c o n t i n u e limit

at most that

our d i s c u s s i o n

(2) the c o r r e s p o n d i n g feN.

for all

Totally

N

large

of

k -a. N . k-N. ~ a. ] i ]

~o

(i) c h o o s e

number

a. : ~(I.). In the ] ] in e a c h i n t e r v a l v a r i e s

of × 's

this gives

=

exp

-

J0"

i

I

f f 0""

_>- exp

Z V,~:I

This means

N ~ ( x v ) d X l . ,. .dx N i

N

0 -v=l ~ l°g~(xv)

(N-I)N

]]I ~(xv)dXl"

"dXN

]

}(x-y)~(x)~(y)dxdy-N 0

~log~dx

0

=>

-

6~(x-y)~(x)~(y)dxdy

-

0 0 @

(4)

The

and h e n c e f(8)

where

@ _> 0

:

There

= lim fN(B)

=

in (4) w i l l ~

i

be d e s i g n a t e d

minimizing

(4)~

equation

2B}*%

:

~n

~n

singular.

+ log%

be a m i n i m i z i n g

minimizing.

is u n i f o r m l y Let

}

C

and

:

by ~

fl + ]0 ~log~dx}


0, E a

and p o s i t i v e

Constant.

sequence.

integrable.

be such a w e a k

Let, for some

~log~dx

4

rI + 0 ~n log @ndX Hence

exists

the n o n - l i n e a r

(5) Let

f

f0i ~dx

exists

and s a t i s f i e s

Proof.

.

0

rl~l inf{[ ] 6 ~ ( x - y ) * ( x ) * ( y ) d x d y ~0 0

and

functional

Lemma.

+ 0(i

0

that

- lim fN(B) for all

= i.

~(xv-x ~) H * ( x v ) d X l ' ' ' d X N i

N

•

~dx 0

limits

By Fatou's

be the

are t h e r e f o r e lemma

set w h e r e

~(x)

%

is > a.

non-

34

Take

@

with

support

z(¢) ~ I(¢+~)

on

Ea~

such

: ~(¢) + 2~ I

that

E L~

@

B¢*¢~d~

+ 6 I

Ea Hence,

we

the

finally istic

: i " Then

(log¢)~dx + 0(62). Ea

have 26¢*¢

Since

[01 @ d x

and

first

that

term

¢6

is

¢ = 0

function =

of

+ log¢

(1-6)¢

c

bounded,

on E0

=

E0 and

on

¢(x)

with

{xl¢(x)>O}.

~ a

>

0

m E 0 > O.

¢(x)

if

@0

Let

> 0.

Suppose

the

character-

be

consider

+ b~0

,

where

=

I(¢)

6 > 0

and

=

b

(mE0)-i

.

Then I(¢6) for

6

small

Let zation

us

enough.

now

I¢(x)I

The

discuss ~ i.

lemma

the

By

+ 0(£) is

(5),

we

< I(¢)

therefore

function

f(B),

proved. and

assume

the

normali-

have

llog¢ i.e.

+ 61og6

- cI ~

26,

,

e c e -28

~ ¢(x)

s e c e 2B

~

S e 4B.

Hence

and e -4B Let it

¢ ( x a) follows

: M a x ¢(x)

¢ ( x b) = M i n ¢(x)

= l-b,

a,b

~

0.

From

(5)

that I

l+a

log

= i + ~

¢(x)

=

l-b

S

26

(¢(Xa-t)

- ¢(xb-t))~(t)dt

~

28(a+b),

0

i.e.

-

log(l+a)

if

i B < --=

I 6 < ~

If

2

then

-

28a

~

log(l-b)

a < e - i -

+

28b

>

0.

~

0

and

]

log(l+(e-l)) We The

conclude

that

following

Theorem.

If

a

= b

theorem

~

if

i B ~ ~,

holds.

I¢(x) I ~ f(6)

= 0

- [(e-l)

0

i

then for

0
_-0 then f ( B ) - 0, 0 < B < ~. n f(6) < 0 for 6 large.

the

theorem except for the rl } _> 0, ~ ~dx : i and ]0

any

@log~dx

last

two

statements.

We

_-> 0.

0 This

follows

tive

definite,

from

our

I(~)

equation

~

0

and

(5) so

with

~ - 0.

f -: @.

H e n c e , if

Assume

%

therefore

c

is p o s i < 0

n

and

choose = Hence,

I1

f -< ~c n ~1 +

i + cos

(l+cos

2~nx.

2~nx)log(l+cos

2~nx)dx

0

Remark.

This

transition",

i.e.,

information above ($)

system

on

is

If we

analytic

assume

distribution

the

for

Let

us

be a

study

the

now

some

as

of

=

in

the

N :

in o r d e r van

consider matrix

the

behavior

sup (m..) 13

e

Tamm

of

at

any

coupling

following positive

we

the

that

time-evolution. we

obtain

time

t

is t h e

theory

potential,

from

B.

the

What

Gibbs'

precise

proves

x = 0

t : 0.

that

with

in

a "phase-

More

of p o i n t s

study

start

Hove

without

B.

of p a r t i c l e s

at

shows

be d e d u c e d

M.

number

is t h e

particles

velocities

in

however

thesis,

a non-

in t e r m s

condition

holds? need

between

only

study

points.

(trivia].)problem. entries.

on

We

wish

Let to

of

~ A A ..... ( i l . . . i N) ill2 1213

distribution

S

can

distribution

points

square

(6)

(7)

here

of the

first

asymptotic S

Exactly

of

enough.

function

f(B)

at a f i n i t e

distribution

k × k

analytic of

all

the

discussion

distribution

A..

large

non-positive-definite

question that

of the

In t h e

~

except

equation

initial

~

B

In a f o r t h c o m i n g

linear

this

with

nature

interesting

e.g.

for

f(B)

the

discussion.

An

< 0

previous

case

of

pairs

of

~ Pi

=

the

main

indices i,

~ (logAij)Pimij Ci,j

A. . 1N_liN

as

N ÷ ~.

contribution

must

come

from

where

[. mij ] -

=

1

[.Pimijlog i,]

If f a l s o d e p e n d s o n a d e n s i t y 0, t h e n finite union of analytic curves.

the

mij

singular

+ o(N)

set

in

(B,P)

is

a

38

mij

are the t r a n s i t i o n

the p r o b a b i l i t i e s We c a n n o w

of

probabilities

i, so that

of a M a r k o v

Pimij

is the f r e q u e n c y

study this as a v a r i a t i o n a l

f i x e d and v a r y

m.. i]

by small

process

quantities

problem.

D.. l]

Let

and of

Pi

are

(i,j).

Pi

be

so that

k U.. 1]

j:l

:

0,

i : l...k

and k [

:

i:l Pi~ij The v a r i a t i o n a l 0

equations =

0,

j

= i

""

.k.

are

i,j[ (log A i J ) P i O i j

- i,j[ P i O i j l ° g

mij"

We o b t a i n (8)

mij

:

xiYjAij .

We can also m a k e a small v a r i a t i o n

qi

of

Pi

so that

k i=l

(qimij+PiUij)

=

qj

, j = i ..... k.

We find U

=

~ log A i j ( q i m i j + P i ~ i j )

-

~ (log x i + l o g

We can v a r y

qi

- ~ qimijlog

mij

- ~ Pi~ij

log mij

yi)qi .

freely,

~ qi

=

m

0,

by c h o o s i n g

~ij'

and h e n c e

K -I xi •

•

i3

=

- -

A . .

=

K x. ]

x.]

i]

•

We o b t a i n k i=l and the m i n i m u m

value

x.A.. i m]

= log K.

If

A.. i]

is a s s u m e d

symmetric

obtain 2

Pi and

K

is the l a r g e s t

The r e s u l t

eigenvalue

is now obvious.

=

xi

of A... i] To e s t i m a t e

S

study instead

we a l s o

37

k ,

S

and c l e a r l y

=

S

[ A ..... (i) ill2

and

S'

due to B e u r l i n g ;

Theorem.

Let

Then t h e r e

K(x,y)

discussion

f(x)

ri

r

It is e a s y to see as b e f o r e

so

f : i/~0

=

(0,i).

on

~f(y)

iff

- ~

that

symmetric

to the e q u a t i o n

~ > 0,

K(x,y)~(x)%(y)dxdy.

s%p I(~)

~ 0 dx -

has a s o l u t i o n

9(x)dx

%0 > @"

] K ( x , Y ) ~ 0 ( Y ) d y + o(I),

is a s o l u t i o n .

Suppose

0

- I(~0))

and

> 0

define, for

: ] log~(x)dx

8-1(I(~0+96)

the f o l l o w i n ~

K(x'-~-~!dy.

= ]0

To get the e x i s t e n c e

I(~)

xi,

let us p r o v e

be e o n t i n u o u s

solution f(x)

Proof.

[

i:l

there is r e l a t e d w o r k by J a m i s o n .

> 0

is a u n i q u e

KN

are c o m p a r a b l e .

In a n a l o g y w i t h the a b o v e theorem

=

A. • XiN mN-IIN

n o w that

:

f

and

g

are

solutions.

K(x,y)

Then

Lg-m~

f(y)] dxdy

since

flf x

0 g - - ~ dx

:

0 f(x)

ff

:

dxdy

f(x)g(y)

Hence, 0 and

=

K(x,y)

g(x)

f(y)

f(x)g(y)

dxdy,

so

f(x)

z

g(x) Remark.

The a b o v e

continuous

result

version

and continuous

on

leads

~

to the f o l l o w i n g

of the c o n t i n u e d (0,i) •

hn(X)

=

(: 1).

f(y)

fraction

problem,

expansion.

which

is a

Let h0(x) > 0

Form

f

l K(x,y) 0 h n - l ( Y ) dy,

n : 1,2, . . . .

W h e n does lim h ( x ) ( = f ( x ) ) e x i s t ? n n~ Our goal

is to

study the p a r t i t i o n

function

S

for g e n e r a l

poten

38

tials.

This p r o b l e m

special

cases: $(x)

=

2.

$(x)

> 0

direct

0''"

with c o m p a c t

method

i observe

times

earlier.

The p r e s e n t

interest.

that we m a y w r i t e

-be

I

x I -x 2 x 2 -x 3 -x N )(e + . . - + e -xN) - ~ e (e +...+e

e

• ..dx l..-dx N

0<xI 0.

0 < x I < ... < x N < L

has b e c o m e

t,

t.

Dt :

e

->_ 0

1

i

--t

+ e

i+l

N

~ t.l

=~ L.

> 2

39

There

is some

small

error

no d i f f e r e n c e .

We h a v e

with

condition

the Markov Let

m(x,y)

near

x = 0

therefore

a problem

in the

be t r a n s i t i o n

which

domain.

is e a s i l y of t h e

It h a s

probabilities

checked

type

the

to m a k e

studied

following

for a Markov

earlier form.

Chain

so

that m(x,y) Let

p(x)

:

be t h e c o r r e s p o n d i n g

(i0)

0,

(x,y)

density

I x p(x)dx

Solve

the

variational

~ M.

and

assume

~ p-i

problem

supE[ m ( x 'P(x)F(x)dx y ) d x d y-~f[' P((x)mm(x)'y)l°g In o u r

case

We

F

was

consider

for a l l

j.

The

given

by

(9);

M is d e f i n e d

i n s t e a d the

finite

problem

variational

result

(8) h e r e

•.

mm]

m.c.

:

,

3 i

c.

Dt

(7) w h e r e

and

p = N/L.

n o w A.. 13

= F. 1

yields mj

:

1

in

i

where

M i : (jl(i,j) The

second

( M}.

type of variation F i - log

yields

c i - log m.

:

a'

+ b'

i

i

if we a l s o The

take

(i0)

continuous

into

consideration.

version

of this

is

m(x)

F(x)

- log

)

fE m ( t ) d ~

:

a + bx

X

where

EX

= {tl(x,t)

6 M}.

Writing

M(x)

our transition

m(x,y)

m(x) M(x)

(ii)

A

matrix

and

b

are

=

I

EX

m(y) = M(x)

_

to be d e t e r m i n e d

m(t)dt

' y

AeF(X)

from the

( Ex'

e

is d e t e r m i n e d

by

-bx

conditions,| J0

p(x)dx

= I

40

and

xp(x)dx 0

m(x).

= p-i

F i r s t , if

It

is

x >_ l o g 2 ,

interesting

then

E

to

= (@,~)

note

how

and

m(x)

(ii)

determines

is c o m p u t e d .

X

Then,

iteratively,

m(x)

is d e t e r m i n e d

X

-X

in t h e

intervals

x0

log

(Xn, X n _ I)

where e One

sees

+ e

=

xn

To

compute

by

e ~(y)

p(x) + e -y

we

equation

:

analytically

on

B,A

on

B

Should

be n o t e d

of

p

quotients Let support

set

2,

for

that

the

of the

~(y)

=

xn ÷

0,

2.

~.

n +

: p(x)/m(x).

~(y)

If

is d e t e r m i n e d

and dx

~(x)AeF(X)e -bx

:

C _ ] l~(Y) 0

by

b. p

the

of t h e

us n o w c o n s i d e r t h e i i (-[,7) and that

iterations.

Whether

for

values

sums

'

< y

easily

and and

n+~

~(x)

I~ J~(y)

is s o l v e d

analytically

tives

=

n+2

log

: 2, t h e n

~(y)

This

n-i

that

(12)

It

n

all

model

or

not

choices

shows

The the

of

case

> 0.

We

depends

is n o t

clear.

in t h e

correspond

assume

We w i s h

depend

energy

discontinuities

2 above.

%(x)

free

(B,P)

x in (12). These n original variables.

in

solutions

to

that

derivacertain

~(x)

to c o m p u t e

has

asymp-

totically ~L

S

=

"--n

Let x

v

I. ] ( I.. ]

(13) S :

N!

be

the

We

e

v

p

then

(j,j+l).

write

S

us

N [ i a _ M , . . . , a M = 0 a-M! " " "aM[

introduce

~ M = 2L + i "

the

notation

Let

a. ]

be t h e

f

e

]I v,j

• x "" .~I. v] 3 v = l , . . . ,aj Xj

: (a.'3,Xlj,...,x a j)

and

dx.. v]

dX.] =

J

, dx lj...dx a . .

Here,

a.

K(X-_I,X.+I)]]

=

aj. We

of p o i n t s

-SX%(Xvj-Xpk) [

-i e

number

as

a _ M + . . . +aM= N

Let

dXl...dXN

~ -n

interval

can

)

~L-B~(x-x

I...~

j]

also

(14)

]

write

= 0,i,...,

l~[a f

e -2

j-i

and

the

xv].

move

_!BT0 e -B[j

dX.

]

e 2 =j+l

in

I..

]

41

where we have fixed the variables to the intervals over variables variables

in

Ij_ I

Xj_ I 0 [~

Ij+ I.

and ~j

and all other variables

concerning

N!e L, the expression

the range of

to estimate

Xj+ I

indicates

in the same interval while lj

by assumption

and

in

that we only sum

is the sum over all

(only in

~).

corresponding

Ij_ I and

Disregarding

lj+ I

the factor

is

L I'''I Nj:I K(X 'Xj+I)dXIj "''dXL"

(15)

The dependence

of

is given in (14).

B

Note that

K ~ 1

for

B ~ 0.

Let Q(X,Y) and let

~

1 ~a--~ a ~

-

be the largest

(16)

K(X,Y),

eigenvalue

I Q(X,Y)f(Y)a(Y)dY

If we rewrite f(XL)dX L

(15) using

we see that

Q

of =

If(X).

and replace

(15) is

a >_- i~

~ CI L.

the last integration

On the other hand,

by

f(X) ~ e -ca(x)

and i e ca(X) which gives an estimate iL

is the asymptotic

in the opposite

behavior.

lated by the following

dX
0 ~0

pending only on

except

0 6 K(x,y)

~ 1

on a set of measure

be the largest

eigenvalue.

s(6)

Iz-lol

so that

< r

on

(0~i).

< s(6),

E(6) ÷ 0,

Then there is an contains

Suppose r

de-

no other eigen-

value. Proof. ~(x)

Let

f ~ 0

correspond

be an eigenfunction

to

D, and assume

corresponding to 10' let i I~I { 2 lo" It is easy to see

that,

q

=

II K(x,y)dxdy,

Clearly,

q __ 6(l-e(d)).

on a set of measure

_ 6

(/l

f(y)dy

Hence

f(x)

is b o u n d e d

below

except

f(x)~(x)dx

=

]

0

,

c ~ (x)

fl¢ldx

f

:l.

i¢[d x

> 4~q f 2

¢2

>= ~2 (1-ct')

dx

'

f>6

so i"

f

i~ldx

-->

~"

Hence,

I~ld× >

6".

(~>0

we h a v e :

# fK(x,y)%(x)%(y)dxdy

< -

We h a v e Theorem.

therefore

If

free

()(x) ~ 0

energy

The

get

K(x,y)lC(x)qb(y)ldxdy-

},

0

%

~,,,

- 8,,,

proved

the

is

following

continuous

is an a n a l y t i c

reasonable

is t h a t

=
0 there

=

(G)

Using

that Clearly {Xnk}~= I such

that

< s , such that

~ I G c is c o n t i n u o u s and has

this

on G c.

continuity

(m,p)-quasicontinuous.

extend

extension Deny-Lions

the d e f i n i t i o n is e s s e n t i a l l y [12],

Wallin

of

f by

unique. [33],

setting We

f(x)

= ~(x).

summarize

Havin-Maz'ja

the

[18]).

80

Theorem

1.8:

Let f E W p.

Then,

after

possible

redefinition

on

m

a set of m e a s u r e and g are two almost

zero,

(m,p)-quasicontinuous

everywhere,

In w h a t

f is ( m , p ) - q u a s i c o n t i n u o u s .

t h e n f(x)

follows

functions

= g(x)

functions

Moreover,

such t h a t

f(x)

if f = g(x)

(m,p)-q.e.

in W p are a l w a y s

assumed

(m,p)-quasi-

m

continuous. or t r a c e

It t h e n also m a k e s

of f u n c t i o n s

Thus

if we w r i t e

f(x)

= 0 (m,p)-q.e.

in W p on a r b i t r a r y m

fIF : 0 for a f u n c t i o n

the m o d i f i c a t i o n s

f in W~,

capacity.

this m e a n s

theorems

mal ~K' and a p o s i t i v e

if ( m , p ) - c a p a c i t y

of o r d e r m.

= inf{llVwl 2 dx; necessary

By c l a s s i c a l

sets of p o s i t i v e

same n u l l s e t s

fined u s i n g o n l y d e r i v a t i v e s CI,2(K)

the r e s t r i c t i o n

that

on F.

If mp < d one gets the

can d e f i n e

sense to t a l k a b o u t

For example,

is de-

if d ~ 3, one

w E CO, w ~ i on K}.

(We omit

if d = 2.) of F r o s t m a n ,

measure

~K w i t h

there

is then a u n i q u e

support

in K and ~K(K)

extre=

CI,2(K) , such that I ~K(x)

The r e s u l t s Theorem measure (a (b

extend

1.9:

U

~E ~E

lx_yld-2

to a r b i t r a r y

For any b o u n d e d

~E ~ 0 w i t h U

d~K(y )

:

support

- u~K(x)

sets.

E c ~d

there

exists

a unique

in E such that

(x) = i (l,2)-q.e.

on E~

(x) ~ i for all x ; ( N o t e

that U

bE

(x) is d e f i n e d

every-

where.) (c

I

d~E

: I U~E d~E

= I(~E)=

CI,2(E)"

~d

U

~E

See e.g.

is c a l l e d Landkof

the e q u i l i b r i u m

U

(x) < i.

potential

for E.

[24].

E is said to be thin bE

or c a p a c i t a r y

(or

More precisely,

(l,2)-thin) we d e f i n e

at the p o i n t s w h e r e

thinness

in the f o l l o w i n g

way.

81

Definition x E [ and

there

i.i0:

A set E is

(l,2)-thin

exists

a positive

measure

U~(x)

A necessary the W i e n e r

~ n=l

(b)

Here

i.ii:

that

UZ(y).

condition

for t h i n n e s s

is g i v e n

by

[24]. E c ~d

CI,2(E

is the

(l,2)-thiek

n A

is

(l,2)-thin

at x if and

only

if

at all

n An(X))

(x)) < m,

< ~,

d >_ 3

d : 2.

n

annulus

is not

thin

{y;2 -n-I


~ CI,2(G\K0)

in one

direction

quasi-open

G = {x;If(x)l>0}

of n e c e s s a r y

= C I , 2 ( G X K 0)

G.

(l,2)-stable

proves

so c a l l e d

f]~K

(T.

CI,2(GXK)

One

following

stable.

1.16:

for

the

= 0.

now

Theorem

K is

by the t h e o r e m

(K0) c

can

we o b s e r v e

1.13.

1.15:

satisfies

In fact, f with

stability

for any

sets

G,

2 f ~ W I.

for all

that

the

open

Now

G.

property

in p a r t i c u l a r suppose

0 > 0

in

to the

(b) e x t e n d s set

f ~ W~(K) ±

i.e.

to

f(x)

= 0 on K c.

If(x)l

It f o l l o w s

= 0(l,2)-q.e.

f r o m the d e f i n i t i o n s .

1.17:

1 1 let I = [- 7,~] B0\I

See

this t h e o r e m

Example

that

stability

[5],

[20].

one e a s i l y

enough

K will be u n s t a b l e .

Then

Theorem

Proof:

(Havin

(l,2)-q.e.

[17])

disks

in

If the B.z are

small

> 0, as is

of t h i n n e s s

(l,2)q.e.

For any

E is thin.

K is ( l , 2 ) - s t a b l e

if and only

if K c

on

~K, t h e n

and of c a p a c i t y In fact,

it f o l l o w s that

easily

CI,2(GXK)

the c a p a c i t a r y

from

=

potential

for

on G \ K 0.

In the c o n v e r s e [9].

~ i ~ Cl'2(Bi)'

on ~K.

C I , 2 ( G \ K 0) for all o p e n G.

exists

and

small.

If K c is t h i c k q.e.

the d e f i n i t i o n s

direction

the p r o o f

set E we let e(E)

Thus e(E)

contains

depends denote

on a l e m m a of

the

the e x t e r i o r

set of p o i n t s and part of the

of E.

Lemma

set.

of I but n o w h e r e

C I , 2 ( B 0 \ K 0 ) ~ CI,2(I)

: Ci'2(~ Bi)

arbitrarily

1.18:

is ( l , 2 ) - t h i c k

boundary

of an u n s t a b l e

On the o t h e r h a n d

can be m a d e

where

i.e.,

easily

in the plane,

at all p o i n t s

~K = IU(U 0 B.).z

In fact,

CI'2(B0\K)

Choquet

= 0.

(a) f o l l o w s

gets an e x a m p l e

accumulate

Let K = B 0 \ ( U I B i ) .

G \ K is i

implies

Let { Bi } , be d i s j o i n t

on the real axis.

else.

which

~ h -1 C I , 2 ( G X K )

Let B 0 be the open unit disk

such that the disks

w e l l known.

CI,}G\K0)

on ~K, q.e.d.

The o t h e r d i r e c t i o n ,

Using

that

1.19:

Let E be an a r b i t r a r y

an open G such that e(E)

The K e l l o g g Corollary

property

1.20:

set.

c G and C I , 2 ( G

is an i m m e d i a t e

For any E , C I , 2 ( E

For any c > 0 t h e r e n E) < s.

consequence.

n e(E))

: 0.

85

To p r o v e is thin,

and s u p p o s e

E : K e there CI,2(GXK0) not

Theorem

CI,2(A)

~ CI,2(A)

By Lemma

> s, so C I , 2 ( G X K )

by T h e o r e m

theory

Theorem

if the part of the fine city

> s > 0.

~K where

1.19 a p p l i e d

1.16.

1.18

< CI,2(GXK0),

(In t e rms

says that

interior

of the

to

< s.

But

and thus

K is

fine t o p o l o g y

K is ( l ~ 2 ) - s t a b l e

of K w h i c h

Kc

belongs

of

if and only

to ~K has capa-

zero.) Theorem

K c and

1.21:

(K0) c are

Proof:

(l,2)-thin

K is ( i j 2 ) - s t a b l e

(K0) c c o n v e r g e

direction

the c o n d i t i o n

as in T h e o r e m 1.22:

K

is

= {y; ly-xl

Proof: Ke!logg's

Assume

lemma

(l,2)-q.e.

on ~K.

K c diverges, Theorem

1.18.

simultaneously. implies

that

CI,2(GXK)

(l,2)-stable

if and only

for

if

(l,2)-q.e.

x E ~K.

K satisfies

1.20)

the above

the W i e n e r

series

so K c is t h i c k

q.e.

on ~K.

The o t h e r d i r e c t i o n results

lim for

inf c o n d i t i o n . (K0) c d i v e r g e s

the W i e n e r

The t h e o r e m

series

follows

Let K 0

= 0.

1.23:

(Gon~ar

is o b v ious.

can b e s h a r p e n e d .

Then K is

[16],

Lysenko

(l,2~-stable

= CI,2(G)

and P i s a r e v s k i i

if e i t h e r

(a)

CI,2(GXK)

for all o p e n G

(b)

lim sup C I ( B ( x , r ) k K ) r ' d > 0 for a.e. r÷0 ,2

or

x.

for

from

v

Theorem

=

1.18.

But t h e n by the a s s u m p t i o n

If K 0 = 0 t h e s e

In the

< r}.

that

(Cor.

if

for all o p e n G, then the

of the t h e o r e m

CI (B(x,r)kK) lim inf ,2 > 0 r÷0 C I , 2 ( B ( x , r ) X K 0) B(x,r)

if and o n l y

at the same points.

for K c a n d

other

Theorem

[23]).

: CI,2(G\K0)

series

CI,2(GXK0)

(Keldys

If C I , 2 ( G X K )

Wiener

Here

of

is an open G s u c h that G n E, and C I , 2 ( G \ K )

(l,2)-stable

potential

1.18 we let A be the s u b s e t

[25]).

By

86

The p h e n o m e n o n Ci,2(B(x,r) also

2.

~ r

d-2

that

(b) are

, is c a l l e d the

[20] and F e r n s t r o m

Generalization

"instability

although See

of c a p a c i t y " .

p ~ 2.

to try to g e n e r a l i z e

conditions

equivalent,

[13].

to WE,

It is n a t u r a l investigate

(a) and

for e.g.

RP(K)

the a b o v e

= LP(K).

to p ~ 2, and

One is led to the

a

following

definitions.

Definition sis

if all

We let i < p < ~, ~ + ~ = i. P q

2.1:

A closed

f ( W ~ ( P d)

Definition o

such that

A compact

2.2:

F c ~d

admits

(l,p)-spectral

flF = 0 b e l o n g

K c ~d

synthe-

to W~(FC).

is ( l , p ) - s t a b l e

if W ~ ( K 0)

o

ff G)) W E is c l o s e d

Beurling-Deny Theorem synthesis,

theorem 2.3:

truncation

given

(T. Bagby

above

for all p, so the proof extends

almost

[5]) All c l o s e d

of the

unchanged.

sets admit

(l,p)-spectral

i < p < ~.

Corollary transforms

under

2.4:

([20])

~ of m e a s u r e s

For any open

~ such that

bounded

~ { Lq(G) a

G c { the C a u c h y

are d e n s e

in Lq(G), a

l i on E 0. This is for example the case if ~,P p : 2 and ~ > 2. However, one can prove the following "boundedness principle". Theorem

(Havin-Maz'ja 2.13:

[18], Adams-Meyers

Let p > 0.

There

[3]).

is a constant

ing on d and p, such that for all x V ]J

~,P

(x)

_< A m a x { V ]J

~,P

(y) ;y

( supp ~}.

A, only depend-

9O

Thus

in p a r t i c u l a r , The

by V.

theory

P. H a v i n

and

D.

and

was

V.

R. A d a m s

[2],

following

given

G. M a z ' j a

results

and

~E

is b o u n d e d

studied

gave

[18],

found

V

was

they

(See

were

many

[19].)

systematically

applications

At the

independently

by A.

same

to

time

by N. G. M e y e r s

[3].

natural

by A d a m s

potential

potentials

in a n a l y s i s .

of t h e i r

The

capacitary

of n o n - l i n e a r

various p r o b l e m s several

the

extension

and M e y e r s

of the

[2] and,

definition

of t h i n n e s s

independently,

by the

author

[20]. Definition or x E E and (a)

2.14:

there

Many

setting.

theorem

of F u g l e d e

Theorem E A = {y;f(y)

See

of

[2].

[14].

(~,p)-thin measure

is not

defined

x ~

that

sets; e x t e n d

following

f E L p. or

~ such

(l,2)-thin

See a l s o

Let

at x if e i t h e r

V ~ p(y). ~'

The

2.15:

to this

is a s p e c i a l

case

more

of a

[20].

For

(e,p)-q.e.

If(y)-f(x)l

x the

~ i}

is

set

(~,p)-thin

at x

A > 0.

A problem generalization in part

cone

is a p o s i t i v e

of the p r o p e r t i e s

general

and

E is

Vp is b o u n d e d ; ~,P Vp (x) < lim inf ~'P y+x,yEE\{x}

(b)

for all

A set

which

has not

of W i e n e r ' s

yet

found

criterion.

a satisfactory The

following

solution is k n o w n

is the ([2],

[20]).

Set

2 n(d-ep)

with

vertex

C

e,p

(E @ B ( x , 2 - n ) )

at x,

then

lim

n÷~

= a

n

(x,E).

a (x,E) n

Note

is f i n i t e

that

if E is a

and p o s i t i v e

for

0 0,

the fact that we

[~,H]f + [H(b2),H]f.

identity Hb2Hf

to d e d u c e

:

are and

HI(~).

in

a

As b e f o r e ,

to be g i v e n

and

Taking

~(~)

this

and consequently

commutator

and c o r r e s p o n d s

of the c o m m u t a t o r .

g(x)

Fourier

ll~ll~.

valid

for all

Finally,

z 0 } F.

z 0 ~ F, we h a v e

g0~z0)

=

lim 2 - ~ ~0

as a c o n s e q u e n c e ,

g0 (z)

-~

z_z0-~

d

;

F

go ~ 0.

For a f u n c t i o n

f

in

A(F)

we d e f i n e

the F o u r i e r

transform

as (3.2)

f(t)

=

It is easy to c h e c k

I

I e-ltZf(z)dz F

= I e-ltXf(x)dx J-~

1 •

that co

f(z)

(3.3) (this r e l a t i o n Another Lemma

(3,4).

is a c t u a l l y

important If

f,g

r

I The p r o o f to

JR.

is t r i v i a l ,

defect

valid

£ A(F)

since then

of this

transform support.

then

if

-

2~

f(t)g(-t)dt.

-~

by C a u c h y ' s

theorem

we can d e f o r m

from Plancherel's

formula

is the a b s e n c e

( w h i c h can be s h o w n

=

F

theorem. of an

L2

transform. that an a l t e r n a t e (3. 2) for,

w a y of i n t r o d u c i n g

say,

one has to i n t e r p r e t

LI

functions

the i n v e r s i o n

(3. 3) as f(z)

F).

is

is by f o r m u l a Then

eitZ~(t)dt

on the strip c o n t a i n i n g

follows

We s h o u l d al s o m e n t i o n

compact

{

relation

for the F o u r i e r

Fourier

i

f(z)g(z)dz

F

The f o r m u l a

The m a i n estimate

=

i I~ lim ~-~

to e x i s t

for

e-~]tl 2eitZ~( t)dt

z 6 F).

the

with

formula

116

Our p u r p o s e

now

is to s t u d y

M(f)(z)

~i

=

operators

Ir~

of the

form

e l"t Z m ( t ) f ( t ) d t

, z E F

J

L2(F).

on

(The

case

p ~ 2

will

be an

immediate

consequence).

if

admits

We have Theorem

V.

M

holomorphic

is

bounded

extension Jim

where

the

Calder6n

Actually M

the

be b o u n d e d The

m(~)

= (I

L

z I
0

0

and

(3.6)

M+f(z)

=

lim 6+0

]

k(z+i6-~)f(~)d~ F

where k(z) The

integral

tinuation

for

k(z)

elsewhere z

=

is d e f i n e d

(see b e l o w ) ; =

i

Re ±~

~ e i t Z m ( t )dt. 0 for also

Re

z > 0, and,

k

is h o l o m o r p h i c

- 00 < ~ < 0 0 +

by a n a l y t i c in

con

119

and

satisfies C _ T~ T

In fact for

if we

let

[8 I < e 0

for

- 88 + a < ~ < 80

kQ(~)

= el8

Re~

> 0, we

and

i ~0 e l"t ~ m ( e ± e" t ) d t , find

which

by an o b v i o u s

is d e f i n e d

change

in c o n t o u r

that k(z) Since tion

ks(~) of

k

is a n a l y t i c whenever

Finally

Re

e

a familiar

k@(ei8

:

in Re6 ie z > 0

> 0 or

integration

Ik<J)(~) I = o llell~, and

theorem,

using

the h o l o m o r p h y

and

k, we h a v e

Before

C.

find

- 80 < arg

elsewhere.

M+f

The

we

Z)-

L2

L2

point

out

needed

If

h

thesis

, f+

estimate that

are

holomorphic

in his

(3.7).

M+f+

Littlewood-Paley

tools

of

=

= for

(M_

by B.

above

F

the

the

below

same is an

case,

see

norms

for the H a r d y

F.

Dahlberg

above

admits

described

to our

equivalent

functions

is h o l o m o r p h i c

M+

the m e t h o d theory

a few

[7] and

C(f).

These in

Stein

were

[8].

proved

~n.

following

norms

equivalent:

i°

( f I h ( x + i ~ ( x ) + i n ) 1 2 d x )1/2

Sup

q>O.

20

(~-~]o n2j-llh(J)(×+i~(x)+in)lSdxdn

Given and

z0 ~ F

.q F* = F + l ~ If

.

we

define

We need

Lemma

(3- 8).

II~' []~ -< 6

I~-z]

> C 6 [ ( X - X o ) 2 + q 2 ] I/2

z : z 0 + iq,q

the

following

then

> 0,

simple

3C 6 s.t.

for

= 1~2, .... z*

= z o + i ~2 '

geometric all

observation.

~ ( F*,

120

,Z

Z

x

= To estimate

the

h(z) we estimate

L2

+ i 7

norm of

r ] k(z-[)f+({)d6 J F

=

h"(z)

+ i~(x)

and then use

tion by parts and change

Lemma

,Im z > ~(~e z) (3.7).

h"(z)

: Jr k"(z-~)f+( ~)d~ : ]F k'(z-~)f+(~)d~

Using

our estimate

on

We have by integra-

of contour,

k'

: ] F ~'r k'(z-[)f~(~)dC.

we get

[h"(z)

I < C

i

I%(c)1

- - F -

ds

F* f~-~1 -< C

d

(I

~ s

r* I~-~1

< c

}1/2(i

2

r*

_m/2 (i

If'(~)l 2 + I~-~12

I%(~)1 2 r*

Iz

_Cl2

)1/2 ds

)m/2 ds

Therefore

oo t~ f_~Jolh"(x+i~(x)+iq)]2n3dxdq = Using

Lemma

oo . 0. the

associated

with

let

i n f {lllFlllp,z0

z0,P

(3.5).

a "norm"

the

this

we are

Ivl

Lemma

introduce

see that

properties z0

D

z ( K.

We

We

of

that

IvI

= 0

if and

:

F E H~, F(z 0) = v}.

only

if

v = 0.

z0,P Proof:

Clearly

v = 0

implies

Ivl

= 0.

If,

on the

F E H

such

other

hand,

that

F(z 0) =

z0,P Ivlz0~ p v

0

and

Ilvll

IiiFilip,z0

:

Since


0,

(3.4),

IIr(z o) II ~ C{z0}llFII ~ s

If

we c a n

with

an

K = {z0},

-< e{z0} IIIFIIIp

can

be a r b i t r a r i l y

small,

IIvll = 0

I ~ p

it f o l l o w s

immediately

from

we t h e n

have

~ e' IIIFIIIp,z ° and,

(3.5),

thus,

and


" 0

12~ ~(8)Ig(ele)l*i82-" d£%0 e

the other hand, linear

2~ I0

satisfying

= ~(e)f(ele). IThen f ~ L#2 and • @n is the function defined above. L(f

by

2 L#

in

In order to see this we choose a m e a s u r a b l e function ~ fies l~(eie)l ie = i and Ig(eie)l~'~i8 = .

f(e ie)

Hence

1 2~ ]2 " 2 de If(ele)l e ie 2~ 0 [l-gn(e)

:

0

be a c o m p l e x

n.

=

2 functional on [# and IILII The r e s % r i c t i o n of L to L 2 is then a bounded such with norm

= IVlz0

d8

=

IF(elS)lei8

IVlz 0

a.e.

Since

by the

141

IG(eiS)l*ie

= i

a.e.,

the

smoothness

assumption

tells

us that

F(e i8)

e

is

a.e.

determined

function

by

G.

But

this

uniquely

determines

the

analytic

F.

If

F

is a n a l y t i c

is a s u b h a r m o n i c erty h o l d s we h a v e function

function.

when

the

We

intermediate

situation

on

of r a d i u s

in a d o m a i n

D.

shall

norms

of t h e o r e m

Let

r > 0

D

z0 ( D

show

are

in

as

that

used.

I and

and

is c o n t a i n e d

then,

F

is w e l l

this More

is a

suppose

the

D.

w(8)

If

known,

loglF I

subharmonicty precisely,

~n-valued closed

prop-

suppose

analytic

disc a b o u t z0 ie then, by

= z 0 + re

definition, 2~

iF(zo)iz °

(


0

(the

integral

IW2(w(8))IllF(w(8))ll p ~ 0

the right

of the

exp { By t a k i n g

logarithms

Corollary

(4.4).

loglF(z) I

i

2~

for

above

defined

2~

since

8 ([0,2~)). inequality

IF(w(e))lw(8 ) Thus,

tends

d0 loglF(w(e))lw(e ) ~}

letting

to

.

0

this

If

w(e)

is well

is b o u n d e d

side

de )

F

shows

is a

(under

the h y p o t h e s e s

cn-valued

analytic

of t h e o r e m

function

on

I):

D

then

is s u b h a r m o n i c .

Z

This

result,

terization

among

of e x t r e m a l

other

things,

functions.

is u s e f u l

Suppose

for o b t a i n i n g

F = F

a charac-

is an e x t r e m a l z0,v

function;then

for any

IF(z)l But,

by the m a x i m u m

IF(Z)lz

that terizes

0

if

~ D F

all F(z).

is

extremal

Corollary z

z

(4.5).

and

~

principal

IVl~o

: for

It

IF(Zo)lzo

:

subharmonic

is

this

functions,

constancy

this

property

means

that

charac-

functions:

If

v E ~n

then

IIIFNI~

constant.

is an a n a l y t i c z ~ D

z E D

F

is an e x t r e m a l

then

IF(z) I = z

~n-valued

Ivl

function

for all

z0

function

it is an e x t r e m a l

corresponding

such

function

for

that each

z ~ D.

to Conversely,

IF(Z)Iz

= c

for

z ~ D

and

vector

142

Proof:

Suppose

the a n a l y t i c

function

F

satisfies

IF(z)l

= c

for

Z

all

z E D.

Then

lary

(4.1).

Thus,

[IIFI[I, . with

z

HWI(Z)F(z)II WIFE

This w o u l d

imply that

and

for each

F(z)

~

IF(z)iz

H'(D;{n). F

= c

for all

We c l a i m that is an e x t r e m a l

z E D

z E D,

F E H~ function

and the c o r o l l a r y

lished.

by c o r o l -

and

c =

associated

would

be e s t a b -

2~

To see this, and put have

G(z)

W2 ( ~

w E cn,

= W(z)-lw.

IIG(=)II Thus,

choose

~

Ilwll e x p

G(z)

z)

Since

~ i

i

let

W(z)

h (8)loglwl* de z eie (by (3.11)) we

0 ~ k2(e)/HwlI

i/lwi*ie e

2~

: exp [

k2(e) P (e)

0

log(

and,

) de

IIwTI

z

therefore,

G E H~*

=

.

IW2(z) I .

Since

G

(e ie

) =

we have W(e lO )

lwI"~ie "IIIGIII~

Hence,

:

ess. sup eE[0 ' 2~)

II



belongs

~

IG(eie)i*ie e

=

=

I.

I w l * ie e

IF(Z)Iz]G(z)l~

to

e

ess. sup @E[0,2~)

H'(D).

~ c-I

=

a.e.

II


= k~c o n v e r g e n c e is that IVklz0

IViz 0

Thus,

F

property

by c o r o l l a r y

A(z,z0)(v).

:

is also



constancy

w E ~n,

of

if n e c e s s a r y ,

IF(z)I and,

(see f o o t n o t e

is a convex 2 H#

thus,

w E Cn

into

has the

II

sequence

weakly.

Choose G ( H#2

H#2

Relabelling, F

is a b o u n d e d

a subsequence

But

2 L#;

{Fk}

of t h e o r e m

therefore, 2 F E L#.

element

Thus,

(4.5),

=

z

Ivl

it must

z E D,

z0 '

be the e x t r e m a l

function

F(z)

=

But the c o n v e r g e n c e

also

l i m < F k ( Z ) , W > = , for each k~lim IIFk(Z)-F(z)ll -- lim IIA(z,z0)(Vk)-A(z,z0)(v)ll k~ k~

implies

0.

We have there

shown,

exists

therefore,

a subsequence

that w h e n e v e r

{Vk. } 3

vk ~ v

clearly

implies

k ~ -

then

such that

lim llA(z,z0)(Vk.)-A(z,z0)(v)ll But this

as

=

0

lim I]A(z,z0)(v k) -A(z,z0)(v)ll

= 0

and the

k-~

desired

continuity

is proved. lim ie

(4.13)

Iv I

:

Ivl

z

i8

a.e.

e

z~e

Proof:

Let us assume

smooth

so that we have

functions dense

{wj

We write

E ~n :

sphere

in

{n.

(a)

lim A * ( z ) ( w . ) i8 ] z~e

(b)

IA*(eie)(wj )I i8 = I, e

the duals

the u n i q u e n e s s

A*(z,z0)(w).

subset,

of the unit

that

space

are also

of the c o r r e s p o n d i n g

extremal

A*(z)

lwjl0~- = I, Then

of the b o u n d a r y = A*(z,0).

j : 1,2.3 .... },

for almost

= A*(eie)(w.), ]

Select

every

0

a countable

of the we have

surface

145

(c)

lim IW2(z) I = k2(e) # 0. ie zm e Let us fix v ( cn. Then

(d)

Ivl

: z

ll

sup

j:l,2,...

]

This is an immediate consequence the density of continuity of

{w.}

of the fact that

IA*(z)l~ : {wl~,

in the surface of the unit sphere of

A*(~):

cn ~ cn

cn,

the

((4.12)) and the onto property of this

. . map ((4.8)) . Let f. (z) = ll / IW2(z) I Then f] is the absolute value of an analytic function in H'(D) since, using (4.1), f.(z) 3

-
- k2(8)h(eie)

h(z)

sup f.(z) = IVlz / lW2(z) 1

'

Consequently, Ivl i8 e

h

:

But, by (e)

lim zt>e

]

lim

IW2(z) {h(z

i@

iO

Ivl

a.e.

z'

z~e

By theorem II, we also have

(f)

Ivl*ie >- iim e

Let us choose

Iv1*Z a.e.

z~el9 w ( {n.

We have just shown

lw e

We shall now use (f) to obtain

lwl ie -< lim_ e 18

i8 >- lim ie

lWlz

~.e.

z~e

lw I 1 Iz

a.e.

Let

e

ie

zme

be a point on the boundary of

D

and denote by

~(e)

a "pointer '~

146

region (See

with

vertex

ie

at

e

;

let

~

(e)

:

{z

(~(8)

:

Izl

>

l-s}.

figure.): Choose {v=}

a countable in t h e

dense

surface

set

of t h e

unit

J

sphere

of

holding

~n.

We t h e n

for

have

(f)

in a set

v = v., ] E c [0,2n)

of measure

2~.

e E E

i > 6 > 0.

Let

j = 1,2,...,

and

V.

We

can

then

find

v =

J e

such

that

l<w'v>l

~

(l-6)lwl

ie" e

For

this

v

using

(f),

(l+5)

=

and

e

s > 0

we

can

such

that

Ivl*ie(l+5)

~

find,

Ivl*z

for

e

all

(1-~)lwl ie

~

l<w,v>l

~

z E ~ g (8).

lwlzlvl [

~

Thus,

lwl (I+5)

e for all z E ~ s (e). Since 6 > 0 ie -< -i+6 - - lw Iz e 1-6 can be chosen arbitrarily s m a l l it f o l l o w s t h a t lwl ie ~- l i m lWlz. e ie This proves (4.12). z~e This

shows

ping

a family

Let

another

that

lwl

now

pass

us

such

extension

of

to t h e

problem

intermediate

family.

to this

We

of

spaces

shall

situation.

interpolation

satisfying

see that Suppose

properties

(2.1)

has

theorem

{B

} : {(~n,N Z

{ ( ¢ n , M z )}

are

operators.

We

two

such

assume

families {T }

and

of operators

these

)}

a natural

and

{C

Z

is a n a n a l y t i c

} = Z

Tz: Bz ~ Cz

that

mapinto

family;

are

we

linear

mean

by this

Z

that

z ~ T v is a n a n a l y t i c z a point z0 ( D and vectors Let

F = Fz0,v

norms

{N z}

w

the

and

be a n

and dual

cn-valued v,w

extremal

G = Gz0 w norms

an

{M[}.

¢(z)

E ~n

such

of

that

function

associated

extremal

function

It

=

function

Z

is an a n a l y t i c (4.1)

to

function

estimate

~(z)

on

D

(we c a n

in t e r m s

see

this

of E u c l i d e a n

most

easily

norms:

if

by using K c D

is

147

compact there eXists a c o n s t a n t

R(z)

!

cK

cN

sup

liT vll z '

Ilvll~l

Then using this inequality, H~(D;¢ n)

again

we can deduce that

tion of

such that

(4.1), and the fact that

TzF(Z)

is an analytic

D.

WIF

Cn-valued func-

z (D).

Suppose the closed disc of radius in

z ( K.

Writing

~(e') = z 0 + pe ie,

s u b h a r m o n i c i t y of

}¢(z) I

p > 0

about

e ( [0,2~),

and C o r o l l a r y

z0

is c o n t a i n e d

using the logarithmic

(4.5),we have

2~

1 I 2w

loglg(zo)l -

log[¢(_

2

IIIhlll] 2,Zo

2

0

'z0

lemma.

v ~

(PG v )(z) is c l e a r l y l i n e a r since P z0 Thus, A ( z , z 0) is a l i n e a r o p e r a t o r on

analytic

in

z.

By C r a m e r ~ s

analytic

in

z.

This

rule,

the

inverse,

is c e r t a i n l y

not

true

which

is

must

also

be

(see the

form

A(z0,z) ,

in g e n e r a l

is a l i n e a r ~n

151

exhibited after

(5.2) for the operator

spaces were LP-spaces). n xn

matrix valued

variables

z

and

the product analytic.

z 0.

a(z) W2 ~

is analytic

From (4.9) we have,

A(z)a(z0);

Ivlz = la(z)vl0.

is a bounded,

have the almost everywhere lim

(5.7)

A(z)

and

is is

we have

analytic

function.

of the non-tangential

a(z)

A(z,z 0) a(z0),

llvll.

matrix-valued existence

to be an

in fact, that

By (4.1)

~

v 0

A(z,z 0)

in either of the two

each of the factors,

a(z)

Thus,

when the boundary

Thus, we can consider

function which

Moreover,

B(z,z 0)

Hence we

limits

a(e ie)

zmei0 W 2 ~

=

W2(eiS)

But Iv]0

=

(

[2~

IA(eie)(v)]

J0 is a Hilbert that

2ie de 1/2 ~-~) e

space norm on

IvI0 = llavll for all

cn.

=

[2~

" d8 1/2 llF(e)A(e~e)(v)II 2 ~ - )

~0

Thus,

v ( cn.

(

there exists a matrix

We must have,

therefore,

a

such

because

of (5.7), lim ie

a.e.

Ivlz

=

lim llaa(z)vll ie

z>e

z>e

On the other h a n d ,

by ( 4 . 1 3 ) ,

lim i8

Ivl

= z

Ivl ie e

=

=

llaa(eie)vll

llF(9)vll a.e.

z~e

This shows that for

m

z ( D

Theorem

and put

(5.8).

8 ([0,2~)

llaa(ele)vll = llF(e)vll a.e. P(e) = F(8)*F(e)

Suppose

such that

P(8)

If We let

b(z)

= aa(z)

we obtain the following result:

is a positive

logllP(e)[l and

definite matrix for each -i logllP(8) II are integrable,

then there exists an analytic matrix valued function

b(z)

on

D

that (5.9)

lim b(z)*b(z) i8

=

P(8)

zbe

almost e v e r y w h e r e .

Moreover,

the

operator

norm

IIb(z)ll

satisfies

such

152

(5.1o)

kl(Z)

for all

z ( D,

_

I log llP(e)-lll 2

where and

This result which

~

Jib(z)]i ~

lOgkl(Z)

logk2(z)

k~(z)

is the Poisson

is the Poisson

is an extension

integral

of the Wiener-Masani

of i

integral of

~ l o g IIP(8)II

theorem

(see [8])

states: If

P(8)

@ ( [0,2~)

= (p~k(0))j

such that

is a positive

p~k(@)

belongs

definite to

n x n

LI(0,2~),

matrix for each j,k : 1,2,...,n,

and 2~

(5.11)

-"


0

fB(x;r) If(Y)Id~(Y)"

Then it follows that there exists a u n i v e r s a l c o n s t a n t

Ce~

such that

llfJl1 ~(Mf(x)

Later on, E. Stein Math.

1960)

> a}

~

C

-

-

(Limits of sequences of operators, Annals.

proved that, under very general conditions,

of

the q u a l i t a t i v e

and the q u a n t i t a t i v e results m e n t i o n e d above are in fact equivalent. It is interesting to o b s e r v e that if one r e p l a c e s balls or cubes in the statement of the Lebesgue theorem by more general families of sets,

for example p a r a l l e l e p i p e d s

in

~n

with sides parallel to the

c o o r d i n a t e axes, then the d i f f e r e n t i a t i o n theorem is false in general for integrable functions

(Saks 1933).

and Zygmund showed that,

in

of

f

~n,

In 1935 Jessen, M a r c i n k i e w i c z

we can d i f f e r e n t i a t e the integral

with respect to the basis of intervals c o n s i s t i n g of p a r a l l e l e -

pipeds with sides p a r a l l e l to the c o o r d i n a t e axes, locally to the space

L(log+L)n-l(~n).

so long as

f

belongs

This result is the best possible

in the sense of Baire category. The theory of d i f f e r e n t i a t i o n of integrals has been c l o s e l y related to the c o v e r i n g properties

of families of sets.

ample is the use of the Vitali covering ferentiation

theorem of Lebesgue.

of this r e l a t i o n s h i p

is given,

A c l a s s i c a l ex-

lemma in the proof of the dif-

In [i] a very precise i n t e r p r e t a t i o n

and [3] contains a geometric proof of

155

the result of Jessen, lemma of exponential

Marcinkiewicz

type for intervals.

Given a positive separately,

function

~

to the rectangular

~2,

sxtx~(s,t),

where

B3,

monotonic

basis

family of parallelepipeds

coordinate s

the differentiation

and

t

of

B~

in

~3

in fact,

B~

behaves

function and covering

Zygmund was the first mathematician 1935 paper in collaboration result and its extensions

defined

are given by In general,

whose sides have

axes and, of course, like

not better than B2

point of view as well as for the estimates

sponding maximal

~3

must be, at least, not worse

the basis of all parallelepipeds

ferentiation

in

are positive real numbers.

properties

We will show that,

B~

in each variable

whose sides are parallel

axes and whose dimensions

the directions of the coordinate B 2.

on

consider the differentiation

by the two parameter

than

and Zygmund by using a covering

properties.

from the diffor the corre-

I believe that A.

to pose this problem after his

with B. Jessen and J. Mareinkiewicz.

to higher dimensions

the behavior of Poisson kernels associated

This

are useful to understand

with certain symmetric

spaces.

Results Theorem.

(a)

B~

differentiates

cally in

L(l+log+L)(~3),

integrals

that is

IRf (y)dp(y)

lim ~ 1 R=x

of functions which are lo-

=

f(x),

a.e. x

REB} so long as

f

gue measure (b)

is locally

in

in

L(l+lo~ L)(~3),

maximal

:

Lebes-

Sup ~

I f ( y ) Idu(y) R

the inequality If(x)~

p{M~f ( x ) ~ > 0 }

for some universal

Coverin~

denotes

function

x(R R6B~)

geometric

p

~3.

The associated M~f(x)

satisfies

where

c Jr s - -

constant

C< ~.

(i + io~

If(x)l] dp(x)

The proof is based on the following

lemma. lemma.

Let

B

be a family of dyadic parallelepipeds

in

~3

156

satisfying

the f o l l o w i n g

the h o r i z o n t a l corresponding

monotonicity

dimensions dimensions

of

RI

of

property:

are b o t h

R2,

strictly

t h e n the v e r t i c a l

m u s t be less than or e q u a l to the v e r t i c a l It f o l l o w s property:

that the f a m i l y

Given

{R } c B

B

one

If

can

RI, R 2 ( B smaller

t h a n the

dimension

dimension

of

and

of

RI

R 2.

has the e x p o n e n t i a l

type c o v e r i n g

select

{Rj}

a subfamily

c

{R }

such that

(i)

~{UR } _< C ~ { U R j } ,

(ii)

f

and

e x p ( Z X R .(x))d~(x)

_< C~{LJRj}

UR. ] for some u n i v e m s a l Application.

R3

constant

Consider

:

{X =

and the c o n e

x3) ,

x3

upper

positive

half-space

For e a c h i n t e g r a b l e integral,"

u(X + iY)

real,

symmetric,

function

:

definite}.

= {X + iY, f

= Py*f(X),

Py(X)

Qn

question: a.e. x

u(X + iY) ~ f(X),

convergence

fails

fact that

T F = tube o v e r

~3

definite}.

we h a v e the

"Poisson

where

For w h i c h

when

if

if

functions

Y ~ 0

y=

without

for e v e r y c l a s s

f

is it

Y ~ 0?.

y-I =ly" 0~- ~ O, ~u YI for i n t e g r a b l e f u n c t i o n s f.

a.e. x,

On the o t h e r h a n d

Then

positive

C [ d e t Y ] 3 / 2 / I d e t ( X + i Y ) I3

true that

It is a w e l l - k n o w n

Y

Y ( F,

and we m a y ask the f o l l o w i n g

u(X + iY) ~ f(X),

2x2-matrices},

x2

F = {Y(IR 3,

F = Siegel's

C ~}

_
0}

~

C(log

3N)

2

2

'

f ~ L2(~2). third

result

is a r e s t r i c t i o n

theorem

for the

Fourier

transform.

163

Theorem

(C).

Let

r e s t r i c t s to an I > 3[i - i] q P '

4 i _< p < ~-.

f (LP(m2), Lq

Then the Fourier t r a n s f o r m

function on the unit circle

S I,

where

and satisfies the a priori i n e q u a l i t y

IIfllLq( sl )

~

C p,q IlfllL P ( m 2 )

(C. F e f f e r m a n and E. Stein [5], A. Zygmund

[8]).

Strategy The m u l t i p l i e r

mx(~)

: (l-I< I2 )+

seems very c o m p l i c a t e d and one

of our first tasks is to find out w h i c h are the basic blocks of the Calder6n-Zygmund

theory c o r r e s p o n d i n g

to

ml.

Since

mI

is radial and

b a s i c a l l y constant on thin annuli it seems r e a s o n a b l e to d e c o m p o s e o0

0 where

ek' k ~ I,

is a smooth f u n c t i o n supported in the interval

2 -k, 1 2- k - 2 ] pendent of k, and [i -

k

!lek~l

such t h a t

on

IDe~k [ 5 Ca2 ka,

[~-, i],

e0

:

i-

k

where

Ca

is i n d e -

!lek.

Then

and the p r o b l e m is reduced to getting good estimates for the growth, as

k ÷ ~,

~k(I~I).

of the norm of the m u l t i p l i e r s For example,

a s s o c i a t e d with the f u n c t i o n

the C a r l e s o n - S j ~ l i n result will follow very

easily if one can show that the o p e r a t o r

T~f(~)

: ~k(l