part of the material is concerned, specifically those of translation, reprinting, re-use of ... fication of expected equilibrium thermodymamic properties and b. the validity of the ...... partly on F to control the error term in (4) and partly on a(x) to control the ..... In the first term we use the assumption on f and find the estimate. N. -3/4).
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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.
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.
.
.
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.
.
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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
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 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 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
le
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
