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.
proved
only
in a case
rather
like
this.
recently The
by R u e l l e
problem
that
f(~)
of r e g u l a r i t y
of
does f
indeed is,
as
however,
still
ing t h a t The
unsolved.
f(p)
is a l w a y s
problems
in p r o b a b i l i t y , by C r a m 6 r well
and
Here
viz.,
shall
analytic
we h a v e
Feller
we
the
some
for small
dealt with
the p r o b l e m and
give
are
results,
related
deviations.
results
which
to a p r o b l e m This
we need
was
studied
later
are
known. Let
tribution
We are
XI,X2,...,X N and
be r e a l
stochastic
E ( e XX )
=
variables
with
identical
assume
interested
F(X)
<
-.
in -N~N(t)
Prob
=
X. > tN J
e
,
t > 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 <<
which
to o c c u r
ly s i m p l e
case,
Boltzmann
motion,
assumed
general
each
in one
of
N states,
as a g i v e n
position
and
velocity.
in s t a t e s
(i,j)
interact
and
go over
proportion
of p a r t i c l e s
can
(v,~)
which
arise
is A ~ Pi ( t ) p j ( t ) & t 13
where
Pi(t)
The m a t r i x
is the A~ m]
(3)
proportion
is a s s u m e d
A~
:
A~Y
i]
We
of p a r t i c l e s
i
in state
at time
t.
to s a f i s f y
:
A ij
]i
>
@
(i,j)
¢ (v,b)
Vb
set
(4)
A~
For
p (t)
we
obtain
=
in this
p (t)
which
is a g e n e r a l
of the
usual
-~
=
discrete
equation,
way
and
A~b.
the
differential
Z A i,j,~
)pj(t),
Boltzmann the
equation.
proofs
are,
equations
It has
of course,
many
all
features
very
easy.
N
(A) Proof.
[ pv(t) 1 By
(4)
it f o l l o w s
:
1.
that
N p~(t)
=
A ~ ] P i ( t ) p j (t)
i
(B) Proof. p
(t)
~'v s
p(t) Suppose
= 0 is
0 ~ t < tO
first
then
p
dense. and
=-
0.
i,j,~,b
that
(t)
~ 0,
Suppose
for all
i.
~
e~ = p
(0) > 0
~ ~ v. now Then
0.
that
By p
for all
~
analyticity (t 0)
= 0
and
and
that
this
set
Pi(t)
> 0
if of for
11
p (t) i.e.,
:
p (t)
an e q u a t i o n
[ (A~ j,~
P% : ~Pv
+ f'
g(t)
is n o n - d e c r e a s i n g > 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
<
~.
set
/Y,~(t)i HN(t)
=
sup s~t
Standard
methods
sun I,,I e
,~
zive
Ibm, I HN(t) From
this
we
is a u n i q u e The denote
see
C sun
lim yv(t~N) N~
q~
Ct
Ic e
= yv(t)
and
that
y(t)
:
{ ~ (t)}
solution.
solution
the
that
~
y(t)
distribution
is e a s i l y
described
explicitly.
Let
b(x)
15
b(x) and
first
A(x)
a distribution
This
~ 6 >
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
<
Const--
-
K2
I. 3 Proof. of
(Theorem) t
f
!
yG,yl...y P
We restrict plify
the
We on
ourselves
formulas.
consider (0,T)
the
distribution
and compute
in t h e
first
function
its c h a r a c t e r i s t i c
place
HT
function.
to the d e r i v a t i v e s
to
sim-
on
17
T h e n we have P :
e
....
Letting
A(x)
dt exp
P u~ ije -13x,
=
the
i~j~
b(x)c6s(a(x)t)ei'~
expression
u n d e r the e x p o n e n t i a l
sign c a n be w r i t t e n
A ( x ) e -i~x e o s ( a ( x ) t ) dx --co
--7[
co
=X
b
~
-~
(AlCOS vx + A 2 s i n
~x)cos(a(x)t)dx.
--co
MT
now
is a f u n c t i o n
of the
; e ~ e i b_u dF(b)
that
u2+o( u 2 )
- ~i (4)
If we o b s e r v e
i n i t i a l values.
=
,u ÷ 0
and also that ~
(AleOS
vx + A2sin
vx)cos(a(x)t)dx
:
- 2~i flA[2 eos2(a(x)t)dx,
we f i n d , u s i n g
(5)
If F
lim E ( M T) T~
lemma, --i ~T 4~ lim T J e T~ 0
=
is the n o r m a l
is no e r r o r zero,
the
t e r m in
it f o l l o w s
IAI 2 c o s 2 ( a ( x ) t ) d x -~ dt.
distribution, (4).
Now,
from W i e n e r ' s
the
if
lemma
a(x)
theorem
= c
is not n e e d e d
since there
only on sets of m e a s u r e
on m e a s u r e s
without
point masses
that
f
= 0 ,
lim liA~ 2 2 e o s ( a ( x ) t ) d x 1 t -~= I -~T if we a v o i d
a set of
t
of d e n s i t y
1
rT
-
zero.
8"~
-~
Hence [A{ 2 d x
lim E(M T) = lim ~ j e T -~ T-~ 0
which
is the F o u r i e r t r a n s f o r m of Gibbs' To p r o v e
the t h e o r e m we m u s t
-4- 0
3
dt = e
distribution.
also c o m p u t e
the
second m o m e n t
of M T.
•
18
The
computation
The
result
is c o m p l e t e l y
analogous
not
be r e p e a t e d
here.
is l i m E ( I M T 12) T-~
From
and w i l l
Tchebycheff's
inequality
=
lim T~
(E(MT))2.
we deduce
that, for a n y
large
given
T
< ~.
For a n y
enough, i MT -
except
on
finite
set of
the
sense The
on
sets
this
a
set l's
the
correct
To p r o v e on
Theorem.
values
same
condition
of m e a s u r e
control
initial
of
is t r u e
probability
and we h a v e
proved convergence
is no d o u b t
that
in
specified.
condition
partly
of
< E
F
the
is
zero.
on
a(x)
I have
however
no p r o o f
a(x)
of the
~ zx + w
lemma
in w h i c h
sufficient.
ordinary
ergodicity,
to c o n t r o l lemma
the
error
uniformly.
If, in a d d i t i o n
The
to the
we n e e d
some
term
(4) a n d
in
following
earlier
extra
assumptions,
partly
theorem
on
a(x)
to
holds.
assumptions,
co
ibl2+6
dF(b)
< ~,
some
6 > 0,
--oo
and
if dim[a'=0]
then
the
system
We n o w
of d e r i v a t i v e s
turn
The
computation
the
similar
to the is t h e
is e r g o d i c .
computation same
characteristic
< I
for t h e
u p to the function
distribution
formula
M}
(5).
of
Here we
y(t). find
for
that
1 i~ IAI 2 si2(a(x>t) dx (6)
lim E(M~)
i lT
=
lim ~ ]
T-~We h a v e Theorem. only
T -~
to d i s t i n g u i s h The
Gibbs
if - g i v e n
two
theory
earlier
One
could
easily
-7
a(x) 2 dt.
0 cases. holds
conditions
IX
(7)
e
4~
describe
f o r the
dx A(x~
the
complete
distribution
if a n d
on a -
< ~.
precise
situation
if
(7) d o e s
not
19
However,
hold.
let us be c o n t e n t A(x)
and A(x)
~ 0
J
x = 0)
convergence
at
case
when
x = 0
particles
E(M~)
therefore
i we have, on the
i
:
line
Y0(t) and the d i s t r i b u t i o n
IIIA(x)I2 s i n 2 x t x2
-~
dx T
to
lim T~ The
2
of the
gT
4~
and we get
x
a study
otherwise. T h e n we s h o u l d c h a n g e the scale of the A's X. X. by 3. The e x p r e s s i o n in the e x p o n e n t of (6) c h a n g e s to
and r e p l a c e (at
=
with
up
1
~
d~.
0
in a r o w w i t h
: Yl(t)
~...:
is a c o m b i n a t i o n o f
other
1 ~iA(0)l 2
-
e
yp(t)
:
,
normal distributions.
hand, i n d e p e n d e n c e
since
the
On the scale
derivatives
are
in-
dependent. In d i m e n s i o n while
in
two a s i m i l a r
3 and m o r e
2.
dimensions,
We n o w t u r n
of energy. question
The as
to the
problem
to
phenomenon
occurs
but
the Gibbs
theory
in g e n e r a l
second
problem,
is s u r p r i s i n g l y
what
extent
that
A(x)
the
on scale
concerning
difficult
conditions
and on
the
l~g
T ,
holds.
distribution
it is an open a(x)
and
b
are
necessary. Theorem. Suppose and
Suppose that
b(x)
b
exist.
= b(-x).
Suppose
Then
finally
~ bnbn+k n:-N
for all
y~(t)
_
that
(IbvJ that
A'(x) ~ C),
the
> 0,0
< × < ~.
that
0,~
correlations
= Pk'
k : 0,1,2...
v i fT
It is of c o u r s e
and
N
lira T÷~ Proof.
5
is a p s e u d o m e a s u r e
I lim 2-~ N+~ all
( C
~ Yv (t)2dt
: i ~ P0"
0
sufficient
to c o n s i d e r
v = 0.
By
i__ [l b ( x ) b ( y ) c o s ( t a ( x ) ) c o s ( t a ( y ) ) d x d y . 4~ 2 JJ
(i)
~ supp(b),
20
Let
w(u)
~ 0
w(u)
= w(-u).
be an element of Let
~
C~,
assuming
be its Fourier
i ]~T - y~(t)2w(t/T)dt 0
~
transform.
w(u)du
= i
Then
8~21 [I b ( x ) b ( y ) w ( T ( a ( x ) - a ( y ) ) ) d x d y
+
+ a(y) ~ 6 > 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'(<)~)d~d~"
x+y : 2~,
21 Observe
1 Thus,
now that f
a'({)
~ 6 > 0
where
h ( q ) w ( 2 T a ' ( ~ ) ~ ) e l m ~" d n
~ ~ 0.
Hence, for all
W ( 2 a ~m- ~ )
=
~ 1
N > 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]
sum termwise
as T + ~
for n : 0,±1,±2 ....
we have
m<s
bn+ m bn_ m : 2 2
pn s + o(s).
Hence,
! :
and our limit
]
r co
Im[ CT
bn+m 2 I
bn+m 2
~(~)ein~
becomes
%(~)eln<
m
W(2a'(6]~)
2--~)T
-~ w(u)du
+ o(i
÷
Pn
%(~)e in[ d6,
22
S(¢)
To
compute
the
this
support
sum,
of
[
let us
b.
~)
=
0n
¢(~)e ~n¢
take
Then, for
- ~
]
":
@
k
with
large
bn+ j bj
d~.
support
enough,
-~
in
I~I
outside
< 6
we h a v e
~ ( ~ ) e In~ d~
< s.
n
The
inner
sum c a n
be w r i t t e n
i k -- ~ k j:l Hence
S(%)
support
~ i~ b. [ bn ] n. . . . .
is i n d e p e n d e n t
of
b.
We h a v e
of the
% ~ i
S(~) Checking
the
constants
.. @(~)e -l]~
for
ein~
definition
d~
of
~
in a n e i g h b o r h o o d
:
S(1)
some
:
trivial
: O.
outside
of the
the
support.
Hence
00. choice
of
b,
our
result
follows. 3. tion,
We
shall
now
i.e., a s s u m e
get t h e
study
that
~ x
for
Y0(t)
we
the m e c h a n i c s
in some
of t h e
suitable
situa-
sense.
To
see t h a t Y0(t)
St
denotes
It is of c o u r s e in s i t u a t i o n s
different. at
the
of this
0,~.
interest
functions
b(x)
f(t)
(LI(-I,I)
to
~
b(x)
study
a"(0)
below,
was
~ 0.
study
the
that
+
Fourier
series
interested
of
b.
in o v e r s h o o t s
We get t h e
x
has
is r a d i c a l l y
its o n l y
extreme
we h a v e
sin a(~)
of this
0nly the
Therefore,
situation
A(x)
C2b(~)
asymptotics
as p o s s i b l e .
and
s u m of the Gibbs
smooth,
CI b(0)
importance.
with
that
simplicity
Then, for
It is o f
a(x)
partial
is b o u n d e d
for
Y0(t)
all
t th
, t ÷ ~,
type.
A(x)
Suppose
x = 0,~ is of
i ~ St(0;b)
~
no c o i n c i d e n c e
If h o w e v e r
points
detail
c o n n e c t i o n w i t h c l a s s i c a l a n a l y s i s let us a g a i n a s s u m e t h a t 2 and is o t h e r w i s e ~ 0. If we c o n s i d e r the f o r m u l a (i)
A(x)
where
in some
b v ÷ 0,1v ] ÷ ~,
type
for as g e n e r a l
l o c a l b e h a v i o u r of a(x) 2 is c o m p l e t e l y g e n e r a l following
problem:
Let
at for
23
f N(x)
This is also of basic bility
ef Fourier
C~
:
interest
It will turn
out
that
even
f ( Ha
on
:
fN(x)
Since
f({)
case.
Actually,
E LI
if
spherical
summa-
the relevant
functions
kernel
is
have pathologies,
classes
compact
]_~ f(~)e ~ x
Ha,
0 < e < i.
support.
Then
~(~)
dE
that
better
If[x)
fN(x) ÷,f(x) uniformly. need not converge. first
ei~ 2 /4N d~.
£(~)el~X .
i~
f~ Ha, ~ > i/2, fN(X)
a slightly
Suppose
Assume
:
of
x,t (~g.
to Holder has
N ÷ ~.
sense, and
(13)
Proof.
study
an example,
continuous
(-~,~)
f(x) in a suitable
the
e iNl×-tl ix_t13/2,
limit our discussion
We may assume exists,
Theorem.
for
integrals. I n ~ 2 , a s
W~
and we shall
i +I e iN(x-t)2 f(t)dt, -I
result
- f(Y)l
÷ f(x)
= o((x-Y )I/2)
If f E HI/2 then fN(x)
the
o-condition.
uniformly
in this
uniformly.
Then
holds.
Fix
is bounded
N
and define
but fN(x)
a
by the
if
~ > O.
relations Na Conzider fN(0)
:
2~I~I,
where
av < 0
and assume
=
N a~+l iNt2 ~ e (f(t)-f(a))dt -N < - a ~
Now observe
second
and
av > 0
We have +
f(a
)e iNt
-a
that IIav+l e iNt2 d t
The
= 0.
~ < 0
x = 0 C~
f(0)
if
sum
is
dominated
( x0
i r = 0 (N-~( I v l + l )
3 - -2 ) .
by
i/4\ / -i/
)
-5/4 :
÷ 0.
d
.
24
In the
first
t e r m we use
the
assumption
N [ o(iAa 13/2 )
on
N_I/4 N ~
:
-N
To
that
see
f ( HI/2,
that
fN(x)
Nj+I/N j ÷ ~
to
the
estimate
-3/4) :
and
we
see
from
not
converge,
see
the
proof take
o(i).
that
fN(x)
is b o u n d N].
a sequence
SO
define
that
=
f(t)
[ NI I/2 i 3
( HI/2.
e
-iN.t 2 3
For
fN.(0)
sion,
we have
the
expres-
]
j-1
fNj(0)
and
find
o(~
need
f(t)
It is e a s y
and
i
If, instead, ed.
f
=
I k=l
C~j
N~
1/2 I +1 i(Nj-Nk )t2 e
-I
~
dt + 2c +
~1/2
~ k=j+l
O(N%/2N ]
the
condition
),
so lim
If we a l l o w be w e a k e n e d
but
fN.(0)
an e x c e p t i o n a l the
following
theorem.
Theorem.
Let
definitive
f(t)
f(0)
result
a
a.e.
On the
other
hand,
lim N-~oo
For the
proof
of
on the
(14)
method,
based
Lemma.
Let
a,b
rs
e i ( a ~ + b ~ 2)
be r e a l
there
we use
eo
the
known.
We
shall
prove
( H(I/8)_ a
such
can the
f(x)
exists
:
zero,
Then
=
IfN (x)I
following
is not
> 0.
lim fN(x)
exists
+ 2c.
set of m e a s u r e
( H(i/4)+a,
(14)
:
f(t)
that
a . e .
Kolmogorov-Seliverstov-Plessner
lemma.
numbers
in
(-2,2)
and
suppose
0 < ~ <
i.
Then (15)
Proof. The
(Lemma)
integral
to
We m a y
dE
assume
estimate
__< Ca
b > 0
becomes
<
ible-i/21ai-@+lal@-i
)
and
and
set
t = bl/2~
.
2A = ab -I/2.
25 r~
b-i/2+e/2
]_~ ei(2At+t 2) _dt Itl ~
If
IAI _< 2,i.e.,
lal _< 4b I/2, then the integral is bounded and (15)
has the estimate 2 < A < ~.
Cb -(I/2)+~/2 _< Clal e-I
We may therefore
assume
First, we have
i
I
i(2At+t 2) dt
- ]tl ~
-1
-<
CA~-I
=
Clal
~-i
bi/2-~/2
which is sufficient. The integral over (i,~) gives the stronger estimate 0(A-I). Finally, decompose the integral over (-~,-i) into (-~,-2A),
(-A/2,-I),
and
(-2A,-A/2).
Since
IaT(2At+t d 2) -the first two integrals have bounds
>_ A
0(A-I).
The third finally is
written A
eiU 2
I
du -A/2
=
0(A -~ )
=
O(lal-~lbl ~/2)
(A+u) ~
and the lemma is proved. Proof.
(Theorem)
of the maximal
+i
is
sufficient
to
get
an
in case f ( C O . We make N I 6(x) = 4N(x)" Using (13) we find
x, N(x), and set
f
It
fN(x)
fN(x)(X)dx
=(I_ The first integral
=
f
IR(~)I21~I1, '
f(~)
f:i
d< .....
A-priori
eiX~+i6(x)~2
d< I~1 ~--M
is bounded by the assumption
estimate
into a function of
dxd~
~
~-1
f ( H(I/~+
.
dx
We write
the square inside the second integral as a double integral and obtain the estimate i+l r~ ei{(x-y)+i~2(6(x)-6(y)) -i [+i ;-idxdy ]_~
<_- C
f+l (+i dxdy ].-i J-llx-yl I/2
l~I ~7~d~
=<
_< C
by the lemma. The theorem is proved. To prove the converse we make the following construction.
26
Let
M ~ /N
(~,a+M-1).
be a l a r g e
We w i s h
to
study
G(x)
If
t : ~ + TM -I
(16)
Let
integer
:
this
the ~
can
e ikN(x-~)2
and
consider
an
inter~al
m
:
function
I
e iNl(x-t)2
be w r i t t e n
g(t)dt.
as
rl e - i 2 1 H ( x - ~ ) m + i l T 2 ]0
g(a+~)dx.
us c h o o s e g(~+
where
~ 1 3 ( CO([, ~)
$
I;
)
and
=
e -2i8T
is such
$(~)eiUTe
e 2iMT
@(T)
lul
8,1ll
that
d~
>i,
-<
< 2.
0
Hence, if (17) the
integral
(16)
Let us n o w 6 = 8
v
= v.
is of a b s o l u t e
consider
Then
for
i.e.,
since
kv
for
suitable
-~ M,
in
Ix-if
and
functions
that
v i-~ l+k
! M -I/2.
obtain are
G(x;l;~)
functions
(i7)
holds
For each
v
choose
if
< 3 M ' Ikl -< /~'
k I
-i+~
constructed
the
Gv(x)
below,
2
independent
translate
functions
n~
Gv(x)
I = X(x)
We can
M
of
v,
construction
so that,
for
in
this
~ i
way
for a n y
by m u l t i p l e s
each
x,
are, x of
M1 / 2
0 < x < i.
define
g(',~) Let
v : 0,i,...,/M.
functions
of
bounded
Let us n o w
>= i.
if
the
choice
value
~ =v/M,
x We c o n c l u d e
ii _< ~4 ,
I = i + k/M v M
x
M- 1 / 2
+ ~8-
[l(x-~)
be the
=
H ~ rv(~) i
gv(" ) H - I / 4 + P , 0 > O.
corresponding
transform,
r v (~)
and
M G(x;A(x);o)
:
rv(o) v=l
Gv(x;l(x))H-i/4+P
are
the
Rademacher
27
Since ~IG
it f o l l o w s
that
M p/2 e x c e p t
for a s u i t a b l e
on a set of
It r e m a i n s the c h a n g e
(x;l(x))12M -I/2+2p
to p r o v e
of scale
choice
x's that
L i p ( ~ - s).
If
intervals
Itl-t21
tl,t 2
belong
r (0),
the c o r r e s p o n d i n e + Mt
> M -I/2
has b e e n
function
to the
same
< M-I/4+P
interval
<
If(t)l
is
clear
< M- 1 / 4 + p
if
creasing
= N- 1 / 8 + p / 2
sequence
4.
by
of
introduce the o r i g i n perties. Let
~(x) 4(0)
i { ~
Itl-t21
clearly
and add the c o r r e s p o n d i n g
functions
to the Gibbs
the t h e o r y was
left
theory
on the p o t e n t i a l
Consider
UN(q)
:
@(( N
trivial.
so that
free we o b t a i n m o d e l s
= 0.
of the h a r m o n i c
essentially
one w h i c h
be a n o n - n e g a t i v e
e the c h a n g e
estimate
a rapidly
which
in-
essentially
continuous
function
IN a _ ~ q ~ q
-NF(B)
f -6UN(q) ~ ] e
dq.
of v a r i a b l e s
xj
2N+I _
q~ cos
2N+I '
close
interesting
we to pro-
is due to Kac. for
the p o t e n t i a l
2
oscillator.
If h o w e v e r
particles have
We w i s h to e v a l u a t e
We m a k e
the
choose
We s h a l l h e r e d i s c u s s
we a s s u m e
If
ll/8-s
]
a restriction are
belong
suffices.
N = N.
N[ 6. ]
that
g-func%ions
I ~CNItl-t21M-I/4+P
series we s h o u l d
We now r e t u r n
We o b s e r v e d
- where
to
then
i < ~.
Itl-t21
To get a d i v e r g e n t
multiplied
is
Itl-t2 II/2-~
CN7/81tl-t217/8+Sltl-t2
case
f(t)
made - belongs
the c o r r e s p o n d i n g
If(tl)-f(t2)l~Itl-t21Maxlf'
so t h i s
G(x;l(x)~)
and
I f ( t l ) - f ( t 2 ~ I ~ 2Maxlg~l
If
of signs
of small m e a s u r e .
• ÷ const
to d i f f e r e n t
~ CH 2p
.
x { 0
and
28
N x0
2N+I _
qv '
and x_j
Using
earlier
notations
N 2~vj 2N+I _~ qv sin 2 - ~ "
:
and
uN(q)
~(Ixl 2) _~~
=
where
A (N)
:
j
Since
the t r a n s f o r m a t i o n
as follows.
Given~
N ~ x~, -N
Ixj 2 :
we o b t a i n
A! N) ×?(~N+I) 3
3
A(N)-2~J
'
.
~2-9n7~"
is o r t h o g o n a l ,
our p r o b l e m
0 s A 1 s A 2 s...s A N s C, N
can be f o r m u l a t e d
we w i s h
to e s t i m a t e
2
-N*(IxI2)~Aj×. (18)
e
Lemma.
Let
creasing
and g(t)
be c o n t i n u o u s
and d e c r e a s i n g , r e s p e c t i v e l y .
S
Proof.
f(t)
d X l . . . d x N.
=
Suppose
fl ~ - ]0 f ( t ) d g ( t )
suPt f(t)g(t)
Clearly, for all to,
functions
to
so that
i
to
0
0
g(t0)f(1)
to e v a l u a t e a large
0 f(t)dg(t).
to
to
0
integral
and w r i t e
1 g
to
+ f ( 1 ) g ( t 0) = S + S l o g ( f ( 1 ) g ( 0 ) ) S
for a fixed
-B. = A. - C ] ]
-N*(r2)Cr2
rN-if
e
value of
so that
JIxl=l
r.
B. > 0. ]
N * ( I x l 2 ) r 2 i B j x ~] e
and w i s h
dXl...dx N = rN-idw
w i s h to e s t i m a t e (19)
Then
T hen
in p o l a r c o o r d i n a t e s :
the s u r f a c e
constant
= S.
gl
g(t 0 )
(18)
= 0.
fl
= - S log(g~)
We w r i t e
= g(1)
in-
we have
FI
Now choose
f(0)
(0,i),
_< S log( f(1)g(0)eS )"
= - ] t O f ( t 0 ) d g ( t ) _~ -
f(to)g(to)
on
dw.
Let
C be
We now
29
Let us set y by a volume bers
e'
: @(r2)r 2. integral
and
two groups: divide
over
Ixl ~ i
0 < j ~ e'N
:
since
and
Let
I e yNIBjx~] Ixt
= -
VN
be the volume
of the
dx
=
e YN['
f(1)g(0) S sup 0
:
Then
=
_ 2
N -N/2
dx'
eYN['
sup
where
g(0).
Observe
also that
CN "
i
the argument
=
for
for some
B
We conclude
e YN['' dx"
I
" Q
I×" I <-lv" p--p2 !
Q
~
CNIogN.
and obtain
P
t
~T
]
YNX(k)B'x~ e
] ] dx (k) Qp,
ix(k)f
k:l
i _< Q
_< (CNlogN) p P
and
2 + p~- +...+
Pl
(20) Taking
N-sphere.
and
~i/~_t eYN~''dx'')
~ C N NBN
i
VNI N
~
Ix' I-<-p
where We can repeat
x = (x',x")
dx' e YN['' dx"
x-I<
and similarly VN
VNI N
B. a 0. Now take two hum] the set of indices into Write
I
integral
f(t)dg(t).
0
f (i) ~ cNV~,N
In the lemma
the surface
~'N < j ~ N.
= - Ii ( / eYN~'dx')d( 0 ix,,l< t
Clearly
replace
e", ~' + ~" = i, and divide
[ = [' + ~".
VNI N
We may clearly
a suitable
mean
value,
B*, 3
p2p = i. of
P yN~ B~p 2+c~jNlogp j VNI N
We can compute
:
i
and obtain
each
P
Try
sup e (0)
the maximum
in
B4 J
.N
c~]
%"
block,
we find
30
I 2yB3* p j + ~Pj
:
Kp.3
(21) pj
for e x t r e m e values, w h e r e We can n o w first (~j+l-~j)
÷ O.
K { 2yB~ for all j 3 N + ~ and then
In this way we find,
l°glN N+~lim - - N provided
let
] , K-2yB. ]
for
i rr B(x) Y 2-~ ]_~ K - 2 r B ( x )
=
P
K
÷ ~, and let
B(x)
i
= C-A(x),
uniquely. Max
that
ir2w
dx -
]@
log(K-2rB(x))dx
that i Iw dx 2~ -~ K - 2 r B ( x )
Going
and (20) gives
back to the o r i g i n a l
t o t i c to
N
_ C r 2 ~ (m2)
variables
=
i.
the l o g a r i t h m
of
(19)
is a s y m p -
times + log r + Cr2@~r 2 )
i -
rw
F£
A(x)
- ~ A+2A(x)
2~
-w
log(2A(x)+A) log(~(r2)r2),
where
~ ~ 0
satisfies
(22) is a f u n c t i o n
(g3)
1 r~
sup
- %-£ ] _ ~
of
~i
I ~-~ 2A(x)-X dx
=
r
and to find
(18) we must
A(x)
r 2@(r)2" take
1 ]r~ _~ log(2A(x)+X)dx-
t+2A(k--T dx - G
log((~(r
2 ) ) r 2 )} •
r
It is c l e a r be e x t r e m e l y occur.
that the d e p e n d e n c e complicated
We a l s o note that
has a s o l u t i o n
or not,
and that d i f f e r e n t the w h o l e
Let us c o n s i d e r ~ ) = i,
the case
r > 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
choices
of
a.. ]
by
It f o l l o w s
E mE N
for
k-N
enough.
Hence, if we d e f i n e
fl 0
<-_ exp{(
- Xajlog(a9k)+E)N}
A converse
Ck (x)
:
inequality
also
follows
a].k, x ~ I.,] it f o l l o w s
~k (x)dx
=
i, ~kd×
~
~k l°g ~k dx}
=
mENe°(N) ,
d~
f r o m the lemma.
that
weakly,
and i exp{-N
I
N ÷ ~
and
k ÷ ~.
0 The r e s u l t measure.
Consider
distribution set
E N.
can be f o r m u l a t e d
lies
Then,
as first mE N
with the
all c h o i c e s in some w e a k
=
interpretation
as f o l l o w s .
of p o i n t s
x
in
e-neighborhood
N ÷ ~, t h e n
e ÷ 0,
f r d~ d~ e x p < - N ] ( ~ log ~ ) d t that
Let
if the
integral
of
d~
be a p r o b a b i l i t y
~, v = I,,..N, dd.
whose
This d e f i n e s
we h a v e + O(N diverges
(in p a r t i c u l a r
a
83
if
~
is n o n - a b s o l u t e l y Going
~aek to
continuous)
f(8)
lim f (6) N~ N
mE N < exp(-AN)
for all
A < ~.
we find
[ fl
--< lim k÷ ~
- ~
#(x-Y)#k(X)~k(Y)dxdy
0
- ]0 %log%dx}. On the other
hand,
let
~ > 0
be a r b i t r a r y
with
1 ]
Then
8
N N [ ~ ( x v - x ~ ) - [ l°g@(xv) [j Ii -Nv,~=I v=l •• e 0
-NfN(8) e
:
>=
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}
<
1(4). is c o n t i n u o u s
Clearly, we have
Const.
The w e a k limit.
a > 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 <
i B < ~-
and
hence
f(6)
~ 0,
i 8 ~ ~.
35
If
~(x)
If
~
is p o s i t i v e
is n o t We
observe
definite,
positive
have
proved
that
for
i.e.,
definite
then
c >_-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<.-.<xN
The
new c o o r d i n a t e s N -x. = [ e 3, i
integral
e
-x. -Yi -Yi+l i = e - e ,
= 0 and
YN+I
becomes ]~
f exp-~
I
"" y
e x
Writing
t i : Yi+l-Yi,
e -~
-Y2 -e
+,. + "
-Y2
-YN 1 e -YN+I -YN e -e
N - " -Yi+l)]-i ~[eYi(e Yl_e dy i . . . d y N. 2
the integral
y.
I
e
is
~F(ti)
dt i . .dt N •
if
(9) The o r i g i n a l
two
'dXl'''dXN
r
J
several
may have a c e r t a i n
;@ e Ze
= N!
consider
support.
case has b e e n t r e a t e d
To s t u d y
We shall here
e -[xl
i.
The first
is still u n s o l v e d .
F(t)
condition
=
6 et_l
l o g ( l _ e -t )
t > 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 <
lemma,
I
direction.
Since the largest depends
We conclude eigenvalue
analytically
that
is iso-
on the parameter
B.
Lemma. that
Let K(x,y)
6 ÷ 0.
Let
K(x,y)
be symmetric,
~ 6 > 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 _<. I0 _< i,
q >_ 6(l-e(d)).
on a set of measure
<~
For
x ~ E
Hence, if
0 < f < i~, and x ~ E
and
mE < g[, then
I~I < -.q2 K(x,y)
~ 6
except
42
lof(X)
>_ 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
=<
ff
<_-
we o n l y
l
,
¢<0
sult
dx
that
/
the
Furthermore,
set.
0
f>6
and
>_ 6 l-W'~ q
in a s m a l l
0
It f o l l o w s
)
- i ~q
0
function
assumption
is p o s i t i v e
on
%(x)
definite.
theorem.
with of
compact
support
then
8.
for t h e However,
validity in t h e
of the re-
above
approach
a bound, K(X,Y)
~ e
Ca(X)a(Y)
and eV~ v[~! However,
if w e a l s o
the
system
and
of c o m p a c t
We
introduce
is n e u t r a l ,
consider
the
then
changes
the
following
support.
Let
diverges.
~
of
signs
in a r a n d o m
way
so t h a t
conclusion still holds. set-up.
Let
be an e v e n
i
e 0 < x -g < N
%(x)
be p o s i t i v e
probability
definite
measure.
dx I • • •dx N •
Define
43
Theorem.
log I N lira N÷
exists
and is a n a l y t i c
Proof.
The p r o o f
Gaussian
variables
F(8)
0 < B < ~.
for
depends
:
N
on a r e f o r m u l a t i o n
and" F o u r i e r
of the limit,
transforms.
using
Define C
(1)N(X)
Then,
:
~(Nx)
:
o
, Ix] < ~-
C
,
:
<
N
Ixl
<
~-
we h a v e co
CN(X)
:
~ Cn c o s 0
nx
where
_
1 ~(~)
e
N
n
In
x v ~ Nx I N we c h a n g e ( d e l e t i n g the p a r a m e t e r ~)
and i n t r o d u c e the F o u r i e r series.
N
IN
NN ~ N--[. J dp(a)
=
We use the
introduce
-ct
2
one v a r i a b l e
-~+itc[
i
e
~ : [n and
d[
for e a c h
sine
and o b s e r v e
sums that
c : Cn,
n : 0,±i,±2,...,
in the e x p o n e n t . Yn
is r a p i d l y
=
some c o m p u t a t i o n s
IN
[ yn([nCOS 0
nx+[_nSin
nx).
we o b t a i n
=limM÷~ ~'~ ( ~ )
]
e-4d~(~
0
P(FN(X~))d
-N
1
]R2M+I Let us n o w w r i t e 1 2~ J0
--
P(FN)dX
:
~
~o
~(FN)dX
:
NXj,
If we de-
decreasing,
set FN(X;[)
After
t~
i
-
Y~ = Cn' Yn ~ 0,
we can
2
([a sin nx ) ] i v v dx.
and e a c h of the two c o s i n e fine
N
2+
- ~ C n [ ( [ c cos nx ) i v v
formula
e
and
t J e
We o b t a i n
44 where
m.]
is an i n t e r v a l
with distribution we c o u l d
X(A),
of l e n g t h and
if__, ~),
use the c o m p u t a t i o n
1 ~-~
E
2zA ~.
__,If(~, NX.3
X(A)
was
of h i g h m o m e n t s
]~(rN)dX N
~
were
independent
in C h a p t e r
exp{N(@A(l)
- 1-10g
independent of
N,
then
I to o b t a i n
I)}
where (17)
~A (x)
i elX(A) K log E( ).
:
and ~i(1)
E(Xe AX )
=
=
i.
A E ( e AX ) The two d i f f i c u l t i e s common
distribution.
the o r i g i n a l
To m a k e
interval
no c o n n e c t i o n
but the limit is s u f f i c i e n t
(0,N)
between
the same l i m i t as
are e a s i l y X. ] into
different
N ÷ ~
~A(A)
for the above
care of.
independent intervals
in
Co). X(A) 1
computation.
corresponds A
As is e a s i l y
A + ~.
uniformly
(a). The
of l e n g t h
intervals.
and t h e n
exists
taken
over
X. have a ] to d i v i d i n g and
seen,
depends
finite
Therefore,
suppose this gives on
N
range which
X(A)
is
I- i 2~A ~ ( F ( x ; [ ) ) d x , : -2~ 0
X(A) where
i
= N+lim Lemma.
F(x;~)
independent Proof.
~ C
if
Ix-yl
F(x;~)
well-known
with >
- sin probability
2~
~.Then ~
is a s t o c h a s t i c
fact.
The
F(x;~)
i, and
integral
It r e m a i n s
to study (0,2~A)
length
>2~.
C0(u).
We d e s c r i b e
Let
~j(t)
support and the
are c l e a r l y
E(F(x;~)F(y;~))
We d i v i d e
has
(17) as
=
normal
and F(y;~) in
are
(-~,~).
first
statement
and
~(x-y).
A ÷ ~.
into i n t e r v a l s
Ul,Vl,U2,V2,...
be some s e q u e n c e
F(x;~)
F(x;~)
of f u n c t i o n s
by its moments,
~k(U) = I F(t'~)~k(t)dt" U
of e q u a l dense
in
is a
45
Given
~l,...,~s
for
u = uI
and
u = u 2.
~lUl~(r)dt K s (I~)(_u
,~(u2))
e
T sf
is c l e a r l y This
:
a bounded
Furthermore, when tor.
!]2~vI [(F)dt
= Eee
where E is e x p e c t a t i o n fine the o p e r a t o r ,
Tsf
Let
under
7~iu 21
dt
• e
the c o n d i t i o n s
~(Ul),e(u2).
We de-
i Ul Ks(~ I, 2 )f( i ) d P ( ~ ) .
operator
from
L2(Ul,dP)
to
L2(u2,dP).
s + ~, T
c o n v e r g e s to a c o m p l e t e l y c o n t i n u o u s o p e r a s i m m e d i a t e l y from K b e i n g u n i f o r m l y b o u n d e d and the s
follows
lemma. T
is not s y m m e t r i c .
which consists measures
and
in c h a n g i n g K(e,8)
K0(~,8)
:
by the
!4 [K(e,B)
nrevious
log I
operator
lemma.
depends
the
an involution,
x-axis.
+ K(~* ~ 8) + K(~,6*)
operator
is e s s e n t i a l l y T0
exists
~ ÷ ~*,
This p r e s e r v e s
We f o r m
T
+ K(~*,B*)] "
has an i s o l a t e d o As b e f o r e ~ we have
E(e IX(A) ) thus
there
the o r d e r on the
= K(~*,6*).
The c o r r e s p o n d i n g value
However,
~
on
eigen-
lA
free e n e r g y
analytically
largest
8
F(6)
in the theorem.
and so the r e s u l t
The
follows.
ON S P E C T R A L
SYNTHESIS Yngve
Uppsala i.
This
for
introduction
smooth Let
B
on
~n.
of
B,
For
every
spaces
sets
We a s s u m e that
of
about
spectral
synthesis
that
the
set
of c o n t i n u o u s ,
Schwartz in
E ~R n
B
we
space
complex-valued ~(~n)
implies
introduce
is a d e n s e
pointwise the
functions subspace
convergence.
followin Z three
sub-
the
(closed)
B2(E) ,
the
closure
in
B
of the
space
of all
f ( ~CIRn)
the
closure
in
B
of the
space
of all
f (D(IRn),
on
of w h i c h
vanishes
BI(E)
for
instance
B,
if
on the
Fourier
we
to
implies
space.
the
f ( B
with
the
which
vanish
on
E,
which
all
well-known
synthesis
i °.
when
E = E-~
2° .
when
n = i
(the or
in
2,
weaker B,
that
B
with
BI(E)
[16],
B
we
holds
space
the
it
are
is
say that Thus
depends
A aR n)
norm
(see
to
= B2(E).
converse
with for
respect
property:
is the
Ll(~n),
definition
S. Herz
if the
postulates
of
inherited
then
fulfilled.
result:
and w e a k
following
by C.
Whether
case
the u s u a l
synthesis
to
each
E.
out
following
functions
Evidently
of the
pointed
synthesis.
discuss of
is of
respect
B.
of
Following
E
the
space
following
A ( R n)
in e a c h
study
of the only
~ B3(E).
As was
weak
transforms
For
of all
neighborhood
say that
synthesis
shall
f r o m that We h a v e
2 B2(E)
natural
choice
We
some
= B3(E).
is of w e a k
synthesis
in
[i]),
BI(E)
very
subspace
E,
Then
ties
is k n o w n
B:
B3(E) ,
E
Domar
University
what
space
convergence
closed
n { 2
~n.
be a B a n a c h
and
~n
BI(E),
vanish
also
in
surveys
IN
synthesis
are
equivalent
cases: closure
of the
interior
of
E),
proper-
47
3° .
when
E
is a subset of an arc of a r e c t i f i a b l e curve.
The proof of i ° follows from the fact that,if implies that all d e r i v a t i v e s of mated in
N0R n)
t o p o l o g y in gives i °.
f
vanish on E. Then
by a sequence of functions
D0R n)
in
statements
f
in
Since the AORn),
The first c o u n t e r e x a m p l e to spectral
(ef., Herz
[15]).
synthesis was given in 1948
by L. Schwartz in [24], where he proves that the sphere if
n ~ 3.
S n-I
is of weak synthesis or not.
answered later,
by Herz and N. Varopoulos.
is a set of synthesis
S n-I c ~ n
does not
This q u e s t i o n was
In 1958 Herz proved that
[15], and in 1966 Varopoulos
the m e t h o d of Herz to higher dimensions, that
sn-i ! ~ n
His proof, however,
reveal w h e t h e r
SI ~ 2
this
follow from a w e l l - k n o w n tech-
nique, d e v e l o p e d by A. Beurling and H. Pollard
is a set of non-synthesis,
f EB2(E)
can be approxi-
B3(E).
is stronger than the t o p o l o g y
The r e m a i n i n g
E = -~, E
is of weak synthesis,
extended
o b t a i n i n g as a partial result
if
n ~ 3
[25].
Thus the
notions of synthesis and weak synthesis do not coincide,
if
n ~ 3.
It is natural to ask if there are any sets at all which are not of weak synthesis. synthesis on
R
imbedding
•
for every
n.
Varopoulos
for every
was c o n s t r u c t e d by M a l l i a v i n
in
constructions
The answer is"yes"
~n,
~n.
A set of non-
[21] in 1959, and, by
his set gives a set of n o n - s y n t h e s i s
in
By 2 ° and 3 ° it is not even of weak synthesis. of sets of n o n - s y n t h e s i s on
[26] and T. K~rner
The c o u n t e r - e x a m p l e s
in
~n,
[19]
~
~n, Other
have been made later by
(cf., R. Kaufman
[17]).
which we obtain in this way, are all
c o m p l i c a t e d sets, w i t h no obvious properties of structure and regularity. It is thus a natural thing to ask w h e t h e r all sufficiently closed subsets of
~n
are of weak synthesis,
ment with this conjecture, favor it: of
E
E !~n
sn-i ! ~ n
and the following known result
is of synthesis
is an ( n - l ) - d i m e n s i o n a l
CI
if
E = --~ E
manifold.
smooth
is in agreeseems to
and if the b o u n d a r y
48
Here is a sketch of the proof.
A p a r t i t i o n of unity shows that
it is enough to prove that every point in the b o u n d a r y has a neighborhood such that functions
in
BI(E)
can be a p p r o x i m a t e d by functions
with support in this n e i g h b o r h o o d
in
B3(E).
Choosing the n e i g h b o r h o o d
small , this can be a c c o m p l i s h e d by first a p p r o x i m a t i n g the function with suitable translates. In view of this result it is natural to turn our a t t e n t i o n to the case when
E°
is empty.
smooth m a n i f o l d in manifold
From now on we shall assume that
~n,
of d i m e n s i o n
S n - i,
E
is a
or a subset of such a
such that it is (in the r e s t r i c t i o n topology of the manifold)
the closure of its interior and has a
CI
boundary.
For such m a n i f o l d s we already have one special result: is of weak synthesis.
sn-i [ ~ n
But the method of Herz and Varopoulos cannot
be g e n e r a l i z e d d i r e c t l y to general manifolds.
The reason is that they
rely on the facts that
( A ORn)~
changed norm, S n-I
if
~
f ~ AOR n)
implies
is a n o n d e g e n e r a t e
is an orbit of a continuous
of r o t a t i o n s around the origin. the weak synthesis property. annihilating
B2(sn-I),
w i t h support in
S n-l.
fo~
with un-
affine mapping,
and that
group of affine mappings,
This is how their basic idea implies
A bounded linear functional on
can be regarded as an element in Using averages of r o t a t i o n s of
~,
possible to c o n s t r u c t a sequence of bounded Borel measures, by late
S n-l,
converging
in the weak* sense to
BI(Sn-I) , and thus
v
does the same.
v .
affine mappings
of course very small
in
~n
it is supported
groups of non-
but this family of m a n i f o l d s
is
(ef., F. Lust [20]).
is that the n o n d e g e n e r a t e
m a p p i n g s w h i c h leave
~'OR n)
The same argument can be
When a t t e m p t i n g to generalize to other manifolds, obstacle
AORn),
The m e a s u r e s annihi-
used for other m a n i f o l d s which are orbits of continuous degenerate
the group
A0R n)
the great
affine mappings are the only
invariant
(A. Beurling and H. Helson [3]).
49
The basic
idea of Herz
can n e v e r t h e l e s s
be r e g a r d e d
as the orbit
even
individual
if the
is still verging
be used
to the
identity
give
in this
to prove weak
This
on
operators
that
certain
operator. case a weak,
may sound
easy,
difficulty
changes
with
the p r o p e r
the
Even
of
norm,
the task
pings
with
Even
the
must
curves
no
the
does
direction)
are
In [6]
as orbits,
line
the
gave
manifold
the
relations
isoclines between
to use
this
smooth
Let us r e t u r n possib l e
is of weak
of Varopoulos.
study of general with
thereof,
C"
being the
spectral
segments. properties
C2
results
in
curvature
this
~2
~0,
is an extension of this
in
~2
paper
to the #0
and
give us i n t e r e s t i n g in
for curves
in the theory
With
using map-
with c u r v a t u r e
in [6]
have
for
curve
sections
of sets
at all
of the o p e r a t o r
Gaussian
This will
where
In [7], b a s i c a l l y
synthesis;
of curves
now to the result
standard
segments.
idea can be applied
families line
with
In the f o l l o w i n g
general
(the loci
a simple
prob-
the orbits
it was proved,
that
~2
vanish
estimation
does not vanish.
a
of the
group of map-
not
isoclines
out.
that
of the
be c h o s e n w i t h care:
such that
curves
boundedness
satisfactory
curvature
to
group of m a p p i n g s , a n d
curve
can be carried
show that
can
difficult
is a point
if its c u r v a t u r e
we shall
which
same way as in the
is of synthesis
of the result
con-
average
in
careful
subset
the
for smooth curves
and
parallel
or a c e r t a i n
there
are operators
uniform
of m a p p i n g s
same m e t h o d
can
A~Rn),
of
to find a good
if there
if
same tangent
choice
in
E then,
it is in general
is to prove
fails
the m a p p i n g s
to form a family
~n,
sequence,
in the
but
its sign m t h e n
can be found.
on the curve,
convergent
If
above.
The m e t h o d
the c u r v a t u r e
adjoints
exactly
given by the averages.
lems arise.
av e r a g e s
The
First, it is n e c e s s a r y
then the e s s e n t i a l
of
are not operators
synthesis
S n-I
programme
carry out.
points
mappings
possibility
discussion
pings
of a group of m a p p i n g s
the
operators
be exploited.
~2 in
of spectral
and ~2.
~. It is
synthesis
50
to find the f o l l o w i n g removal
of a certain
pieces
which
curvature natural every
extension:
let
denumerable
point
are either
~0;
then
E
to ask w h e t h e r
simple
general
C~
line
segments
the c u r v a t u r e
or
all
E
simple
C2
sufficiently
curves
is needed
This would smooth
the
into d i s j o i n t
In view of this,
assumption
of synthesis?
that
be a set such that
set splits
is of synthesis.
curve
conjecture
E c ~2
with it is
at all.
Is
then agree with one
sets are of weak
synthesis. However,
it is p o s s i b l e
the graph of w h i c h time a set
not of weak
desired
in
were
needed
things.
is,
AaR)
example
weak
folds
function
synthesis). on
~
AaR 2)
shows
that
~2
This was done
~2
to
in
and extending so that
from
•
(thus
to
~,
at the
in [12],
same
starting
the c o r r e s p o n d i n g
it vanishes
on a set of the
type.
This yields
C~
is a set of n o n - s y n t h e s i s
from a c o u n t e r e x a m p l e functi o n
to find a
synthesis,
in
has
order
in that
is not true.
for our method
One
to
to
case,
our conjecture,
to
work
restrict
seem
to
be
approach
which
continuity
assumptions
in
say,
for an a f f i r m a t i v e
different
C~
curvature
the conjecture,
have hope a
The
that
the
which
nature
of
to analytic
mani-
answer--maybe avoids
there
curvature
assumptions. The n o n - s y n t h e s i s respect notion find,
that
A.
in
~2
is of some
it can be used to disprove
of synthesis using
curve
in
~2
(thus also
interest,
too,
certain
conjectures
for weak
synthesis).
in the
for the Thus we
I °,
A set of synthesis
in
~2
may have
a boundary
of non-
synthesis. B.
The
intersection
of two
sets of synthesis
in
~2
may be of
non-synthesis. C.
A
synthesis.
C~
map of a set of synthesis
in
~2
may be of non-
51
The
special
manifolds
problems
of d i m e n s i o n
in the d i s c u s s i o n parall e l
curves,
proved
by
sn-
of curves
operators
led to the desired
torsion
40,
family
of orbits.
ments
while
plane
should
problem:
that loci
can every
be i n c o r p o r a t e d conditions
that a s o l u t i o n does
not
ator norms metric
question
Thus metric
manifolds,
of
A a R n)
with
the
giving where
parallel
smooth
curve
in
~3
dimensions. geometric
but after that
problems
involved.
This
and related be seen
? (a,t)
the m a p p i n g
with
a different
the m e t h o d in
~n
seems
spaces
reply
is injective
problems
of the
geo-
work.
differential
in [8],
y(a,t))
and in
Similar
of the oper-
in their r e l a t i o n
~ (x(a,t)
( R 2
C"
40
be stressed
to be a general
from the results
geometric
type?
Estimates
would
seg-
to any fixed
construction
where
have
are line
It should
a positive
work
with t o r s i o n
where (i)
of the
the orbits
of the m e n t i o n e d
of our results.
is hope
x [-2,2]
as
~3,
should
Let us form the m a p p i n g [0,1]
AaRn),
in
same binormal
the tangent
higher
extensions
as can also
curve
method
of mappings,
of curves
to be done,
are
on
modification
C3
look for a set-up
of the d i f f e r e n t i a l
there
a simple
are
but they do
operators
0nly a certain
choice
sufficient
we are left with
concepts
the study
2.
have
case;
of w h i c h
Thus we are led to a d i f f e r e n t i a l
in still
guarantee
in this
that the original
with
in a family
appear
the orbits
apparent
[i0].
for points
be planes.
to treat
are already
Mappings,
result:
for points
n ~ 3,
are bounded
[14].
should
loci
~n,
~3.
which
subtle
One
when one a t t e m p t s
to try also
however,
with a more
the p r o p e r t y
in
is of synthesis
It is believed,
arise,
in
R. G u s t a v s s o n
approach
unchanged
2
are natural
not give average was
which
geo-
feature
in
to smooth
[9] and
[ii].
$2
(2)
xaY t
~
xtY a
(3)
xtYtt
~
xttY t
(4)
x(a,t)
and
y(a,t)
are
x(a,t)
=
ag(t)
+ j(t)
y(a,t)
=
ak(t)
+ g(t),
of the
form
where g'(t)g'(t)
We
shall
Properties
make
(i) and
is an i n v e r s e obtained
in
smooth
(3)
says
C®
that
(2)
and
of the
the
curvature
direction
of t h e s e
curves
E [0,i],
or f i n i t e
that
a mapping.
two
that
curve
a = constant
and
intersection
angles.
family
Condition only
such
implying
the
(the
(4)
on
t.
Thus
families,
Condition
of
that the
there
t = constant,
images
implies
from
Conversely,
every
all
isoclines
of this
the
form.
equation,
ag(t o)
+ j(t o)
y
=
ak(t o)
+ g(t o) ,
C®
The r e a s o n
in S e c t i o n
i,
such
to any curve
of the
solutions
of
unions
a "generalized
giving
remarked
=
[0,i]
the
images
I x
unions
the m a p p i n g
fied w i t h
itly,
first
depends
of
a =
the
tangent
isoclines
are
segments
(5)
with
non-zero
~0.
properties
implying
in the
.
conditions,
segments
with
curves
have
on the
also
line
families
constant)
that
remarks
k'(t)j'(t)
are r e g u l a r i t y
C~
as maps
are
line
some
=
of
Clairaut
family, line
the
given
with
segments,
by
(5),
curvatures can
to o b s e r v e is affine. ~0
and
be l o c a l l y
a = constant,
for
corresponding
differential
equation,"
of the d e s i r e d it is thus
It is i m p o r t a n t
isocline,
segments
is that
sets.
form.
natural
can be In v i e w
to m a k e
identi-
some m a p p i n g
integrated of w h a t
a thorough
explic-
was
53
i n v e s t i g a t i o n of our m a p p i n g in r e l a t i o n to Let us first introduce some notation. A(F)=AORn)/BI(F). of
AaRn).
Let
s E [-I,I], T s
E'
be the image of
is the m a p p i n g of
(x(o,t), y(~,t))
If
be the image of the whole r e c t a n g l e
and let
into
" t r a n s l a t i o n by
For any closed set
s
E'
(x(o,t-s),
into
E
y(o,t-s)).
A(E').
w h i c h sends We may call
it does not follow in general that
T
s
foT s E A(E').
[3], w h i c h
[4] has shown to hold in a local version,
a p p l i c a b l e to
His result says that the implication holds only if
the r e s t r i c t i o n of an affine m a p p i n g of We
For every
along the curves."
f E A(E),
Brenner
F
[0,I] x [-2,2]
[0,1]x[-l,l].
This is seen from the t h e o r e m of Beurling and Helson P.
F ~ ~n,
Thus it is the Banach space of r e s t r i c t i o n s to E
for our mapping,
A~R2).
shall go one step further,
Ts
is
~2.
i n t r o d u c i n g a concept w h i c h can
be d e s c r i b e d as " c o n v o l u t i o n along the curves."
For
f E A(E),
E N([-I,I]), we form i T f
=
I
f°Ts~(S)ds • -i
Thus
T f
is the function on
E'
S
I f(x(o,t-s), -i
at the point It
y(o,t-s))~(s)ds
(x(a,t), y(a,t)).
turns
out
that
a more general result. function on
taking the value
~
T f E A(E'),
and
Let us denote by
such that
Under our assumptions,
we have
T f (A(E').
If
[
J~
~(t)dt
=
i,
then
~h'
~h(t)= (i/h)~(t/h).
holds:
Theorem i.
we
shall for
in
fact
h E ]0,I],
prove the
Then the f o l l o w i n g
54
T~hf ~ f E' '
in
A(E'),
as
h ~ 0.
This is a general theorem that can be used in d i s c u s s i o n s of spectral synthesis problems on
~2,
as sketched in Section i.
Section 4 we shall see how T h e o r e m i can be exploited
In
in this
direction. The proof of Theorem i is given in Section 3.
Here we shall
just give some indication on the ideas in it, and prove a simple lemma. In the proof we use the following e l e m e n t a r y
facts and tech-
niques: i)
If
then
dual 2)
×
is a bounded continuous
f (A(F)
implies
kf
c h a r a c t e r on
(A(F),
:
(In fact, m u l t i p l i c a t i o n with
X
If
and
HfrIA(r) corresponds to t r a n s l a t i o n on the
and this does not change the norm.)
9
is a bijective affine map of
fo9 -I ( A ( % ( F ) )
F S R2
and
f (A(F),
(Follows from the invasiance of
:
NfIIA(F).
AaR 2)
and its norm under non-
degenerate
affine
3)
be a nice and not too large compact subset of
bounds of
F
then
and
IIfo¢-lllA(¢(F))
Let
F c ~2,
and
11×fllA(r)
R 2,
R 2
mappings.)
f E C2(F)
~2
then
t o g e t h e r with its derivatives of first and sec-
ond order give a bound for
IIflIA(F)
which is independent of
F.
(This is made more precise in Lemma i, which follows.) 4) pose
Let F
(Fn) ~
be compact subsets of
is closed.
~2.
Put
Under certain conditions,
F = U Fn,
and sup-
implying among other
things that there is a substantial o v e r l a p p i n g between the sets, there
55
is a constant
C,
such that if
flF, ( A ( F n ) , n
then
f (A(F)
tlfllA(m)
f
is defined on
F
and
and
~ C -[ r[flFn/IA(Fn
)•
(The needed details are given in the beginning of the proof in Section 3.). As a preparation
for Lemma i, we shall prove an inequality,
to F. Carlson [8] (who proved the corresponding
one-dimensional
inequality),
Beurling
dimensions),
and B. Kjellberg [18] (who made such extensions):
Inequality:
Let
[3] (who found a new proof, extendable to higher
f' fxx' fyy ( L2aR 2 )
defined in distributional
Then
f ( A(IR2)
f
are
can be altered on a set of
0,
(6)
llf!IA (jR2) S C[IIfllL2t(jR 2 ) • ( IIfxx IIL2 aR 2 ) + NfyyllL2 (jR2) I C
so that
sense.
where the derivatives
measure
where
due
and /2 '
is an absolute constant.
Sketch of the proof:
By Schwarz'
inequality we have, for
X > 0,
~ < l~Id~d~ = ~- 2 ( X + ~ 4 +~4)I/21~I 2 " ( X + ~ 4 +~4)-I/2d~d~
-< (/m2(X+~4+n4)l~12d~dn)l/2"C0" x-l/4 :
= C0(XI/2
where
CO
" fIR21f12d~ d~ + X -I/2
is an absolute constant.
member is minimized
and
~]R21f12(~4 +~4)d~ d~)i/2 '
Choosing
X
so that the last
then using Parseval's relation, the desired
inequality is obtained. Lemma i
Let
C > i,
and let
F
be a compact subset of
~2
56
satisfying pair
F = F°
Zl,Z 2
length
and
of p o i n t s
-< C l Z l - Z 2 1 .
exists
a constant
diam(F) in
F
Then K,
_< C.
can
be
Suppose joined
f (C2(F)
depending
furthermore
by a c u r v e
implies
only
on
in
such
F
every
with
and t h e r e
f (A(F),
C,
that
that
IIfNA(F) -< KIIflIC2(F)' where
Ilfllc2(F )
Proof:
F
is i n c l u d e d
it is o b v i o u s value
i
that
on
F,
=
~
in a c i r c u l a r
there
exists
vanishing
suplf~l •
disc
of r a d i u s
a function
outside
a circle
C.
Since
C ~ i,
~ (C2aR2),
with
the
with
2C,
and
radius
satisfying
I1~11c2~2)
(7)
where
K1
is
an absolute
By a q u a n t i t a t i v e there
is
extended
to
~2
constant.
version
a constant
S K1,
K2,
of
Whitney's
depending
on
extension
C,
such
theorem
that
f
[27],
can
be
and
(8)
_< ~2 - llfllc~ (F)
Ilfllc2~2)
Then
f~
is
cular
disc
of r a d i u s
also
an
extension 2C,
and
of
f.
in the
It v a n i s h e s
disc
outside
it s a t i s f i e s ,
by
a cir-
(7)
and
(8), IIf~Hc20R2 ) -< K311flIc2(F) , where the
K3
is a c o n s t a n t
inequality
(6)
to
depending
f~
only
on
C.
gives
IIf e l l
-< KII ftl A(IR 2 )
, C2(F)
An a p p l i c a t i o n
of
57
where
K
depends only on
C,
and the lemma is proved.
It should be m e n t i o n e d that in the applications which we shall make of the lemma, W h i t n e y ' s cases
theorem is d i s p e n s a b l e since in these
it is easy to make explicit extensions of the considered
func-
tions.
3.
We shall now prove T h e o r e m I. Let us first point out that it suffices to show that
T Q f E A(E')
and that there exists a constant
(9)
C
such that
lIT hfllA(E,) s CNflIA(E ) ,
i n d e p e n d e n t l y of
f
and
h.
The reason for this is that AOR2).
Hence
C'(E)
is dense in
~aR 2)
is a dense subspace of
A(E).
For every
f E C'(E),
it is easy to see that the condition,
I implies that u n i f o r m l y to
T
f - fl ~h E' 0 on E'
~(t)dt
h
~
0.
i,
and its derivatives of all orders converge Thus, by the lemma,
T~hf ~ fiE' ' as
:
By the density,
in A(E),
the same result holds for every
f E A(E). Furthermore tinuous)
it suffices to prove
character
X.
f(x,y)
on
E,
with
In fact,
=
~ E LIoR2),
T~hf
:
(9) when
f
in that case,
if
is a (bounded con-
~IR2 e-i(x~ + Yn)~(~'n)d~ d~ then
~]R T 2 ~h
(e -i(x~ + Y n ) ) f ( ~ , n ) d ~ d~
,58
on
E';
and
hence
liT ~hf! [A(E' ) 5 C Varying
f,
Thus
we o b t a i n
we m u s t
(9).
prove
that
HT~h×NA(E' ) is u n i f o r m l y ~2,
and We
and,
bounded,
h
varies
shall
moreover,
case,
reviewing various
the
proof
ditions lowing
to
assume
that
proved
obtain
and
that
~ e iay
stated
certain
uniform
on
how
the
set of c h a r a c t e r s ,
special
boundedness
conditions
for this
general
situation
uniform
bound
set of c h a r a c t e r s
where
in the
a > 0},
beginning
:
(ii)
i,
and
under
that
of this
g(0)
xt(o,t)
yt(~,-t)
<
:
> 0,
0 < yt(o,t),
(13)
hold
particu-
simply
depends
discussion
the m a p p i n g
section
j(0)
for
to
find
the v a l u e
an u p p e r
of w h i c h
[
at
bound
o
by
on the
is
fulfills
as w e l l
as the
the
con-
fol-
~(0)
:
0,
(~,t)
E [0,I],
t ( ]0,2] .
> 0 .
of the
(x(a,t),
eiay(~,t-
:
every
if
Ytt(~,0) We w a n t
(14)
a one-parameter
it in the
seeing
the
k(0)
T hXa,
set of c h a r a c t e r s
conditions:
(i0)
(12)
the
involved.
We a s s u m e {×a:(X,y)
through
ourselves
Having then
varies
X
[-I,i].
shall
we can
data
in
restrict
for the m a p p i n g . lar
as
A(E')-norm
y(o,t))
s) I
~(
s K )ds .
is
of the
function
59
The b o u n d
has
to be u n i f o r m
We c a n n o t ently
apply
in d i f f e r e n t
up the
the
in
lemma
parts
of
a
and
h.
directly,
E',
and
for
T
× ~h a reason
for that
behaves
differ-
we h a v e
to
split
set•
Let
N
be an
{0 •
integer
I o = [-2-N,2 -N]
We put
and,
for
0 < n _< N,
I
=
[2-n-l,2-n+l],
I
n
Then
U In
forms
[0,i] × I . n lapping We
a covering
Thus
:
-I
-n
U E
of
forms
n
[-i,i].
n
Let
a covering
of
En E'
be the m a p by m e a n s
of
of over-
sectors. shall
now make
a partition
we c o n s t r u c t
a function
such
N ~n
that
for any
n,
function
with
a constant,
i
there
~n on
exists
C 2-norm
(C2(IR2), E'
This
an a f f i n e
~ C
independent
of unity:
of
for
which
vanishes
can be d o n e mapping
and
support
n.
The
takes
area
of this
n,
on
in such
which
having
proof
every
-N S n ~ N,
E'\En, a way ~n
sC ,
is left
that,
into
where
a C
is
to the
reader. By the the
lemma
functions
in S e c t i o n
~n
are
2, this
uniformly
means
bounded
by
that
the
some
AOR2)-norms
constant
since
Ilmm[I
we o b t a i n ,
for
~ Ilmll
A(R 2 )
every
A(R 2 )
g ~ A(E')
llgllA(E,)
=
I!~!1
,
~,
9
(AQR2),
A ( ~ 2)
,
II~ g~nIIA(E ,) ~ ~ llg~nIIA(E,)
_< [ IIglIA(En)II~nlIAOR2 ) _< D ~ llglIA(En ) "
Applied
to
T
X , ~h a
this
gives
D.
of Hence,
60
N
(15)
lIT hXalIA(E,) ~ D -N ~ IIT~hXaNA(En) "
Thus it suffices chosen
to show that,
for every
a
and
h,
N
can be
in such a way that the right hand member of (15) is bounded,
uniformly
in
a
and
h.
We shall choose
N
as the largest
integer
{0,
such that
2 -N ~ Max(4h, a~) If no such integer exists, The estimate
we take
N = 0.
of the right hand member of (15) is organized
follows:
first we estimate
the term with index
i__ > 4h, ah -
then we take the
n-th
term I.
n = O, Let
we can, without
in the case that
(in both cases), ~ and finally the
1 ~-~ < 4h.
when
i >- 4h, aT
term
0,
as
and put
b -- 2 -N
By
change of norm, multiply
property
the function
(2)
in Section
(14) with
a
2,
'
giving fir
=
e
ia(y(o,t-s) - y(~,t))
I~
and we have to estimate Changing
the
e
~(~)
i a ( y ( o ~ t - h s ) - y(o,t))
its norm in
x-variable,
ds
~(s)ds;
A(Eo).
affinely,
and at the same time changing
the
t-parameter,
we see that this is the same problem as estimating
the
A(F) - n o r m of the function with value I eia(y(o,bt_hs ) -y (o,bt)~(s)ds '
where
(16)
x
=
[ x(a,bt) i
y
:
y(~,bt)
i
61
and
F
is the image
Due
to
the
of
[0,i] × [-i,i]
conditions
under this mapping.
(i0) - (13)
the m a p p i n g
w i t h each individual d e r i v a t i v e u n i f o r m l y b o u n d e d Furthermore away from
its functional d e t e r m i n a n t 0.
F
o,t
and
C ",
b.
is in similar fashion bounded
Hence the inverse is bounded in
This means that the r e g i o n
in
(16) is in
in the
C'.
x y - p l a n e where the
A-norm
shall be taken satisfies the c o n d i t i o n of the lemma in Section
2 in a
u n i f o r m way.
C2-norm
Thus the
is u n i f o r m l y bounded, of the function, bounded in
A ( F ) - n o r m is u n i f o r m l y b o u n d e d if the which
in
turn
is
equivalent
c o n s i d e r e d as a function of
[0,i] × [-i,I].
to
the
C2-norm
(o,t), being u n i f o r m l y
Since b
2 , 2 -N < ~-~
=
it suffices to show that the function
~(o,t,s)
=
c o n s i d e r e d as a function of
l(y(o,bt-hs)
(o,t),
-y(o,bt)),
is u n i f o r m l y bounded in
u n i f o r m i t y now c o n s i d e r e d w i t h respect to
s, b
and
C 2,
h.
This is easy to prove, using the explicit e x p r e s s i o n for and Lagrange's mean value theorem.
For instance,
y(o,bt)
the b o u n d e d n e s s of
the function follows from
~(o,t,s) where
0 < 8 < I,
the boundedness. II. of
=
i. hb
and since
(-hs). yt(o,bt-Shs)
Isl~
i, h < b,
The r e m a i n i n g v e r i f i c a t i o n s
We proceed as in I, the d i f f e r e n c e En
is made
sponding to
in a
t = 2 -n,
direction
being
condition
(12) gives
are left to the reader. that the affine m a p p i n g
o r t h o g o n a l to the isocline corre-
while keeping this line segment fixed.
before e v e r y t h i n g proceeds
in the same way up to the proof that
I eiahb~(~,t,s)~(s)ds
As
62
is u n i f o r m l y
bounded
[0,I] x [-I,I]. and we h a v e
This
(13)
assumption
9(a,t,s)
of
as
9s
the
b(
discussion
considered
considered
function
is not
function
of
comparable
(o,s)
with
in
I/(ah),
way. shows
of
that
(a,t,s)
bounded
(3) on the
introduced
as
9(a,t,s)
is u n i f o r m l y
can be
2 -n)
in a d i f f e r e n t
Furthermore and
C 2,
time
to p r o c e e d
A further uniformly,
in
away
C3
( [0,i] x [-i,i]
from
curvature
it is in
0
due
of the
as a n e w v a r i a b l e
u
to
(12)
curves.
of
2 and
Hence,
integration,
giving
f e l°a h b ~ % ( a , t , ~ ) d ~ where
~
bounded
( C3
in a u n i f o r m
interval
Standard function
of
of the
and v a n i s h e s
outside
a uniformly
~-axis.
estimates (a,t)
way
,
(with
partial
( [0,i] × [-1,1]
integration)
has
its
show
C2-norm
that
bounded
this by
< C < c2n-N - ahb where
C
is a u n i f o r m
constant. -I
This
shows
N
+ ~ lIT hXallA(En) -N 1 III.
Thus
it r e m a i n s
c o n v e n i e n t to c h a n g e
only
to d i s c u s s
the p a r a m e t e r
y(~,t)
This
is d o n e
simply
A(a,t)
( i ) - (4), putation
(i0)shows
is a p o s i t i v e (13)),and that
the
t
=
=
Jacobian
to a n e w p a r a m e t e r
~
It is so that
2
that
in
C"
,
(due
~ = tgA-~,t) does
i 4h > ~ .
when
a + t2A(a,t)
function
putting
~ 4C.
Eo,
a +
by o b s e r v i n g
y(a,t)
where
that
not
vanish
.
to our An
and
easy thus
assumptions comthat
63
t where
B
is in
Hence
=
~B(a,<)
SiReiay (~, t-s ) ~i(
h )ds =
,
C" .
we o b t a i n
: [ e iay(a's) [I ~ ( JIR
)ds :
= [ eia(a+~ 2) i t-~B(a,~))C(o,~)d ~ = J]R h ~( h : e iaa - Ii~ e iah2[ 2 ~ ,t -
where
C
is in
Put ments
each
factor
2 -N = b,
,
C~ .
By the s u b m u l t i p l i c a t i v i t y sider
~B(a,~))C(a,~)d~
at a time.
which
as in I and
of the n o r m
in
B(E
Let us start w i t h
is b e t w e e n
4h
II we find that
and
8h.
it s u f f i c e s
o
),
we can con-
the second Using
to have
factor.
the same argua bound
in
for the f u n c t i o n (a,t) ~ + I
(a,t)
E [0,i] x [-i,I].
The f u n c t i o n
(a,t),--+
(17)
where
eiah2~2~(~t-
D
~ C a, w i t h
uniform
~B(a,~))C(a,~)d~
can be w r i t t e n
S eiah2~2D(t,~,a)d~
bound
transform
of
iah2~ 2 taken
in
the d i s t r i b u t i o n
sense,
,
on each d e r i v a t i v e
support. The F o u r i e r
is
,
and
uniform
C2
64
iu 2 i u~-+C o hqa -
where
C
implies
is an a b s o l u t e
o
that
the
-
constant.
C2-norm of
(17)
ah 2 e
Via P a r s e r v a l ' s
is
relation
this
b o u n d e d by I
CI " hV~ where
CI Thus
is a c o n s t a n t , it s u f f i c e s
independent
of
a
to s h o w that t h e r e
and
h.
is a u n i f o r m
constant
C2
such that
IIela°llA(E
) ~ C2hl/a. O
Let
m
be the l a r g e s t
We f o r m the
integer
SHY[.
intervals Jk
=
[-b + m~ 2 b ~ -b + k+2m 2b],
0 ~ k -< m - 2 .
By the o r i g i n a l
cove~ing
U Fk
of
constant
C3
Eo,
and
m a p p i n g , the [0,1]x Jk
it is easy to see that t h e r e
into a
is a u n i f o r m
such that •
m-2
llela°llA(Eo ) -< C3 Hence
are m a p p e d
it s u f f i c e s
to p r o v e that
.
0[
Nela°IIA(Fk ) "
for e v e r y
interval
J
of l e n g t h
where
is the map of
4b m
on
[-i,i],
is u n i f o r m l y
the n o r m of bounded.
e ia~
in
A(G),
B e t t e r yet, we p r o ve
of l e n g t h
I _ < - -i -32 - (for -, b ~ 8h, v~ m hV~ iao lle IIA(Eo ) is uniformly, b o u n d e d ) . This
is h o w
as the n o r m of
IlelaOIIA(j)
meaning (x,y)
in
J.
that the t a n g e n t - p l a n e s
E J.
This
geometric
the same for all if
can be e s t i m ated.
e iao- X , where
T a k e any i s o c l i n e
and
×
G
a < 322 '
The n o r m
intervals
then
is the same
is any c h a r a c t e r .
The s u r f a c e
z : ~(x,y)
is d e v e l o p a b l e ,
are the same for all p o i n t s
property
J,
is v e r y e a s i l y
proved
for w h i c h
f r o m the
65
special the
form
tangent
of our m a p p i n g . plane.
Then
Let
we
be the
equation
of
choose
X(x,y)
so that
z = $(x,y)
=
e
-ia~(x,y)
consider
lleia(°-g(x'y))TIA(J ) •
For
simplicity
segment y
and
the
[0,I]
suppose
on the
enlarges
invariance
x
the
y-axis.
by the
of the
that
isocline
We t a k e
factor
v~g.
n o r m we are
left
and fact
it is easy that
~
to o b t a i n
J
o,
assumptions
the u n i f o r m
stated
in the
Let us n o w
look
at t h e s e
the
turn
their
general
curves case
original
E
to this
by a s u i t a b l e
and
the p r o b l e m
for
defined of
in the
[-2,2],
[0,i] × [a',b'],
J'.
By
using
the
of o r d e r
It is v e r y
easy
to
discussion
as well.
image
a'
see that
beginning
way,
2.
one
fixes
The d i s c u s s i o n
the
are
earlier
they
<27.
where
in the
[a,b]
relation of
applies
cor-
[-2,2],
instead
interior
discussion
the
subsets, of
is
f r o m the
represents
subintervals
smoothing
the
demand
To go
T h e n we o b t a i n
the
under
section.
thing
simply
[0,i] × [a,b],
b'
norm
of o v e r l a p p i n g
subset.
to p r o v e and
of this
in an a n g l e
union
of
of the
One
of o v e r l a p p i n g
each
and w a n t
where
boundedness
is easy:
finite
union
interval
new region
in the u s u a l
assumptions.
situation
to a f i n i t e
function
that
IpA(J, ) ,
error
direction
responding solves
mapping
,y))
bound,
with
the
is the
is similar.
Thus we have p r o v e d
that
Call
~g
a uniform
approximates
for a r b i t r a r y
special
~
affine
above
to c o n s i d e r
ia(~( x ,y) _ ~ ( x
Ple
the
mentioned
a
is a subin [a,b].
to that
66
One the
further
tangent
dicular the
to a curve,
to the
general
mapping
is that
then
tangent.
case
(with
Thus,
assumption
the
This
for e v e r y
is,
however,
bounded
e,
character
corresponding
can be t r a n s f e r r e d
determinant
if the
away
we o b t a i n
isocline
no
to that
severe
along
is p e r p e n -
restriction,
situation
from
uniform
is c o n s t a n t
for
by an a f f i n e
0). boundedness
for the
char-
acters (x,y)
a
( ~+
of the
(just proof
parameter
4.
We
giving
choose reveals
e,
shall
give
T
ends
of c l o s e d
define,
for
(i),
(2),
every
~
2.
Under
~(
),
if
of T h e o r e m
boundedness
of T h e o r e m
I, v a l i d
An
inspection
as w e l l
in the
i.
for c e r t a i n
mappings
take a m a p p i n g
The
(C'(~),
by
2~h,
properly).
~ (a,t) ~ - ~ ( x ( a , t ) , y ( a ~ t ) )
T
=
i r 2~ ~w
assumptions,
I~ ~ ( t ) d t
Itl <
proof
(3), and (4).
our
supp ~ ~ ] - n , ~ [ ,
Let
our
uniform
let us
T f(x(d,t),y(a,t))
Theorem
have
system
curves.
and
[0,i] x ~
satisfying
we
a variant
= 2w ~ / ~ ,
ia(x cos e ÷ y sin e) ,
coordinate
that
and this
families
Let
the
~e
~h(t)
= 1,
= 0,
T
image
E ~2 ,
is d e n o t e d
by
E.
f(x(a,t-s),y(a,t-s))~(s)ds
f ~ A(E) and,
if
for
t
implies
.
T f ~ A(E).
0 < h ~ 1,
is not
We
~h(t ) =
on that
arc.
Then
f ~ f ~h
in
E
A(E),
as
The
theorem
and
ports.
h ~ 0. follows
a splitting
of
~
directly into
from
Theorem
i after
a sum of f u n c t i o n s
with
a partition smaller
sup-
of
67
that
In p a r t i c u l a r ,
we can take
if
t h e n the
f (A(E)
Of c o u r s e
the o b t a i n e d
parametric
"mean-value
function
depends
it can be p r o v e d
in T h e o r e m
i can be e x t e n d e d
ping where
the c u r v e s
to r e s t r i c t
do this
along
of f u n c t i o n s norm defined
the ares"
the r e s u l t T
on the p a r t i c u l a r
f (A(E). ~o c h o i c e of
~,
locally
at each p o i n t
Thus
the B a n a c h
s u c h that
set
AI/2(IR) ,
Let us d e n o t e
space
AI/2(]R)
~ ( t ) ( l + I t l ) I/2
=
F ~,
S~
are closed,
of F o u r i e r
E LI(IR)
transforms
and w i t h the
on
AI/2 (E)
=
by
h
l~(t)I(l ÷ Iti)i/2dt "
11/2(F)
vanishing
t h e n have the
F,
is the c l o s e d
[13],
subspaee
following
and we put
AI/2(]R)/II/2~E ) •
proved
graph theorem
a E [0,i] .
theorem.
is e q u i v a l e n t
and a p r o o f
to
goh
E A(E).
in the ease of c o n c e n t r i c in the
implies
general
situation
that the n o r m s
~ E C'(T) ~(t)
=
by its F o u r i e r Cneint
=
~~
circles
was g i v e n
by in [9].
in the two s p a c e s
equivalent. Representing
of func-
the m a p p i n g
g ( Al/2[0,1]
This t h e o r e m was
The c l o s e d
to a m a p -
it is no loss of g e n e r a l i t y
E } (x(o,t),y(o,t))~+
Gatesoupe
(o,t)
by
In e v e r y c l o s e d
3.
of the f o r m considered
to the case w h e r e the c u r v e s
IIf IIAI/2 (JR)
Theorem
obtaining
f r o m n o w on.
Let us i n t r o d u c e
We
T,
that a m a p p i n g
are closed.
the d i s c u s s i o n
and we shall
in
on
representation.
Conversely,
tions
9o ~ i
series Cn~n(t ) ,
are
88 we o b t a i n ,
for e v e r y
(o,t)
T f(x(o,t),y(o,t))
:
~
One
= where
the c o n v e r g e n c e If
~n
= 2~
int
e int • T
is a b s o l u t e
~o
( $ n" f ) ( x ( o , t ) , y ( a , t ) )
since
C
as a f u n c t i o n
on
E
its n o r m
in
2 gives us that
submultiplicativity
f E A(E),
$-n(S) f (x((~, s) ,y(o, s) )ds
IlSn • fIIA(E) by the
and
~(t-s)f(x(~,s),y(o,s))ds
-2~
-~ Cn
is c o n s i d e r e d
the l e m m a in S e c t i o n
( [0,i] × T
:
= 0(n -p)
n
by
,
for e v e r y
p .
(o,t) ~ + ~ n ( t ) ,
A(E)
is
O(n).
Hence
O(n),
of the norm.
By T h e o r e m
IiT~o($n " f)II
:
O(n) ,
ll~nT~o($ n " f)ll
=
0(n 2) •
2,
giving
Hence
the
series
of the f o l l o w i n g
Theorem
4.
In
A(E),
h-l(G),
tions
of f u n c t i o n s
to an i s o e l i n e , The
second
every
for some
o n l y on
For e v e r y
in norm,
which
proves
the first part
theorem.
form
depends
converges
G,
~ - ~, t,
and
f
vanishing
on a c l o s e d
can be a p p r o x i m a t e d where ~
~, ~ ( A ( E ) ,
vanishes
~ E C~TIF)
and
considered
as f u n c t i o n
f E A(E),
part of the t h e o r e m
lIT o ( $ n "
on
of
=
by l i n e a r e
depends
F
of the
combinaonly on
F. the r e s t r i c t i o n o,
is p r o v e d
f)IIA(E)
set
is in
T f
AI/2([0,1]).
as follows.
O(n),
of
~,
6g
as proved above.
(~ • f) is constant along the curves, and, 4o n its values along an isocline, considered as a function of
hence, has
T
Al/2([0,1])-norm
restricted
of the order
to an isocline,
is in
O(n).
[]C n In < - , T f,
Since
AI/2([0,1]).
The second part tells us that the convolution which
smooths the functions
on each individual
function,
also
f E A(E)
implies that its restriction
of
~)
smooths
is in
along the curves,
curve to a
to some extent the function
A([0,1]),
a,
whereas
C"
on every isocline.
to every isocline (as a function
the r e s t r i c t i o n
of
T f
is in
A1/2([0,1]). We shall that a set vanishing
finally prove a result on spectral
F E~
is of synthesis
F
can be approximated
on
on individual Take Theorem if
neighborhoods
F ! [0,i]
5.
F
h-l(F)
of
for
AI/2aR) ,
by functions
synthesis.
We say
if every
f (AaR)
in
vanishing
F.
and form
h-l(F).
is of synthesis
with respect
is of synthesis with respect
to
to
AI/2aR)
By the first part of Theorem 4, synthesis
h-l(F)
is equivalent
to the approximation
property
holding
for functions
in
e(a)~(t)
its turn is equivalent form
e(~)
restrict
in
For the subspace of these
e(a)
in
Thus, by Theorem
for
h-l(F)
for
F.
Remarks.
to the property holding
to approximating
of the form T o"
F.
[22,23],
by functions which
3,
the a p p r o x i m a t i o n
which for
in
F, which in of the
we can
with test functions
is seen by applying
the operation
problem in
AaR 2)
problem in
S is somewhat related ~2
for
(see Section i)
functions,
to the approximation
in Theorem
(AaR2))
for functions
coinciding
F,
is equivalent
The result
by H. Reiter
of the form
if and only
AaR2).
Proof:
~2
DaR)
AI/2aR)
to a result
can be formulated:
E
is of
70
synthesis
for
A~R)
Following
if and only if
E ×~
is of synthesis
ideas in [13], one can carry over the problems
this section to
~3,
by introducing
curves which lie in planes parallel the orthogonal
projections
to the
the t w o - p a r a m e t e r to the
xy-plane
we have just considered.
Theorems
for sets in
seem within reach,
~2
and
could provide methods able surfaces
for
in
R 3.
~3
xy-plane
and
for which
in the family
between
spectra
and in particular
to analyze the synthesis
of
family of all
are curves
giving relations
AaR2).
property
this
for develop-
71
REFERENCES
[I]
J. J. Benedetto,
[2]
A. Beurling, Sur les int6grales de Fourier absolument convergentes et leur a p p l i c a t i o n ~ une t r a n s f o r m a t i o n fonctionelle, IX Congr~s des M a t h 6 m a t i c i e n s Scandinaves, Helsinki, 1938(1939), 345-366.
[3]
A. Beurling and H. Helson, F o u r i e r - S t i e l t j e s t r a n s f o r m s with bounded powers, Math. Scand. 1(1953), 120-126.
[4]
Ph. Brenner, Power bounded m a t r i c e s of F o u r i e r - S t i e l t j e s transforms, Math. Scand. 22(1968), 115-129; Corrections, Math. Seand. 30(1972), 150-151.
[5]
F. Carlson,
[6]
Y. Domar, Sur la synth~se h a r m o n i q u e des courbes de Acad. Sci. Paris 270(1970), 875-878.
[7]
Y. Domar, On the spectral synthesis p r o b l e m for (n-l)d i m e n s i o n a l subsets of ~ n , n ~ 2, Ark. Mat. 9(1971), 23-37.
[8]
Y. Domar,
synthesis,
New York,
Une in@galit6, Ark. Mat. Astr.
Estimates of
a mean-valued
[9]
Spectral
IIeitfllA(F),
function,
when
Israel J. Math.
1975.
Fys.
25, BI(1934).
F ~n
12(1972),
~2,
and
C. R.
f
is
184-189.
Y. Domar, Harmonic analysis of a class of d i s t r i b u t i o n s on n ~ 2. SIAM J. Math. Anal. 3(1972), 230-245.
~n,
[io]
Y. Domar, 39(1976),
ELl]
Y. Domar, On the Banach algebra A(F) for smooth sets Comment. Math. Helv. 52(1977), 357-371.
[12]
Y. Domar, A C~ (1977), 189-192.
[13]
M. Gatesoupe, Sur les t r a n s f o r m 6 e s de Fourier radiales, Soc. Math. France, M6moire 28(1971).
[14]
R. Gustavsson, On the spectral synthesis p r o b l e m for curves in ~ 3 , Uppsala University, Department of Mathematics, Report 1974:6.
[is]
C. S. Herz, Spectral (1958), 709-712.
[16]
C. S. Herz, The ideal t h e o r e m in certain Banaeh algebras of functions satisfying smoothness conditions, Proc. Conf. Functional Analysis, Irvine, Calif. 1966, London (1967), 222-234.
[17]
R. Kaufman, A p s e u d o f u n c t i o n on a Helson set II, A s t 6 r i s q u e (1973), 231-239.
,[18]
On spectral 282-294.
B. Kjellberg, A2(1943).
synthesis
for curves in
curve of spectral
synthesis,
R 3
Math
Scand
F c ~n,
Mathematika
Bull.
synthesis for the circle, Ann. Math.
Ein m o m e n t e n p r o b l e m ,
Ark. Mat.
Fys. Astr.
24
29
68
5
72 [19]
T. K6rner, A p s e u d o f u n c t i o n (1973), 3-224.
on a Helson
set I, Ast~risque
[20]
F. Lust, Le probl~me de la synth&se et de la p-finesse pour certaines orbites de groupes lin@aires dans Ap~Rn), Studia Math. 39(1971), 17-28.
[21]
P. Malliavin, Sur l'impossibilit6 de la synth~se spectrale sur la droite, C. R. Acad. Sci. Paris, 248(1959), 2155-2157.
[22]
H. Reiter, 133(1957),
Contributions 298-302.
to harmonic
analysis,
II, Math.
[23]
H. Reiter, Math. Soc.
Contributions to harmonic 32(1957), 477-483.
analysis,
III, J. Lond.
[24]
L. Schwartz, Sur une propri~t~ de synth~se spectrale dans les groupes non compacts, C. R. Acad. Sci. Paris 227(1948), 424426.
[25]
N. Th. Varopoulos, Spectral synthesis on spheres, Cambridge Philos. Soc. 62(1966), 379-387.
[26]
N. Th. Varopoulos, Tensor algebra Math. 119(1967), 51-111.
E27]
H. Whitney, Bull. Amer.
and harmonic
On the extension of d i f f e r e n t i a b l e Math. Soc. 50(1944), 76-81.
5
Ann.
Proc.
analysis, functions,
Acta
S P E C T R A L SYNTHESIS AND STABILITY IN SOBOLEV SPACES Lars Inge Hedberg U n i v e r s i t y of S t o c k h o l m 0.
I n t r o d u c t i o n and p r e l i m i n a r i e s The a p p r o x i m a t i o n problems in Sobolev spaces w h i c h here will be
called the spectral synthesis and s t a b i l i t y problems arise different ways.
The stability problem goes back to the fundamental
work by M. V. K e l d y ~ [ 2 3 ]
on the s t a b i l i t y of the solution of the
Dirichlet p r o b l e m and u n i f o r m a p p r o x i m a t i o n by h a r m o n i c
functions.
In greater g e n e r a l i t y the p r o b l e m was raised by I. Babu~ka also
noticed
polyharmonic
its c o n n e c t i o n with a p p r o x i m a t i o n functions.
approximating
in several
V. P. Havin
[4], who
in L 2 by h a r m o n i c and
[17] proved that the p r o b l e m of
in planar L 2 by rational
functions
is also e q u i v a l e n t
to stability. The closely related appears
implicitly
a uniqueness
synthesis p r o b l e m in Sobolev spaces
in the work of S. L. Sobolev
[31], in the form of
t h e o r e m for the Dirichlet p r o b l e m for the p o l y h a r m o n i c
equation A m u = 0. [7],
spectral
It is explicit in the work of Beurling and Deny
[ii] on Dirichlet
f o r m u l a t e d by Fuglede
spaces.
In more general Sobolev spaces it was
(see [30~ IX,
§5.1]).
It also arises n a t u r a l l y
in the context of L P - a p p r o x i m a t i o n by analytic or harmonic
functions,
w h i c h was the m o t i v a t i o n behind the work of T. Bagby [5], J. Polking [28], and the author
[20],
[22] on this problem.
The purpose of this paper is to explain the connection between these problems and to give a survey of work done on them in recent years, without going into too much t e c h n i c a l detail. new results
in the paper;
it is purely expository.
There are no It is in the
nature of the problems treated here that the methods used are rather heavily potential-theoretical, theory is assumed.
but no previous k n o w l e d g e of p o t e n t i a l
74
It has not been my purpose
to write the history of the subject.
This would be quite complicated,
since results have often been dis-
covered i n d e p e n d e n t l y by different people.
Neither has any attempt
been made to give a complete bibliography.
Several of the papers
quoted contain c o m p r e h e n s i v e bibliographies,
e.g.
[18],
[19],
[22],
and [26]. I am grateful to the U n i v e r s i t y of Maryland and the U n i v e r s i t y of C a l i f o r n i a at Los Angeles for giving me the o p p o r t u n i t y to give the series of lectures, Our n o t a t i o n a l
of which these notes are a write-up.
conventions will be the following.
of a set E is DE, its interior E 0 E c.
its closure E, and its complement
K is always going to be a compact
space ~ d
F is closed
with compact
,
and G is open
support in G.
•
set, usually in d - d i m e n s i o n a l C0(G) ~ means the
Various constants,
within the same sequence of inequalities, The Sobolev space w P ( R d) m
C0(~d)
The boundary
C ~
functions
whose value may change
will be denoted by A.
= W p is defined as the closure of m
w i t h respect to the norm II " llm,p defined by
! i llfl] 'P = O~ I l~m ~ d
Here ~ = (el,...,ed)
l ~ f ( x ) l p dx"
is a m u l t i i n d e x of order
I~I = el+...+~d ,
x = (Xl,...,Xd) , dx = dXl...dx d is Lebesgue measure, integer,
m is a positive
and i ~ p < ~.
For any open G c ~ d
we denote by wP(G) the subspace of w P ( ~ d) m
o b t a i n e d as the closure of C0(G) with respect to II Equivalently,
m
llm,p.
W p can be defined as the functions in L p, all of m
whose partial d e r i v a t i v e s
in the weak
order s m are also functions It is a basic fact that tinuously
sense of
in L p. llfll~ ~ Allfllm,p, so W d ~) (_ ~
Can be con-
embedded in C ( ~ d) , if mp > d, but not if mp ~ d.
book by E. M. Stein facts.
(or distribution)
The
[32] is a good r e f e r e n c e for these and r e l a t e d
75
In the f o l l o w i n g synthesis
and
stability
2 we g e n e r a l i z e
to WE,
linear potential
I.
Spectral
s e c t i o n we
in the c l a s s i c a l
synthesis
and
point
principle.
f E W~(IRd) .
Consider
closed. shows
The
that t h e r e
Let G c ~ d
fG ~ C~(G)
solution
argument
problem
using
a unique
open
of f i n d i n g
functions
fG'
that
°2 f - fG ( W I ( G ) .
I.e.,
say.
fG is the
proved
and
by let
inf(llVgl 2 dx;
g is c o n v e x
problem with boundary
It ia e a s i l y
problem set,
the p a r a l l e l o g r a m
extremal,
and Af G = 0 in G.
of the D i r i c h l e t
to a n o n -
of the D i r i c h l e t
IVf G • V~ dx = 0 for all ~ ~ C0(G) , w h i c h by W e y l ' s that
In S e c t i o n
3 we study W~, m ~ 2.
be a b o u n d e d
set of c o m p e t i n g
exists
spectral
2 s t a b i l i t y i n W I.
the e x t r e m a l
By a s t a n d a r d
2 in W 1.
h o w this leads
in S e c t i o n
is the s o l u t i o n
the D i r i c h l e t
°2 f - g ~ WI(G)}.
Finally,
and d i s c u s s
context
p ~ 2, and d i s c u s s
theory.
Our s t a r t i n g
shall d e f i n e
and
identity
one
fG s a t i s f i e s lemma
implies
(generalized)
d a t a f in the
that this
solution
sense is
unique. Thus,
if we use IVf
W (IR d)
splits
D~(G).
Here
• Vg dx as an i n n e r p r o d u c t ,
into two p e r p e n d i c u l a r
D 12(G) = {f ( W ~ ( ~ d ) ~
We c a n now f o r m u l a t e and s t a b i l i t y Consider measure.
potentials
e n e r g y are d e n s e
represented If the
first v e r s i o n
= WI(G )
in G}
of the s p e c t r a l
= flx-yl 2-d d ~ ( y ) ,
we a s s u m e
by log ~ i
IU ~ d~ = A I IVU~I 2 dx.
in G.
U~(x)
(For s i m p l i c i t y
= IU ~ d~, w h e n e v e r
finite
f is h a r m o n i c
W I2(IR d)
synthesis
problems.
Ixl 2-d is r e p l a c e d I(~)
the
subspaees,
we see that
.)
that d ~ 3.
The e n e r g y
IU i~l dl~ I < ~. It is e a s i l y in W ( ~ d ) .
I(~)
where
~ is a s i g n e d
For d = 2 the k e r n e l is d e f i n e d
By a c l a s s i c a l
formula
seen that p o t e n t i a l s In fact,
all
by
with
CO functions
can be
in this way. support
The p r o b l e m
of ~ does of s p e c t r a l
not
intersect
synthesis
is:
G, t h e n U ~ is h a r m o n i c Are the p o t e n t i a l s
U~
76
which belong
to D (G) dense
If this
is the case,
synthesis.
A fundamental
([7],
is that
[ii])
synthesis.
This
different problem
form)
be p u s h e d
that K is
property.
give
sequence
Set
+ Vf G
identity
- Vf G
(l,2)-stable, w h i c h will
dx ~ fIVfGi 2dx.
following
Then
also be
for e v e r y
= M.
so that the
Then we increasing
For n > m we have °2
[2 dx
12dx = 2 f l V f G
It f o l l o w s
, and by the p a r a l l e l o g r a m
12dx + 2flVf G m
- 2f]VfGmI2dx 2 in WI,
12dx - flVfG
~ 0, as n,m ÷ ~.
the limit
+ n
n
l i m n ÷ ~ fG
Thus the = fK e x i s t s
n
: M.
fK is h a r m o n i c
of the D i r i c h l e t proposition
Proposition 2 all f ~ W I.
but
In fact,
K c G, G open)
n
solution
we say
of the s t a b i l i t y
i
m
{fG }i is C a u c h y
flVfKl2dx
synthesis
m
VfGm 12dx ~ 2 f l V f G n f ] 2 d x
and
supp ~ c K c
f - ~(fG ÷ fG ) E WI(Gm). n m
I2 dx ~ 4flVf G
n
sequence
T h e n the
is the case,
K be given.
to M as n ÷ ~.
m
flVf G
(in a s o m e w h a t
U ~ with
condition
formulations
flVfK012
~ W I ( G m) and thus a l s o
n
If this
sufficient
flV fG 12 dx c o n v e r g e s n
flVf G
solved
( G n ]i' K c G n c ~ n e Gn_l,
n that
1.13.).
sets are not
s u p ( f l V f G 12 d×;
°2
f - fG
(l,2)-spectral
in the s p e c t r a l
f E W I2 and a c o m p a c t
K we have
can find a s e q u e n c e
and J. Deny
Let K be compact.
~K?
some e q u i v a l e n t
E W~(G).
fG
fK 0
(l,2)-spectral
1.21).
Let a g a i n
open G containing
and
and
the p o t e n t i a l s
All c o m p a c t
found a necessary
We first
[23].
off the b o u n d a r y
(Theorem
(Theorem
was r a i s e d
Are
Beurling
sets admit
can the m e a s u r e s
(l.2)-stable.
given below
due to A.
below
by M. V. K e l d y ~
l.e.
say that G c a d m i t s
closed
be p r o v e d
can be f o r m u l a t e d :
problem
Keldy~
result,
of s t a b i l i t y
in D I2(K0)?
dense
we s h a l l
in fact all
will
The p r o b l e m
2 in DI(G)?
i.i:
problem
now follows
in K 0 and c a l l e d
the e x t e r i o r
for K w i t h b o u n d a r y
values
f.
The
easily.
K is ( l , 2 ) - s t a b l e
if and only if fK = fK 0 for
77
K is thus
(l,2)-stable
K 0 that have extensions functions
harmonic
if and only if all harmonic
to W ( ~ d )
We define D (K) : open G containing
in
in W I2 by
can be approximated
on neighborhoods
functions
of K.
(G), where the union is taken over all
K, and the closure
2 is taken in W l ( ~ d ) .
Then it is
easy to prove the following. Proposition
1.2:
K is (l,2)-stable
if and only if D~(K)
=
D~(K0). A function °2 g E N G WI(G), taining K. that W °2
WI(K)
g E W I2 is orthogonal
where the intersection
• D (K).
The following
Proposition
1.3:
In other words,
prove,
It is easily seen
We can thus write W ( ~ d ) =
K is (l,2)-stable
K is (l,2)-stable
is immediate. °2 if and only if WI(K)
=
if and only if for every
is a sequence
{ ~ n ~ ' ~n ( C0(K0)'
limn+~ IIVg-V~n I2 dx = 0.
Keldy~ originally gence.
°2 by WI(K).
proposition
g E W 12 such that g = 0 off K there
if and only if
is taken over all open G con-
We denote the intersection
(K) = {g E WI; g = 0 off K}.
such that
to D~(K)
studied
It is a non-trivial that the definition
the one given by Keldy~. [24; Ch. V,
§5].
the Dirichlet the following.
stability
fact,
problem
which it would take us too far to
of stability See Keldy~
If f E c ( ~ d ) w e
in terms of uniform conver-
given here
[23], Deny
is equivalent
to
[i0] and Landkof
again let fG denote the solution of
in G with boundary values
(Note that by Tietze's
extension
f.
The result
is
theorem any function
in C(K) can be extended to c ( ~ d ) . ) Theorem 1.4: which is harmonic
K is (l,2)-stable
if and only if every f E C ( ~ d)
in K 0 can be uniformly
tions fG' where G is open and G contains
approximated K.
on K by func-
78
In the c o m p l e x an i n t e r p r e t a t i o n by LP(K) a K 0.
the
LP(K)
in t e r m s
subspace
By RP(K)
with poles
plane
Clearly
RP(K)
c L~(K).
RP(K)
w h i c h are a n a l y t i c We w r i t e
allow
If K c C we d e n o t e
of f u n c t i o n s
in LP(K)
theorem
synthesis)
f u nctions.
consisting
the c l o s u r e
By R u n g e ' s
of the f u n c t i o n s
(and s p e c t r a l
of a n a l y t i c
of LP(K)
we d e n o t e
off K.
stability
analytic
in
of the r a t i o n a l f u n c t i o n s is also
the c l o s u r e
on some n e i g h b o r h o o d
in of K.
x = x I + ix2, x = x i - ix 2, dx =
dx I dx 2 • Theorem K is
1.5.
(V. P. H a v i n
R 2 (K) = L2(K) a
[17]):
if
(l,2)-stable. Proof:
Let g ( L 2 ( K ) .
We can a s s u m e
that g(x)
T h e n I fg dx = 0 for all f in L2(K) if and only K a °2 (WI(K0). This is e a s y to p r o v e u s i n g W e y l ' s that
if and only
If ~
dx = 0 for all ~ ( C 0 ( K 0 )
= 0 for x ~ K.
if g
~-~ for some
lemma,
i.e.
if and only
the
fact
if f is a n a l y t i c
in
K0" For the
same r e a s o n ,
G n K, it f o l l o w s in R2(K) L2(K) a
if and on l y
if K is
(l,2)-stable. R2(K)
if f fg dx = 0 for all f in L2(G) for some ~K °2 a that g = ~-~, ~ ~ WI(G). Thus I fg dx = 0 for all f if g = ~,~ ~ ( n G n K WI(G)=°2 KW~(K).
(l,2)-stable. In fact,
Conversely,
if R2(K)
(~-~)
= 0, and
~-~
= L2(K), a
°2 ~ let ~ ~ WI(K) , and let g = ~
= L (K), we a l s o have g = ~-~ ~ , where has c o m p a c t
~ ( W ~ ( K 0)
support,
Thus R2(K)
.
=
t h e n K is
Then,
if
But then
so ~ = ~, and thus
°2 (WI(K0). In o r d e r functions (f(x)
to get f u r t h e r we have to d e f i n e
in W ~ ( ~ d )
are not
= log l o g l x I for
The n a t u r a l continuity
Ixl < e, 0 for
way of m e a s u r i n g is by m e a n s
More generally, (m,p)-capacity
in g e n e r a l
capacities.
continuous
The
if d ~ 2.
Ixl ~ e is an e x a m p l e
by h o w m u c h the f u n c t i o n s
in ~ 2
deviate
)
from
of c a p a c i t y .
to e v e r y
Sobolev
by the f o l l o w i n g
space w P ( ~ d ) m
definition.
we a s s o c i a t e
an
79
Definition
1.6:
(a)
If K is c o m p a c t ,
Cm,p(K)
: inf{II~N p
m~p'
"
( C 0, ~ ~ i on K}. (b)
If G is open,
(e)
E is a r b i t r a r y ,
A property if it is true
is said
for all
Cm,p(G) C
= sup{Cm,p(K); (E)
m,p
= inf{C
m,p
to h o l d ( m , p ) - q u a s i
x except
those
Then
following
K c G,
(G);
K compact}.
G n E, G open}.
everywhere
belonging
((m,p)-q.e.)
to a set w i t h
zero
(m,p)-capaeity. Let
f ( W p @ C.
the
inequality
is an
immediate
m
consequence
of the
definition
of c a p a c i t y .
Cm,p({X;[f(x) I One
can
maximal
prove
function
a similar
Mf,
defined
inequality
for
,
the
x >
o.
Hardy-Littlewood
by
lf(y)ldy
= sup - ~ r>0
1.7:
[ i f l l mp ,P
1 !
Mf(x)
Theorem
_< iA p
A}
>
r
(D. R. A d a m s
IY-
Isr
[i]).
Let
f ( W p.
Then
m
Cm,p({x;Mf(x)
Now
let X ( Co({Ixl
< i}),
X ~ 0, and
}~
approximate
identity
{Xn i' by Xn(X)
that
( C ~ and
that
f * Xn
Theorem
1.7,
and
l i m n + ~ f * Xn(X) f(x)
= ~(x)
for any
f * Xn k ÷ ~(x)
property Thus Moreover, result.
= ~(x)
a.e.
given
A function
standard
which
this (See
for
there
ndx(nx)
one
outside (m,p9
also
(m,p)
= i.
.
Define
G.
G,
proves
- q.e.
C
Thus
- q.e.
m,p
an
If f ( WPm if f o l l o w s
x.
is a s u b s e q u e n c e
is an o p e n
is d e f i n e d
is c a l l e d we can
arguments
Moreover,
uniformly
I X dx
llf - f * Xn llm,P ÷ O as n ÷ ~ •
exists
e > 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
<
we
say that
interior
shall points.
ly-xl
<
2 -n} it is thick.
In p a r t i c u l a r
U
~E
Any
(x)
set
is
= i every-
in E 0 . We can n o w Theorem
give
1.12:
belonging
to D 2(G) I
such
fiG c
that
Proof: In o r d e r gonal
A set
~ such
x ~ E or
2n ( d - 2 )
If a set
where
See
[ n CI,2(E n:l
An(X)
lim inf y+x,y(E\{x)
sufficient
criterion.
Theorem (a)
and
<
at x if e i t h e r
:
Let
are
We k n o w
complement that
dense
that
the of
be o p e n
in D (G)
to W the
theorem
{U~;
flGc
f ( W 12 .
interpretation
Let G c ]I{d
0 belongs
to p r o v e
f ( W 12 such
a dual
of s p e c t r a l
and
bounded.
if and
only
orthogonal
it is e n o u g h
to
consists
show
Formally
identity
U~
f in W I
I×-yl d f(y)d~(y).
of M.
Riesz
to D2(G) i that
of the
Ix-yl d
But by a c l a s s i c a l
if e v e r y
complement
0.
: A
Potentials
(G).
2 U ~ ( DI(G)}
_ ff
synthesis.
the
is W ortho-
functions
(G)
82
It f o l l o w s
from
integration Using to
show
theorem
that
is j u s t i f i e d
if I(~)
< ~.
the r e g u l a r i z i n g
the
sequence
= lim
] fn d~
f E W I2 and
for all
change
{fn } = {f * Xn}
all
: lim A ; Vf
~ with
I(~)
VU~
< ~.
If F is an a r b i t r a r y
closed
as our
(l,2)-spectral
chosen
above
of o r d e r
it is n o w
of
easy
that
I fd~
1.12
Fubini's
definition
to c o n s i d e r
restriction We
of
the
equation
to b o u n d e d
can now
prove
G would the
dx : A I V f
The
theorem
set we take
synthesis.
theorem
have
been
dx
follows
the p r o p e r t y
easily.
in T h e o r e m
(If we had
- Au + u = 0 i n s t e a d not
• VU~
of Au
= 0 the
necessary.)
of B e u r l i n g
and
Deny
referred
to
earlier. Theorem
1.13:
Every
closed
set
in ~ d
admits
(l,2)-spectral
synthesis. Proof: under and
I
10+I 2 dx 2 f E Wl,
that
f can
neighborhood function support. Let f (x)
crucial
truncations.
Let claim
The
we
For =
example,
I {f(x)>0]
and
suppose
assume
It is s u f f i c i e n t let
f
= 0 in a n e i g h b o r h o o d
f tvf
that
then
space
by f u n c t i o n s
that
= (f+
that
and m u l t i p l y i n g f is b o u n d e d
to c o n s i d e r
set F.
vanish
2 E W I,
0
on a
and
has
compact
f+.
- ~ )+ .
f
We
by a c u t - o f f
If f is c o n t i n u o u s ,
of F, and
vfl 2 dx :
is c l o s e d
f+ = m a x ( f , 0 )
fl F = 0 for a c l o s e d
By t r u n c a t i n g
can a l w a y s
s > 0 and
if f E WI, 2
this
IVfl 2 dx ~ /IVfl 2 dx.
be a p p r o x i m a t e d
of F.
of W 2 I is that
property
l fl 2 dx
0,
then
83
which
proves
the
In g e n e r a l an open and
set
such
~n(X)
CI,2(Gn)
f(x)
case.
< nl , such
: 0 on FkG
of f one
Then,
quasicontinuous.
n
.
easily
n is an ~
There
shows
for any n t h e r e
fig c is c o n t i n u o u s
that
i and IIV~Jn 12 dx < n'
= i on Gn,
boundedness
in this
f is o n l y
Gn,
that
theorem
W2 such i
~
n
0 < ~n (x) -< I.
that
f (i-~)
is
on G c, n
that
Using
the
is the r e q u i r e d
n +
approximation In the analogous we m e a n
to
complex
to T h e o r e m
the
function
Corollary Cauchy
f .
(See plane
the
1.5.
theorem
By the
~(x)
1.14:
transform
[20].) has a dual
Cauehy
transform
= /(~-x) -I d~(~),
([20]).
For any
~ of m e a s u r e s
with
interpretation of a m e a s u r e
x = x I + ix 2.
bounded
support
open
G c C the
in G c are
dense
in
L2(G). a
Returning corollary
to
to T h e o r e m
Corollary o2 f (WI(K)
fl
We
for a set
(a)
(b)
give
K to be
all
K is
immediate
(l,2)-stable
if and o n l y
if e v e r y
= 0. W
a number
Bagby
(l,2)-stable all
K is
open
(K 0) c o n s i s t s
exactly
of the
functions
and
sufficient
conditions
[5])
if and
only
if C l , 2 ( G \ K )
if and
only
if for some
> ~ 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
This
a different
The d u a l distributions
space
an e q u i v a l e n t Corollary with
proof,
is true
to W~ is d e n o t e d
as l i n e a r
In terms dual 2.5:
W!I.
which
For any c l o s e d
It c o n s i s t s
extensions
of W!I D e f i n i t i o n
formulation,
but this re-
[6].
combinations
supp T c F can be a p p r o x i m a t e d
supp ~ c F.
for q : i also,
due to L. Bers
w h i c h have c o n t i n u o u s
can be r e p r e s e n t e d Lq-functions.
corollary
of first
The e l e m e n t s
derivatives
2.1 can e a s i l y
leads
F c ~d
to W E.
of t h o s e
to the
be g i v e n
following
all T £ Wql,
in W!I by m e a s u r e s
of
corollary.
i < q < ~,
~ with
87
The
following
two
theorems
are
proved
(Havin
[17]):
Let
K c {.
much
as T h e o r e m s
1.5
and
1.6. Theorem
2.6
i < q < ~,
if and
Theorem p > d.
If 1 < p < d,
CI,p(G\K)
(b)
there open
If one
= Lq(K), a
All
(l,p)-stable
are
if and only
for all that
K c ~d
open
(l,p)-stable
to
runs
~K s a t i s f i e s
generalize
into
if e i t h e r
G
CI,p(G\K)
which
case
What
as a p o t e n t i a l more
there
~ ~ CI,p(G\K0)
1.21
is a p r o p e r extremal
for all
complicated
and
1.22
one
generalization
~K is h a r m o n i c
of a p o s i t i v e
measure.
equation
For
of on K c,
p ~ 2
d i v ( I V ~ K Ip-2 V~ K)
= 0,
as a p o t e n t i a l .
is a d i f f e r e n t
originated
1.18,
p : 2 the
be r e p r e s e n t e d
Fortunately,
Theorems
problems.
In the
the m u c h
it c a n n o t
in w o r k s
approach
of
B.
to the
Fuglede
[15]
potential and N.
theory
G.Meyers
[27]. If f
E W m' p m < d,
then
f(x)
where W p.
g ~ L p. The r e a s o n
However,
f can
=
f
be r e p r e s e n t e d
g(y)dy
] l×_yld-m
if g ~ L p,
for t h i s
is t h a t
- R
then
R
m
R
m
m
as a Riesz
potential
* g,
* g is not n e c e s s a r i l y
(x) t e n d s
to
zero
too
slowly
in at
m
~.
This (6)
corresponds
= AI{I -m has
to the
fact
a singularity
that at
@.
the
Fourier
Therefore,
transform for any
~ > @ one
m
defines
the
so c a l l e d
transform
of G
(l-A)-~g.
The
e.g.
Stein
if
G.
is r e p r e s e n t e d
of WE,
K is
[5]):
= Cl, p (G\K 0)
(112)-thinness?
and
Baghy
is an ~ > 0 such
tries
immediately
Rq(K)
if K i s ( l , p ) - s t a b l e .
2.7(Havin[17],
(a)
and
only
Then
(~)
= (i +
Bessel
[32].
Bessel
kernel
G (x) as the
1612) -~/2.
kernels
have
the
In o t h e r
inverse
words,
following
G~
Fourier * g
properties.
z
See
88
a)
G (x) > 0;
b)
G
c)
G(x)
:
d)
G(x)
m Alxl ~-d,
Gd(X)
i % A log ~ x T
e) Using
, GB : G + B ~
0(e-ClX)),
the t h e o r y
Ixl
~
÷
Ixl + 0, o < ~ < d; ' IXl ÷ o.
of s i n g u l a r
integrals
the f o l l o w i n g
theorem
is
n o w e a s y to prove. Theorem f -- G
m
2.8
* g, w h e r e
that A -I
g ( L p.
~'~ g;g
2.9:
f E W p,
Moreover,
Definition C
Note Gc~ * g ( x )
The B e s s e l
( LP(IR d) }, and
We now m o d i f y
ous
[8]):
i < p < ~, if and only
there
is a c o n s t a n t
is
For any
inf{I
of
capacity
set E c ]Rd
gP d x ; g _ > 0, G~
the d e f i n i t i o n
defined
space
We s h a l l
makes
everywhere
capacity.
investigate
for
the
and
Let g >_ 0 a n d G
L pa .
(~,p)-capacity
* g(x)
sense
> i
on E}.
for a r b i t r a r y
g >~ @.
the e x t r e m a l
Let K be c o m p a c t ,
on K.
LP(]R d) =
by using
Again,
(E) > 0 for all n o n - e m p t y @,P i n t e r e s t i n g for our p u r p o s e s .
support
A > 0 such
llfIl~,p = IlgIlp-
definition
2.10:
p(E)=
that
the
potential
for ap > d, C
is not
if
llgHp -< Hfllm,p -<- Allgllp-
Definition {G
(Calder6n
continu-
and the c a p a c i t y
in the d e f i n i t i o n
let ~ be a p o s i t i v e * g > i on K.
since
Gc~' * g i s
sets,
function
E,
measure
of
with
Then by F u b i n i ' s
theorem
Thus
sup
m
Applying I(G
U(K)
IIGG * gllq
the M i n i m a x
* ~)g dx one
can
Theorem
s inf g
llgll
P
: c
(K) I/p G,P
to the b i l i n e a r
show that
equality
functional
holds
~(g,~)
in the last
=
inequality.
89
(Fuglede
[15], Meyers
so that G
* gK ~ i (~,p)-q.e. f : ] (G
UK(K) It follows gK = ( G
that
* pK )q-I
The function
Moreover
G
Choosing A = i we have
and = I (G~ * PK)q dx : ; gkP dx = C~,p(K).
* gK = G ~ * (G ~ * pK )q-I = V pK ~p ZK"
The results
extend
2.11:
is called a non-linear
(If p = 2 the n o n - l i n e a r i t y
VP6,2 = Ge * G~ * p = G2~ * p , w h i c h
there
~K and gK
fIG * ~Kllq llgKllp.
* pK)ZK dx :
of the measure
Theorem
there are extremal
on K, and
(G~ * ~K )q = Ag~.
~K(K)
potential
[27]).
is a classical
to arbitrary
sets
disappears;
potential.)
(See [27]).
We summarize.
For every bounded E c ~ d , I < p < ~, ~ > 0,
is a unique measure ~E { 0, the capacitary
measure,
with support
in E such that (a)
V~Ep(x)
(b)
PE V ,p(X) ~ 1 f o r
(c)
/
ddPE
~ 1 (~,p)-q.e.
all
on E;
x ( supp ~E;
= ; V~Ep d~E : I (G~ * pE )q dx : C ,p(E).
It is easy to see that C
(E) : inf{C (G);G o E, G open} ~,P ~,P any E. One can show that C satisfies the axioms of Choquet's ~,P theory of c a p a c i t a b i l i t y (Fuglede [15], Meyers [27]). Thus Theorem C
~,P
2.12:
(E) = sup{C
~,p
For every Borel
(or Suslin)
for
set E
(K);K c E, K compact].
In general
V pE (x) > 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<~p
2.16:
i < p < ~,
(a)
If [ a n ( x , E ) q - i i o < ep ~ d.
an(x,E)q-i
If n:l
< ~'
p > 2 - ~,
= ~,
then
then
E is
E is
(~,p)-thick
(e,p)-thin
at x.
91 0o @
(c)
If
[ n:l
a
(x,E) " - - ~ ' ~ ' - - ~ "
(x,EA)}q-I
log
n:l
to
(b)
the
breaks
above
now
corollary
the
properties
together
one
(l,p)-thick
3.
true
2.17:
(l,p)-q.e.
Theorems
and
1.21
and
the
then
E is
estimate
to
L q-l.
the
Choquet
is
2.16
one
and
can as a
Kellogg
unknown.
synthesis
generalization
K c ~d
leads
[24]).
p > 2 - ~, and
~ remains spectral
which
For p = 2
(See
Theorem
case
is b e s t
theorem
of T h e o r e m
(l,p)-stable
1.18.
if K c is
~K.
the 1.22
without
(c)
criterion.
Whether
following
on
belongs
in t h e
-
in
the
in p r o v i n g
lemma
A compact
because
Wiener
involved
the
then
Higher
the
results
exponent
longer
i < p i_ < 2
for
obtains
the
property.
If p > 2 - ~,
What
Let
Kellogg
generalizes
no
Choquet's
these
Theorem
1.23
estimates
are
Putting
G
contains
extend
that
2 - ~ appears
down when
the
easily
([2])
number
theorem
Using
p = 2 - d'
at x.
It is a l s o k n o w n The
< ~,
an
(~,p)-thin
possible.
E is
at x.
~[ i an (x ,E)
If
then
i < p < 2 - ~, <1
(e,p)-thin
(d)
< ~,
n
converse
is a l s o
generalize
true.
in t h e
restriction.
(See
same way.
Theorem
[20]).
derivatives. should
F c ~d
be m e a n t
be a closed
by
set.
(m,p)-spectral
If
a function
synthesis
f
in
if m > i?
wP(~ d) , m > 1, m
o
to b e l o n g that
fIF
t o w P ( F c) m
= 0.
V~ n + V f i n (m-k,p)-q.e.
it is c l e a r
In fact,
if ~n
WPm-I' V 2 ~ n + V 2 f on F,
means
the
vector
tives
of f of o r d e r
i.e.
valued k.)
vkflF
that
one
~ C0(FC)'
in
to r e q u i r e
~ n ÷ f in W~,
Wpm_2, e t c .
= 0 for
function
has
Thus
consisting
of
all
than
then
vkf(x)
0 ~ k ~ m - i.
more
= 0 (Here v k f
partial
deriva-
is
92
Definition (m,p)-spectral
3.1:
(Fuglede,
synthesis
see
if e v e r y
[30~
IX,§5]
and
[221).
f ( W p such that v k f I F
F admits
= 0
m o
0 < k ~ m - i, b e l o n g s
to wP(FC). m
In o t h e r words,
F admits
(m,p)-spectral
synthesis
if the
o
obvious
necessary
condition
for f to b e l o n g to w P ( F c) is also m
sufficient. Conjecture:
All c l o s e d
F c ~d
admit
(m,p)-spectral
synthesis
for i < p < ~, m = 1,2, . . . . The r e a s o n why WPm are no l o n g e r distribution
closed
The c o n c e p t
is a p r o b l e m
for m > i is t h a t the
under truncation.
derivatives
Some p o s i t i v e
way.
this
Even if f ( CO, the
of V(f +) are not,
results
will
of s t a b i l i t y
be g i v e n
can also
spaces
in g e n e r a l ,
functions.
below.
be g e n e r a l i z e d
in a n a t u r a l
As b e f o r e we w r i t e o
o
wP(K)
=
m
Definition
if ~ P ( K )
:
3.2
n wP(G) m GnK
= {f ~ W p" f(x) ,
= 0
on K c}
•
m
(Babuska
[4]):
K c ~d
is c a l l e d
(m,p)-stable
~P(K0).
m
m
Clearly, spectral
K can be
synthesis.
(m,p)-stability
(m,p)-stable
General
only
necessary
are not known,
but
some
if (K0) c a d m i t s
and
sufficient
sufficient
(m,p)-
conditions
conditions
for
will
be
g i v e n below. Again, given
the
several
spectral
equivalent
synthesis
and
formulations.
stability
properties
For s i m p l i c i t y
can be
of s t a t e m e n t
we let m = 2. Thus,
F admits
distribution
(2,p)-spectral
T in the dual
synthesis
space wq2 w i t h
if and only if e v e r y
supp T c F can be a p p r o x i I
mated
in wq2
measures
with
by l i n e a r
combinations
supp ~i c F, ~0
For any E we let L~(E)
~0 + ~
E wq2,
denote
~i/~xi
~l,...,~d
the
subspace
' where
a~e
E Wql. of Lq(E)
consisting
93
of f u n c t i o n s admits
harmonic
(2,p)-spectral
be a p p r o x i m a t e d where
~i• are m e a s u r e s
general
expect
for e x a m p l e
potentials
E ~(G),
problem
are two c o n t i n u o u s
domain
In the g e n e r a l
c o p y the p r o c e d u r e
stability
i"
in
in L~(G).
If
w i t h a singu-
if and o n l y
if all
harmonic
with given
we u s e d
belongs
find a f u n c t i o n
u E W (~d) can a g a i n
on
for the
extremal,
of the D i r i c h l e t the
such t h a t
~u/~n
problem
fact that
One
Laplace
problem
standard
equation.
boundary way
the
is to
Let G be o p e n
a n d let f be a g i v e n
T h e n the D i r i c h l e t
function
problem
is to
= 0 in G and u - f E W2(G). an e x t r e m a l
problem,
g E W ( ~ d ) , g _ f E W (G)}. denote
complement
in this
if f and g
A2u : 0 in G,
a regular
be found by s o l v i n g
It is e a s i l y
to
= g on SG.
such that £2u
the o r t h o g o n a l
and
on ~G, then the D i r i c h l e t
arbitrary,
w h i c h we a g a i n
related
A2u = 0.
boundary,
differently.
to find inf{/[AgI 2 dx;
= far Ag dx.
equation,
smooth
n C4(G)
to W 2 ( ~ d )
w h i c h we a s s u m e
of f to D (G),
are c l o s e l y
case w h e n G does not have
but o t h e r w i s e
solution
and
derivative
has to be f o r m u l a t e d
However,
that one c a n n o t
in Lq(K) by f u n c t i o n s
u E CI(G)
uI~ G = f and the n o r m a l
(f,g)
E W
and q < 2, t h e n
for the b i h a r m o n i c
functions,
is to find a f u n c t i o n
The u n i q u e
~l'''''~d
supp ~ c G c, to be d e n s e
set is ( 2 , p ) - s t a b l e
synthesis
If G is a b o u n d e d
case
can
of K.
the D i r i c h l e t
in this
f E ~(G)
of log i/ix I .
a compact
(2,2)-spectral
and b o u n d e d ,
E W q 2'
from e x a m p l e s
then G c
U ~0 + [id ~U ~ i /Sx., l
but the only p o t e n t i a l s
can be a p p r o x i m a t e d
neighborhoods
The
in G c ' ~0
G : {0 < IxI < i] in ~ 2
Similarly,
problem
U ~,
if e v e r y
combinations
with support
0 are m u l t i p l e s
f E L~(K)
if and only
by l i n e a r
seen d i r e c t l y
= X l / ( X ~ + x~)
l a r i t y at
If G is o p e n and b o u n d e d ,
synthesis
in Lq(G)
It is e a s i l y
f(x)
in E 0.
fG'
is f o u n d
of W2(G)
seen that
by p r o j e c t i o n
with
fG is the u n i q u e
solution
formulation.
°2 f - fG E W2(G)
does not
in i t s e l f des-
94
cribe
very
values,
clearly
so a m o r e e x p l i c i t
is d e s i r a b l e . formulated on ~G.
Now,
"fine
the most defined
Dirichlet
in G, U]G c = f]Gc,
ordinary
problem"
case
clearly
However, has
This
Theorem imu
3.3
G c satisfies
the u n i q u e n e s s
to be p r o v e d
(m,2)-spectral
In the case w h e n dimension)
synthesis
problem,
were
given
theorems
[28].)
[30]):
solvable
arbitrary
The
such that A2u
= 0
for p a r t s
is for e x a m p l e
of the
is that
solution
the
is no
if u ~ D 22(G)
the e q u a t i o n
and
Amu
if
= 0.
solved
those
Dirichlet open
problem
for
set G if and o n l y
if
synthesis.
the u n i q u e n e s s
[28],
fine
in the b o u n d e d
~G is a f i n i t e
was
by P o l k i n g
are a m o n g
[30].)
immediate.
(Fuglede
= 0 is u n i q u e l y
of
since u : fG s a t i s f i e s
u I = 0 and Vu I = 0, then u ~ 0. Gc Gc All this is e a s i l y g e n e r a l i z e d to e.g. is now
(See
on Vu is v a c u o u s
zero.
has a s o l u t i o n ,
The f o l l o w i n g
is in terms
leads to the f o r m u l a t i o n
the c o n d i t i o n
(l~2)-capacity
What
who
if d < 4, and thus UlG c is the
This p r o b l e m
obvious.
the trace
Find u E W ~ ( ~ d)
in ~ d .
longer
314],
of the t r a c e of f and Vf
by Fuglede:
for a ( d - 2 ) - m a n i f o l d
the r e q u i r e m e n t s .
[31~
problem
VUlG c = VflG e.
Also
with
and this
given
c c(~d)
restriction.
by S. L. S o b o l e v ,
in t e r m s
boundary
the D i r i c h l e t
way of d e f i n i n g
be given.
that W ~ ( ~ d )
of the b o u n d a r y
problem
natural
and
Vf G take the r i g h t
way of f o r m u l a t i n g
functions,
Let f ~ W ~ ( ~ d)
Note
fG and
This was u n d e r t a k e n
the D i r i c h l e t
of p r e c i s e l y the
in what way
union
problem,
by S o b o l e v
[31;
and the a u t h o r
proved
of s m o o t h m a n i f o l d s
in [22].
and thus the §15].
[22]. (Theorem
spectral
Further
The 3.4
(of
results
following (a)
is in
95
Theorem (a)
3.4:
All m
(b)
closed
F c
: 2,3,...,
F admits
~d
admit
(m,p)-spectral
synthesis
for
if p > d.
(m,p)-spectral
synthesis
for m = 2 , 3 , . . . ,
if
i < p ~ d, a n d {2 n ( d - p ) n:l
(F N B ( x ~ 2 - n ) ) } q - I
= ~ for
(m,p)-q.e.
CI~p
xEF. (Recall subset
the
Kellogg
of F w h e r e
capacity
the
property,
above
which
infinite
i if p > 2 - ~,
says
series
converges
has
the
(l,p)-
zero.)
Theorem spectral
3.5:
A closed
synthesis
(a)
F < ~d
with
for m : 2 , 3 , . . . ,
CI,p(F)
provided
= 0 admits
(m,p)-
either
2p > d
or
C2 (b)
l i m inf 6+0
(F N B ( x , 6 ) ) 'P
(A s l i g h t This replace one
theorem
the
lim
in T h e o r e m Using
hard
G in ~ 2
the
or
(m,p)
q.e.
x
E F.
in
quite
is n e e d e d
if d = 2p.)
satisfactory.
(b) by the d i v e r g e n c e
One w o u l d of a s u m
like
to
similar
to the
theorems
and
the
Kellogg
property
it is not
following.
3.6:
All
closed
F c ~d
admit
if p > m i n ( d / 2 , 2 - 1 / d ) .
problem
(m,p)-spectral
In p a r t i c u l a r ,
for £ m u = 0 is u n i q u e l y
solvable
the
for a l l
synthefine bounded
~3
The p r o o f (See
two
= 1,2,...,
Dirichlet
for
3.4.
Corollary sis,m
modification
is n o t
inf
these
to p r o v e
> 0 6d-2p
of T h e o r e m
3.4 d e p e n d s
on the
following
estimate.
[22].) Lemma
3.7:
0 ~ k ~ m - i.
Suppose Then
that
for a l l
f E wP(~d) m balls
B(x,6)
and that
that
vkfIF
intersect
= 0 for F
96
If(y)I p dy < A6 mp
Ivmf(y)I P dy
S
B(x,6)
if
p > d,
B(x,26)
and If(Y)]P
dy-
C1
(FflB(x,6))
B(x,6)
'P
The tion
proof
of the
u in C O such
theorem
that
is no r e s t r i c t i o n
Ivmf(y)I p dy B(
then
consists
in c o n s t r u c t i n g
0 S w ~ i, w = i on a n e i g h b o r h o o d
to a s s u m e
that
F is c o m p a c t ) ,
and
Then
w has
6d-P/CI,p(F
To
show
theorem
how
under
belongs
in such
fl B(x,6))
this
more
leads
to w P ( F c) and m
a way
in the to the
restrictive
that
Lemma
x
[n:l
a
that
nq-1 : that
that
Let
wn
f.
match
the
series
condition, (These
we are
prove
the
satisfied
in
there
is a s e q u e n c e
{an}~,
an
> 0,
such
that
for
(F N B ( x , 2 - n ) )
~ a
, n
~. a
n
is b o u n d e d
below
if F s a t i s f i e s
for e x a m p l e
a
condition.)
We c l a i m
n
is
E F
(Note
G
(it
3.7.
l,p
cone
F,
approximates
its d e r i v a t i v e s
conditions.
2n(d-P)c and
of
applications.). Suppose
all
(l-w)f
to be c o n s t r u c t e d
factor
most
clearly
a func-
IiwflI m,p
o small.
if i < p - < d .
,26)
f
F admits
E W~,
vkfiF
(m,p)-spectral : 0,
0 ~
k ~ m - i.
= {x;dist(x,F)
< 2-n].
E C O such
0 ~ Wn < - i, w n (x)
G n,_cl
that
iVkwn(X) I ~ A2 kn,
easily
applying
sees
that
the
to vm-kf
B(x,2 -n)
= 0 on
to e s t i m a t e
caBe
(which
k = 0 is e a s i l y
disposed
belongs
one
to W~),
n
Ivm-kfiPdY _< A2-kn a -1 n
let
a function
x E G n I\G --
f
construct
n we
on F.
lemma
for any
integer
= i on G n , u n (x)
The
= 0 a.e.
For any
can e a s i l y
0 ~ k _~ m.
of,
Now,
one
We h a v e
dx for
vmf(x)
Then
i ~ k _ < m.
IiVkwnlPivm-kfiP since
synthesis.
! B(x,
I vmfl p dy. -n+3)
97
But Gn_I\G n can be covered point
in ~ d
belongs
B(xi,2-n+3).
to more
It follows
I
by balls
B(xi,2 -n)
than a
fixed
in such a way that no
number
that
IVken IP Ivm-kflP
dy S A a~l I
Gn_IXG n Since large
IV m f[P dy. Gn_ 4
[~I aq-i
= ~ and a n is b o u n d e d
we can for a r b i t r a r i l y
N q-I i <_ [M a S A.
M find N > H so tha%
w = X M oo
A = A d of balls
-1
We then
set
n
e
Thus,
~ ( CO,
~ : i on Gn,
~ : 0 on GH_ I, 0 S e S i, and
IVk~l
<~ aq-llVk~n n I on Gn\Gn_ I.
Hence
Nf
IvkojlPlvm-kflP
IvkwtPlvm-kfl
dx = [ M
fir d
P dx s
Gn\Gn_ I N -< ~ a q ] M n
IVk~n Ip Ivm-kfl p dx Gn\Gn_ I
N
Z M
-1
Ivmfl p dx G
n-4
N
A Z
IvmfIPdx_
-1
M
GH- 4 This
last
integral
is a r b i t r a r i l y
small,
since
/Ivmfl p dx = 0.
o
To prove
Theorem
3.4 in general,
tion of e to a s i t u a t i o n
where
divergence
of the
"Wiener
[22,
3.2].
Finally,
E c F where function
the W i e n e r
using
independent
Thus
F
f E wP(FC). m
Lemma
GM- 4
the
series", in order
series
following
interest.
there
one has to adapt is no u n i f o r m i t y
which
to a l l o w
converges, lemma
(In fact,
gets
([22],
quite
the c o n s t r u c in the
technical.
for an e x c e p t i o n a l
one first Lemma
has to adapt
5.2),
which
J. R. L. Webb has given
has
See set the some
an i n t e r e s t -
98
ing a p p l i c a t i o n self-contained
in the theory
of non-linear
proof of the lemma(without
PDE.
See [34], where a
the set E), is also
found.) Lemma C
m,p
3.8:
(E) = 0.
such that
Let f E W p, and let E be an arbitrary m
set with
Then for any s > @ there exists a function ~ E W p m
0 < w _< i, ~ = i on a neighborhood
of E,
(l-~)f ( W p N L~
--
II~ofNm,p
m
s.
<
To prove Theorem useless. replaced
Also,
3.5~ where
Cl,p(F)
= 0, Lemma
3.7 is off course
the assumption vm-lfIF = 0 is vacuous.
The lemma
is
by the following:
Lemma
3.9:
0 _< k _< m - 2.
Jl
'
Suppose that f ( WPm and that vkfiF Thenfor all balls
IfiPdy -
B(x,6)
B(x,~)
that
1vm- ifl Pdy+6P
intersect
£
[B(x,26 )
= 0 for F
Ivmfl p d Y l (x,26)
if 2p > d, and
I
iflPdy~A~(m_1) p
I
C2,p(FNB(x,6))
]
B(x,6)
I
m
IPdy +6p
B(x,26)
IVmf[ p B(x,26)
if 2p _< d. As before the theorem is proved by constructing such that strueted
(l-~)f
~ wP(F c) and approximates m
by means
for a n e i g h b o r h o o d
of "smooth truncation" of f.
The integrals
mated using the Whitney d e c o m p o s i t i o n ~, maximal
functions,
derivatives
of the capacitary fivk~iPivm-kflP
potential
dx are esti-
of F c, a Harnack property
and an interpolation
of Theorem
inequality
for
for intermediate
3.4 and 3.5 can also be applied
sets, and then they give sufficient
for example:
This time ~ is con-
from [21].
The proofs
consequently
f.
a function
for Lq-approximation
conditions
by solutions
to open
for stability, of £mu = 0.
and
We have
99
Theorem and
3.10:
A compact
K c ~d
is
(m,p)-stable
for a l l
p ~ d,
f o r p ~ d if {2n(d-P)Cl,p(B(x,2-n)\K)}
q-I
=
n=l for
(m,p)-q.e.
x E ~K.
Different used
in p r o v i n g Theorem
synthesis,
synthesis
and
and
Theorem
(b)
K is
Thus
m,p
in o r d e r
necessary
done for
is to p r o d u c e some
G, w h i c h
One c a n C
m,p
for,
(G\K)
= C
even m,p
the
simplicity
we
[28]).
mp
open
Let
the
not
following
K 0 = ~.
Then
if m p > d.
~ d,
if a n d
only
if C
(G\K)
m,p
=
G. the
conditions in ~ 2 ,
such that
ask
if t h e
(GXK 0) for all
(m,p)-stability
in T h e o r e m all
K @ = ~ and
that
3.11
has
are
to be
~,2(G\K)<
CI,2(G)
are
condition open
G, is not
in the all
in T h e o r e m
general
K c ~d
both case
3.13, necessary when
(m,p)-stable
and
K has if m p > d,
w h e n m = i?
following
presence
one has
are
is easy.
In p a r t i c u l a r ~
The
interior,
conditions
condition.
a K c ~2
interior~
case
(m,p)-spectral
on ~K for k = 1 , 2 , . . . , m .
sufficient
(2,2)-stability
for
in t h e
these
show that
sufficient
as
(m,p)-spectral
(K0) c a d m i t s
(k,p)-q.e.
if K has no
for a l l
say
(K0) c a d m i t s
if
(m,p)-stable
to
if
(m,p)-stable
(m,p)-stable, (G)
by the argument
for k = 1 , 2 , . . . , m .
(Polking
K is a l w a y s
C
not
sufficient
(a)
obtained
= Ck,p(G\K0)
direction,
In fact,
3.13:
are
(m,p)-stable
(k,p)-thick
converse
necessary.
necessary
is
K is
K c is
conditions
1.16.
Ck,p(G\K)
3.12:
and
In the
Theorem
3.11: K
Theorem
all
sufficient
of a n
example interior
choose
shows
that
really
m = p = d = 2.
this
changes
is not the
so if p r~ d.
situation.
For
Thus,
1O0
Theorem
3.14
(2,2)-stable.
([22]):
Thus
There
is a c o m p a c t
the h a r m o n i c
functions
K c ~2
which
is not
on K are not dense
in
L~(K). Proof:
to find a K and a ~ E W.2 2 such that ~ = 0 on K c
We have
but V~(x)
~ 0 on a part
B 0 = ~IxI
e i~, and
Let
of ~K w i t h
positive
discs,
Let
i [- ~, ~] on the xl-axis.
let I be the i n t e r v a l
~Bk~ I be d i s j o i n t
(l,2)-eapacity.
B k = ~Ix-x(k) I e rk) , with x (k)
E I,
and let K = B0\(UIBk). Let R k ~ r k, Xk E C ~ ( ~ +)
Then
such that
r ~ Rk,
0 ~ Xk ~ i,
Ix~(r)I
-~ A / r 2 ( l o g
Set ~k(X) ~0(x)
choose
= ~0(i-~i Every
(Rk)l,
Then
= x 2 near
capacity.
appears
for
to be found.
calculation
~ E W 22 and ~(x)
not c o m p l e t e l y
measure
I, and thus
that
to give
a different
a modified
on I.
Set
~ 0 on K c. set of
~ ~, and thus to I t h r o u g h
capacity
which
concludes
necessary
positive such a of
the proof.
and s u f f i c i e n t
way of m e a s u r i n g E. M. Saak
sets
[29] has
in the case p = 2, d > 2m, but the understood.
so
~ / ~ x 2 = i on a subset
~ ~ W (K0),
in order
(x(k))
to ~K, and the
line p e r p e n d i c u l a r
(m,p)-stability, Using
shows
and dense
Rk/rk )-I < ~.
Lebesgue
Thus
that
such a c o n d i t i o n
is still
A short
~i Rk ~ ~, and c h o o s e
~(log
clearly
But on any
= ~0(x)
It thus
given
so that
has 1 - d i m e n s i o n a l
~K of p o s i t i v e
conditions
E C0(B 0) be such that
Rk/rk)-l.
(rk) I so that
(l,2)-capacity. ~(x)
of I.
let ~0
x E l \ ( u ~ ( i x - x (k) ] ~ R k) b e l o n g s
such points
= 0 for
~ Ar_ (log R k / r k )-I, and
(Ix-x(k) I s R k) are all disjoint,
~k ).
a function
Rk/rk)-i
~k)I 2 dx ~ A ( l o g
the balls
Finally
has
Ix~(r)I
construct
= i for 0 ~ r ~ rk,Xk(r)
= X k ( i x - × ( k ) I~ , and
Now choose
point
Xk(r)
= x 2 in a n e i g h b o r h o o d
fIV2(~0
that
one can easily
situation
101
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J. Polking,
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B. W. Schulze and G. Wildenhain,
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FOURIER
ANALYSIS
CALDERON'S
OF M U L T I L I N E A R
THEOREM, A N D
ANALYSIS
CONVOLUTIONS,
ON
LIPSCHITZ
CURVES
R. R. C o i f m a n Washington University and Y. M e y e r de P a r i s - S u d , C e n t r e
Universit@
The
purpose
of t h i s
methods
of Fourier
are
convolutions,
not
The m a i n
point
transform much
in t h e
same
In the by Fourier introduce As
first
§i.
and
a Fourier
and
real
we
are based
Calder6n
Let
in
is to
of their
indicate
in the a n a l y s i s study
of
Lp
a treatment
generalize
Fourier
the
of t h e s e
Fourier operators,
of t h e
first
In t h e
p ~ 2. commutator
second
half we
curves. algebra
the
on C a l d e r 6 n ' s
that
which
arguments.
multipliers
methods.
Lipschitz
certain
of o p e r a t o r s
some
a multiplier
s o m e of t h e
Cauchy
of o p e r a t o r s integral
result.
We p r e s e h t
analysis
developed
on
studied a proof for t h e
H
Commutator
denote
the H i l b e r t
Hf(x)
a n d let
Af
Calder~n
commutator
is t h e
proved,
:
= A(x)f(x)
=
commutator
by using
later),that
C(f)
transform, f(t) x-t
lim S÷0
be t h e
is t h e
C(f)
A'
for
obtain
study
linear
variable
operators
using
in the
is to d e s c r i b e
commutator.
The
This
in t h e
transform
These
theorem
as
tool
p a r t we p r e s e n t
a consequence
Calder6n,
of h i s
even
presentation
spirit
first
of l e c t u r e s
useful
or not
a powerful
analysis
such curves. by
analysis
o f this
remains
series
d'Orsay
operator
of m u l t i p l i c a t i o n
by
A.
The
operator
i I lim ~ ] e+o
Ix-tl>c
between
d H ~
complex
dt,
Ix-t I>S
variable
is a b o u n d e d
A(x)-A(t) f(t)dt (x_t) 2 "
and
A.
methods
operator
A.
(which on
L p, i
P.
Calder6n
will
be
[i]
explained
e p e ~ , whenever
( L~ Like
the Hilbert
be a b a s i c Calder6n
"building
was
transform block"
led to c o n s i d e r
the
Calder6n
for various it in h i s
commutator
operators precise
turns
out
in a n a l y s i s .
calculus
of p s e u d o -
to
105
differential naturally
operators
in v a r i o u s
Before should
other
studying
this
dimensions.
contexts, see operator
It a l s o
[2],
in
[3],
some
appears
quite
[5].
detail,
various
comments
be made:
a)
The m a i n
estimate, can
in h i g h e r
say,
extend b)
difficulty
in
L2(R).
it to o t h e r
Any
estimate
in s t u d y i n g Once
classes on
such
C
is in o b t a i n i n g
an e s t i m a t e
of f u n c t i o n s
L2(~)
of the
by
some
is o b t a i n e d ,
"real
variable
one methods".
form
llC(f)II 2 -~ N(A)Ilfl[ 2 , where
N(A)
is a c o n s t a n t
i0 I0
derivatives
to be
equivalent
of
A,
depending,
can e a s i l y
say, on the
be s h o w n
L~ norm
of the
(by d i l a t i n g
the
first
functions)
to
ilc(f)li~ ~ c0rIA'II~llfll2 We are that,
thus
forced
by c o n t r a s t ,
to obtain; c) stems
the
The
first
convolution d)
here
difficulty
to
is
with
is the
is a b i l i n e a r
form
Fourier
multiplier
We are now of th~
=
ready
(i.i)
L2
( L
for
of
C(f)
linearly
translations
is not a
convolution
on
a
of a,f.
and
'
of f and ~ is a d i s t r i b u t i o n . symbol
C.
T~e
to a n a l y s e
C
to c o n s i d e r
the
of idea
C
as a b i l i n e a r is to use
by b r e a k i n g
~
(o
operator).
a representation
into
simpler
objects. It is c o n v e n i e n t [A,H]f'
writing
[A,H]f'(x)
commutator
C(a,f)
- H(af)
it as
=
c
II
eiX([+~)(sgn([+~)-sgn~)
f;
Formally
of w r i t i n g
fl e i X ( ~ + ~ ) ~ ( ~ ' ~ ) a ( ~ ) f ( ~ ) d ~ d ~
or the
( L~
analysis.
as a b i l i n e a r
to the p o s s i b i l i t y
transform
A'
kernel
Fourier
depends
trivial
is not
(x-y)2 the
are
~.)
estimate
since
(Observe
A"
on
A(x)-A(y)
to use
simultaneous
to s t u d y
and
A'
involving
how
a = A'
"equivalent"
C(a,f)(x)
f
an
kernel
to t h i n k
with
estimate
Second,
C = C(a,f)
it c o m m u t e s
(i.i)
the
x-y it is not c l e a r
In fact,
observation
where
that A'(y)
It is c o n v e n i e n t
moreover,
estimates
is a g l o b a l
in o b t a i n i n g
fact
"close" kernel
operator.
an e s t i m a t e
on the c i r c l e
problem
f r o m the
sufficiently
this
to c o n s i d e r
~ f(~)a(~)d~d~.
=
106
Here, ~(~,~) is q u i t e we
can
easy
:
to a n a l y Z e .
represent
this
!
In
(sgn(~+~)-sgn~)
fact, s i n c e
quantity
[ (We
used
the
Combining
formula
the
two
~
identities
! 1
whenever
o ~ 0,
iv
-~
l+y
f : c] e -isY _~
e -s
I~I
as
we
dy l+y2
2
, s ~ 0).
obtain
(sgn(<+~)_sgnt)l~l-iv~(~)i~]iv~(t) dtd~dy 2 l+y
:
where
leI=~,S
iy = [a,H]f
:
At
stage
this
If
f
( L2
we
so
dy l+y2
[a_y,H]fy
= l~l-1~S<~),
-y
a(Hf)
f~ ].~
c
- H(af)
:
can
a few
is
make
f
and
] a(x)-a(y) _oo x-y
, with
f(y)dy.
observations:
the
same
norm.
Unfortunately
if
Y
a
( L
are
a
is
Y f o r c e d to
to a s s u m e , L4
(I~I iY
[a_y,H]fy some
in g e n e r a l study
say~
[a,H]f
f ( L4~
being
a
for
in This
( L 4.
see
by r e a l
variable
belongs
( B.M.O.
In t h i s
L 2, w i t h
L2
yields
estimate
an
norm
to
B.M.O.
Another
case
multiplier
IIEA,H d ] f l l s which
and a
a Harcinkiewicz
is t h e n
0 < ~ < i.
unbounded
fy, on
not of
thus
possibility
ay
L p)
are
also
is in
and
exceeding the
We
cIYI 6
for
form
-< ellall~llfll 4
techniques
can
be
extended
[a,H]f
aHf
- H(af)
for
other
indices~
[4]. We
with
now
return
f ~ L2,
a
to
E B.M.O.
This
:
operator
is
in f a c t
with (1.2)
II[a,H]fI[ 2 _< cIIaIIB.M.o. IIfll2.
bounded
on
L2
107
(1.2)
can be p r o v e d
"good
~
inequalities",
inequalitiesjsee Perhaps B.M.O.
in v a r i o u s
This a p p r o a c h =
either
or as a c o r o l l a r y
directly
by the
of the w e i g h t e d
so c a l l e d norm
[5].
the s i m p l e s t
a
ways,
proof
permits
b 1 + H(b 2)
of
(1.2)
uses the d u a l i t y
us to w r i t e
a E B.M.O.
Ilblll ~ + IIb2H ~
with
_<
between
H I and
as
cllallB.M.O.
and so [a,H]f We now
e m p l o y the
Using (See
[6])
Theorem
- H(Hb2f)
the b o u n d e d n e s s
on
I.
Calder6n's
H2 <_ c ]IA' rlB.M.O.[IfPr2,
equation,
(1.3)
h'
is i n t i m a t e l y
related
to the a d j o i n t
f,g h
and
g
( L2 in
of b o t h terms.
obtain
Jl [ A , H ] f '
f
L2
= b2f + H ( H f b 2)
llayI]B.M.O " _~ C (Iyl+l) (I/2)+E ,re > 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
f'g,
to the C a l d e r 6 n
assumed f(x)
=
equals
=
:
are
functions 0
for
transforms
~_~
in
H2(~)
(i.e.,
x < 0) and we look for
we find that
g((-n)nf(n)dn
108
:
h(x)
where
(Myf)A(~) Since
c
M _~
( -Y
gMf)
dy l+y 2
:niYf(n).
M
preserves
L2
and
HI
(see
[4]) with norm
Y e(1+iyl) (I/2)+~ we have the desired does
not
in general
have
a solution
result. in
Observe
L I when
that the e q u a t i o n
f and g are m e r e l y
in
L2 " We will
return
to this
equation
later
using
it's
conformal
in-
variance.
§2.
CarlesonMeasures, The p r e c e d i n g
study
of
bilinear
took
(~,~).
to c o m m u t a t o r s
and
operators
we would
defined
like to have
operators.
blocks
the analog
To obtain
certain
basic
we proceed
be two test
functions
such that
$,~
~.
i = ~ ¢(x/t)
:
I
(f*~t)(a*~t)
~
co
and
m
~ L (IR).
We have
II. llB(a'f)II2
depends
involves
as in
which the
to the one d i m e n s i o n a l
0
c
~.
case.)
0o
where
symbols
operators
formulation
our a t t e n t i o n
for the general
B(a,f)
Theorem
our a n a l y s i s
of a m u l t i p l i e r t h e o r e m
such results
(an e q u i v a l e n t
we r e s t r i c t [5]
0 ~ supp
(2.1)
~t(x)
to
as a
theory).
(see ¢,¢
by r e a s o n a b l e
We define
where
~
or even to general
For s i m p l i c i t y
support
analysis of
to extend
as b u i l d i n g
We let
and Mellin
the h o m o g e n e i t y
operators,
case by c o n s t r u c t i n g
case
Fourier
to be able
Littlewood-Paley
biline a r
both
into account
It is d e s i r a b l e
for m u l t i l i n e a r the m u l t i p l i e r
using
Operators
of p s e u d o d i f f e r e n t i a l
(multilinear)
In other w o r d s
serve
method
the c o m m u t a t o r
functi o n
Commutators, and M u l t i l i n e a r
on
¢,9,m
<- cllall B . M . O .
alone.
IrfllL2,
dt
have
compact
109
We w i l l (varying
see l a t e r t h a t a s u p e r p o s i t i o n
~,~,m)
yields
various
bilinear
of
the o p e r a t o r s
operators,
B
and in p a r t i c u -
lar the C a l d e r ~ n c o m m u t a t o r . To p r o v e
the t h e o r e m ,
Carleson measures. ~+2
= {(x,t)
Recall
E ~ 2 : t > 0}
S = I x ( 0 , 1 1 I) (here
that a m e a s u p e
cIll
with
c
geometric
ous f u n c t i o n
argument 2 on IR+
f(x,t)
facts
and
III
if for e a c h its
independent
shows t h a t
concerning
on
of
measure)
square we h a v e
I.
for any lower
semi-continu-
leo
II If(x,t)I P
(2.2)
~
is a C a r l e s o n m e a s u r e
I is an i n t e r v a l
v(S) ~ A simple
we need a few d i m p l e
d~ -< Cp J-~
2 ZR+ where
N(f)(x)
function,
see
=
sup
[8].
We a l s o n e e d the
(2.3).
Lemma
If(y,t)I
is the n o n t a n g e n t i a l
Let
~
following
lemmas.
be d e f i n e d
as above.
0 if~tl 2 Tdt dx This
is i m m e d i a t e
Lemma
(2.4).
as a b o v e
d~(x,t)
measure
then
dx.
a ( B.M.O. ,
i~t,a(x) i2 dxdtt
with
2
v(s) ~ cllallB.Z.O, lZl see
f E L2
theorem.
and
=
If
: e _~If(x)I2
by P l a n c h e r e l ' s
With
is a C a r l e s o n
maximal
Ix-yI
and
o
depending
o n l y on
4,
[6]. This
geometric
Proof
is an i m m e d i a t e
consequence
of
(2.3)
combined
with simple
arguments.
of T h e o r e m
0 E supp $0
II.
but
We
start by w r i t i n g
0 ~ supp 30+
support
of
$0
~0' ~0
in
c~(~).
supp ~
to be s u f f i c i e n t l y
and
small);
~ = ~0 + 40 0 ~ Supp ~0
where (just c h o o s e the
here we can take
110
Correspondingly we have B(a,f)
=
B0(a,f) + Bl(a,f
where B0(a,f)
=
and
f*$0,t)(a*@t
, m(t)dt
eo
Bl(a,f)
=
10(f:'~0,t)(a*$t
m(~---_£)dt.
To study B 0 we introduce an even test function @I whose Fourier transform @i is i on supp @0 + supp @ and such that 0 ~ supp ~I" Clearly Pl,t*E(f*$0,t)(a*@t)] since
@l,t ~ i
=
(f*$0,t)(a*@t),
on supp((f,@0,t)(a,@t)) A
We can rewrite
B0(a,f)
B0(a,f ) To estimate the
=
as f
0
@l,t,[(f,~0,t)(a,@t) ] m(~) dt.
L2
norm of this term it suffices to estimate
I~
~B0(a,f)dx
=
I
for
g ( L 2.
We have f~ =
f~
J-Jo (g*@i't(x))(f*$o't(x)(a*$t(x))
-~ Iolg*$1't(x)I2 dxdtt i/2
-~
IIg112
If*(x)12dx
m(t)dxdt
J-~]Or~ r~If*$0,t(x)I21a*@t(x)]2
dxdtt i12
.M.o. ~ lla11B.M.o.Ilgl1211f112'
_oo
Where the first integral is estimated by Lemma (2.3), while the second integral is estimated by Lemma (2.4) and (2.2) using the fact that (Here, f*(x) is the Hardy-Littlewood maximal N(f*@o, t ) ~ ef*(x). function, and la,$tl2 dxdtt is a Carleson measure). The estimate for B I is similar. We choose el even, such that ^ ~i ~ i on supp ~0 + supp ~ and rewrite Bl(a,f) as
Bl(a'f)
=
I ~l,t*((f*@0, t)(a*@t)) m(t)dtt
111 t
We again
estimate
I
=
~gBl(a,f)dx J
= ff
as before
dx__mtdtt
by i n t e r c h a n g i n g
A variation
on this
the roles theorem
B(a,f)
~
S
:
of
f
and
g.
involves dt -~-
(f*~t)(ae~t)m(t)
0
where
a
is assumed
We have
the
to be bounded.
following
(2.5)
estimate:
I I B ( a , f ) l T 2 <- cllalloollfl! 2.
~ : 40 + ~0 with is used to deal with II
This estimate is obtained as before. We write 0 ~ supp ~0 + supp {.. The technique of T h e o r e m
I
0 (f*~t)(a*~0't)m(t)
dt
~
"
The term oo
I can be r e w r i t t e n
@'
For
is 1
g ( L2
t
as l(x)
where
(f,@t) (a,~0,t)m (t) d/_t
0
:
on supp
we make
@t:.[ (f,~t) (a,~0,t)]
J0
50 + supp
@
and
dt t
0 ~ supp
@'.
the estimate, co
I g(x)i(x)d x
E
dx
Ig*~l
:
[1o(g,~@%).(f,¢t)(a,e0,t)m(t
If**t
2
0 and ( 2 . 5 )
Theorem Define
Prall~ ~ cllfll211gll2llall~,
0
follows.
As an a p p l i c a t i o n Calder6n
]2 --
) d~_tt
commutator
III.
Let
the o p e r a t o r
we p r o v e
the
following
generalization
of
the
estimate,
m(~) H
satisfy
Im(k)(~)l
by Hf
=
(mf) V
_~ Ckl~l -k , k : 0,1,2 . . . . .
112
Then the commutator of order i, i.e. ,
between
M
II[M,A]f'll
Proof.
and m~itiplication
by
A
is smoothing
2 ~ allA'jloollfll 2.
We write a s before
[M,A](f')(x)
=
ff
c
eiX(~+~)(m(<+e)
- m([))
and introduce a "partition" of unity permitting relative size of ~ and e. First we write
~
f(~)a(~)d~d<~
the comparison
where $0(t) is supported in [tl<10 and ~0(t) in split the corresponding integrals into I + J where I
:
ff eiX([+~)m([+~)~0(~)
If eiX(~+~)m(<)$0(~)
=
II
-
:
if e ix(~+~) (m([+~)
Itl>s,
of the
and then
[ f([)~(~)d[d~
~ f(<)a(~)d~d~
12
and J
To study I I (or 12 ) we choose in l
i
=
]
~
~2(t ) d_~t = 0 t
m(<)) ~ ~0([)f([)a(~)d
-
even in
r¢°
1
0
@2(Gt)
supported
dt t "
2(~t)
Clearly, for any (since
C~(~),
~l(t) in C0(~) which is equal to i for ItI<30 I(I
=
~ ~2(~t)$2((t)£(()£(~)d(d~ JOJJ
We now observe
that the function
80(rs-) rs ~(r)6(s)
dt
as
113 2
is in
C0(]R )I and can be represented II
(where
q
e
i(su+rv
as
)~(u,v)dudv
is its Fourier transform).
introducing
this in the expression
Taking
for
II
s = ~t, r = at
and
we get
ii = lq(!u,v)dudv I IIeiX(~+~)m(~+~)~(at)~l(~t)ei(~tv+~tu)~(~)~(~)d~d~dt 0 t f =
where
J~2
n(u,v)M(B
u,v
Mf = (mf) v^
(a,f))dudv
'
and Bu,v(a,f )
with
~u(~)
=
=
~ u ~ v dt f 0 f"~t a"~t t-
~l(~)ei
=
~(~)ei~V.
Using Theorem II and the rapid decay of estimate
for
As for
II J
and
12
(12
we take ~
with
~2(~t) dt --~- .
= Jo
have
~O(~)@2(~t) where
we obtain the desired
as before and write
l We a g a i n
q
is studied by the same argument).
$(u)
is
J
f[ff
=
a(~,~,t)
Observe that
i
for
=
lul<30.
@O(~)~2(~t)*2(et) We get
eiX(6+~)a(~,~,t)~(~t)$(~t)f(~)~(~)d~d ~ -~
=
(m(~+~)
~k+j [s - ~ r D
m(~))
@O(~)~(~t)~(~t).
a(~,~,t) I ~ Ck, j
uniformly in t, and that this expression has support in Thus using the Fourier inversion formula we can write s r a(~,~,t)
=
II IR 2
e
i(su+rv)
q(t,u,v)dudv
IsI+Irl~4@.
114
q(t,u,v)
with
_<
s = It , r = ~t
c
Substituting
(l+lu 2+lv12)N
in this
formula
by
we get J=
l~
dudv
(a f)
~2n (l+lulhlvl2)N u,v where Bu'v(a'f) is e s t i m a t e d We theorem Theorem
=
J0
using
end
this
on
IRn
IV.
(f,~)(a,~)n(t,u,v)(l+lui2+ivl2)N dtt
(2.5).
section which
Let
by quoting
can easily
~(D) (a,f)
a general
be r e d u c e d Ir
:
J
l
IRn
ix(~+~)~(~,
e
]
bilinear
multiplier
to T h e o r e m
III.
~)f(~)a(~)d~d~
~9n
where
la~(~m)l
~ Cm~q(l~l+lml)-Ip1-1ql,(~,m) ~
then
[lo(n)(a,f)l! 2 < clia11~llfil2 Moreover,
if
a(0,~
~ 0
then
~(D)(a,f)ll 2 <_ cilallB.M.0.1ifli2, see
[5].
§3.
F o u r i e r A n a l y s l s on L i p s c h i t z o__n the C a u c h y I n t e g r a l We n o w w o u l d
on L i p s c h i t z
like
curves.
of p s e u d o d i f f e r e n t i a l a proof As
ii~'ii~ s first
of
Calder6n's
in C a l d e r ~ n ' s 60
that
H~'II~).
We
is d e n s e
in
(6 0 9'
such
operators theorem
small
study
(L2(F)
= {f(z)
the
curves
A(F)
If(x+iy)I
dx
with
for m i n
that < cs
We w i l l
also with
simplicity
will
transform
a new algebra
connection
For
is i d e n t i f i e d
holomorphic
obtain
F : {y:~(x)]
estimates
a class
Theorem
of a Fourier
curves.
constant).
(our f i n a l
Calder6n's
we will
indicating
paper we
and
the n o t i o n
on s u c h
by introducing
L2(F).
A(F)
introduce
As a c o n s e q u e n c e
is some (C0(~)
start
Definition. and
to
Curves
depend
sketch §i.
where we a s s u m e only
of f u n c t i o n s
on which
L2(~)).
~ - s < Im z < m a x
~ +
115
in this r e g i o n } ,
i.e.,
A(F)
in some o p e n h o r i z o n t a l h o r i z o n t a l line. L e m m a (3.1). A(F)
is the
strip c o n t a i n i n g
is d e n s e
in
0
=
By a~alytie for a l m o s t
go
=
continuation all
F, w h i c h are
holomorphic in
L2
on each
L2(F).
In f a c t ~ a s s u m i n g that g is f dz ] g0 (z) z-z 0 '
space of f u n c t i o n s
orthogonal i ~
g(z)
to
and
this r e m a i n s
A(F),
we have
IIm z01>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 <
inequality
the
Cauchy
as
will
certain
on
curves
m
some
B <
60
is n e c e s s a r y
with
a given
verifying
A simple
in
L2
integral
to be v a l i d problem
above
a bounded
if we r e q u i r e
Lipschitz
this
application
constant.
hypothesis of
that
Theorem
is V gives
~ ~ 0
The
open
m(t)
ll~'II~ < 6 0 , is s a t i s f i e d .
multiplier
Mf(z)
known
BIRe zlfor
condition
~ > @ 0
L2(F)
domain
condition,
for all
simplest
on
to the
=
i 2wi
for
this
seen
results
operator
on L i p s c h i t z
curves
II~'II~ ~
for
if
for a g e n e r a l
be
[ f(~) JF ~ - ( z + i s )
lim s~0
~0
Lipschitz
of C. K e n i g
is C a l d e r d n ' s [3].
some
This
small
curve.
to be a c o n s e q u e n c e
d~
of
The
inequality ~0 > 0.
theorem
Calder6n's
concerning
Hardy
theorem
is
It is an stated
theorem,
Spaces
for
as w e l l
on L i p s c h i t z
domains. We n o w minor proof
follows The
the
idea
Hilbert
~'(x)
sketch aproof
changes
= a(x)
(designed Calderdn's
paper
is to v i e w
the
transform. is in
simplicity
estimate
here,
depending
on
=
theorem.
it to the
Except
analysis
of
for §i)
some the
[3].
Cauchy
Letting L ~, we
C~(f)(x) (For
of C a l d e r d n ' s to r e l a t e
integral
z~(x)
study -i 2~i
~ ( CO, and
= the
lim
x + il~(x), variation
of
where
of the
operators
~ f (t)dzl(t) ] zl(x)-zl(t)-is"
we are
II~'II~ alone).
as a p e r t u r b a t i o n
only
interested
in an ~ prior~
117
Calder6n's
idea is to show that
(3.5)
~Xx-xlicxil _< clicxll 2
where
IICAII
universal
is the
L2
operator
n o r m of
and
CA
c
is a fixed
constant.
Integrating
this
inequality
one o b t a i n s
H c o II llcxll-s z-xcF~pT which gives
is b o u n d e d
as long as
the t h e o r e m To o b t a i n
a-~ C A
where
f0
rewrite
small
Lipschitz
e(x)-m(t)
c lim E+0
if(t)
To e s t i m a t e Calder6n
for s u f f i c i e n t l y
is s u f f i c i e n t l y
small.
This
constants.
(3.5) we c a l c u l a t e
:
=
A ~ 60
2 f(t)dzA(t)
+ CA(f0)
(zX(x)-zA(t)-is)
~'(t) l+iXm(t)
the
first term,which
commutator,
we can use the
looks
"curve
formally Fourier"
like the transform
to
it as F(z)
:
c if e i Z ( a + t ) t x ( t ) - ( t + ~ ) X ( t + ~ ) ~
f(t)a(~)dtda
where -i~z(x) a(~)
=
I e
, ~ (x)dz(x)
and
x(t)
=
J
~i
t > 0
t < O.
We n e e d to e s t i m a t e I
f I g(z)F(z)dz rA
=
for
g ~ L2(FA).
J
Using
(3.4)
we o b t a i n
I where
G(e)
We now w r i t e
:
:
I
c [ g(~-t)f(t) f = f+ + f
G(z)9'(z)dz F tx(t)-(t-~)X(t-~)~
dt
.
,g = _ ~+ + g
where
f+ We have the
=
two t r i v i a l
first the
CA(f) cases
integral
has
Fourier
corresponding
reduces
to
transform
fx.
to
and
g_,f+
g_(z)f+(z).
In the
g+,f
.
In
s e c o n d the
118
integral
is
8.
The
case
g+, f+
h(~) that
i g^ + ( ~ - t ) f + ( t )
:
is, h(z)
This
is C a l d e r 6 n ' s
case
-,The
f+,g+.
gives
tz ] f'(u)g+(u)du.+
=
equation
~ t dt,
for
functions
holomorphic
above
F.
The
is similar. function
We
just
h
above
observe
of v a r i a b l e s , i . e . ~
is e a s i l y
that
the
equation
estimated
in terms
is i n v a r i a n t
under
of changes
if =
h ~
T. f+g+
and H then
H'
=
onto
the
region
ever
the
estimate
If we
=
ho0,
F
:
f+o0,
be
the c o n f o r m a l
G
:
g+o~,
F'G. let
@
above
follows
(see
[ 7 ])
find
from
our
map
the
from
the
equation
upper
half
plane
studied
in
§i.
lacx)l
I
C×)Id
How-
is
e j Ir(x)j
easily and
we
we n e e d
j IH(x)lj~'(x)ld;:-< This
F
the
I
(×)ldx
fact
treatment
of
that the
J~'(x) J
equation.
is an
A2
weight
Altogether
we
find
IzL ~ ~llf±ll2jlg±H 2 .~ ~llcxll211fll21rgll2. Consequently
a~-Ilcx and
so
(3.5),
We are n 0 < 8 0 < ~, M+
+ M_
s II
and,
cxil ~ c(JICxpI2 + IlCxlt) < cllCxll 2
hence,
Calder6n's
theorem,
now ready
to p r o v e
Theorem
and
that
operator
claim
where
M+
the
corresponds
is proved.
V.
We w r i t e M
can
0 0 = tanB,
be r e a l i z e d
to the r e s t r i c t i o n
of
m
to
as t>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
(3.6)
Observe decay
of
follows
continuation also
that
proving
t r e a t m e n t ) we adaptation
basic
space
H2
Keiig
Lemma
Lemma
by C a u o h y ' s
the
should
of the
by
are
by parts
- 00 < arg
from
the
analytic
continua-
z < 80 + ~.
argument
gives
z < 80 + ~ - a 0.
(3.4)
whenever
Im z > 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 .<-Cfo Ifq2 If+(t+i~(t)+i~ ~)'(t-x)2 +
q I2
dtdxdq
C' IO I n]f'(t+i~(t)+iq)]2dtdn.+
(3.7)
we find
NH+(f) II2 -<- Clif+ll 2 (This result
is Valid without
the restriction
II~'II~ s ~0).
121
Using
Calder6n's
theorem,assuming
II~'II~ ~ 60 , we
IIM+(f)ll and
this We
completes conclude
the
proof
s Cllfll 2,
of T h e o r e m
by o b s e r v i n g
get
that
V.
the
same
proof
gives
the
follow-
ing r e s u l t . T h e o r e m VI
Let
(V 0 is a f i x e d such
K(z,6)
wedge
be h o l o m o r p h i c
s.t.
z + V0
lies
z,[
for
[ ~ V0 + z
F
for
all
above
z)
and
that <
C
IK(z,e)l_ T ~ T (V I ~ V~ If
in
is a l a r g e r
II~'II~ ~ 60
for
% ~ VI
+ z
wedge).
then I
k(z,~)f(~)d~ F
is b o u n d e d
on
L2(F). z+V 0
~+ z
A natural
example
is
k+(z)
= ~
I
~ - ~ < argz
where
<
3~
7"
z
A combination
of this
with < T~
37 2 < argz
determination
the k e r n e l gives
the
corresponding boundedness
to the L 2 of the on
operator, r
M~(f) It is a s i m p l e that
these
exercise
operators
We c o n c l u d e ~I
admit
permits apply
natural
one
them
using
are
also
by o b s e r v i n g extensions
to d e d u c e to v a r i o u s
the
f(~)
dy.
: ]r I~-~Il+i~ the
Calder6n
bounded that to
the
~n.
corresponding
questions
on
Zygmund L p,
operators The
studied
so c a l l e d
estimate
in P.D.E,
theory
see
to
show
i < p < ~.
on [2],
so far
rotation L P ( ~ n) [3],
[8].
in
method and
122
REFERENCES [i].
Calder6n,
A. P.,"Commutators
Proc.
Nat,
Acad.
Proc.
Symp.
[2].
, "Algebras Pure Math.
[3].
53 (1965)
of singular I0 (1966)
~'Cauchy integrals operators",Proc.
[4].
of singular
Sei. U.S.A.
Coifman,
Nat. Acad.
singular
operators",
integral
operators",
18-55.
on Lipschitz
Sci. U.S.A.
R. and Meyer Y.,"On commutators
and bilinear
integral
1092-1099.
integrals",Trams.
curves and related
74 (1977)
1324-1327.
of singular A.M.S.
integrals
212 (1975)
315-
331. [5].
, "Au dela des operateurs entiels",
[6].
Fefferman, Acta.
[7].
Asterisque
Kenig,
57 (1978).
C. and Stein,
Math.
129
pseudo-differ-
(1972)
E.M.,"H p spaces of several variables",
137-193.
C.E.,"H p spaces
of Lipschitz
domains",Thesis, U. of Chicago
1977. [8].
Stein,
E.M.,
of functions,
Singular
integrals
Princeton
Univ.
and d i f f e r e n t i a b i l i t y
Press,
Princeton,
N.J.
properties (1970).
THE
COMPLEX
METHOD
ACTING
FOR
INTERPOLATION
ON F A M I L I E S
OF O P E R A T O R S
OF B A N A C H
SPACES
by R.R. C o i f m a n , Washington
R° R o c h b e r g , and G° W e i s s (1) U n i v e r s i t y , St. Louis
M. C w i k e l I n s t i t u t e of T e c h n o l o g y
Israel
Y. S a g h e r U n i v e r s i t y of I l l i n o i s at C h i c a g o C i r c l e
§i.
Introduction It has
ied
not
been
in a n a l y s i s
tions, case
are
z ~ Tz,
that
the
sufficiently particular
defined
emohasized
values
domain
in a s m o o t h
way.
of
many
of a n a l y t i c
on a d o m a i n
"natural"
that
D c {.
T
operators
operator-valued
Moreover,
is a B a n a c h
Fourier Izl
z
transform.
~ i
and
Re
To
we t h e n
define
see this
z ~ 0}.
i p(z)
=
This
For
space
B
consider
the var-
for e x a m p l e ,
the
domain
for the
D = {z ( {:
let
and
by d e s c r i b i n g
T
that Z
is true,
z ( D
Re{l+--~}
func-
it is o f t e n
Z
ies w i t h
stud-
i q(z) its
Re {~ }z
:
action
;
on the H e r m i t e
polynomials
Z
{H }: n
T
It turns
out
that
T
z
: H
~ znH
n
LP(Z)(d~)
maps
n
.
into
Lq(Z)(db)
with
operator
Z
norm
~(z)
= i
for
where
z ( D,
d~(×)
~
is the m e a s u r e
I :
-
-
-x2/2 e
on
~
defined
by
dx.
g2~ The
functions
system
L2OR,dx). fore, can
(x)e -x2/2
n
vectors
n = 0,1,2,...,
of the
The c o r r e s p o n d i n g
that,
identify
variable = I
H
of p r o p e r
by a p p r o p r i a t e F
proper
form
a complete
transform,
values
multiplications
are
F,
.n
l
by the
orthogonal
acting
on
It follows, function
there-
e -x2/2,
and
argument,
is e q u i v a l e n t
(1)This r e s e a r c h and N S F g r a n t
Fourier
we
T.. In fact, B e c k n e r [I] s h o w e d by a c h a n g e of l that w h e n z = iy, 0 S y S i, the fact that ~(iy) to his
was MCS
remarkable
sharp
s u p p o r t e d in part 76-05789-A01.
inequality
by NSF
grant
MCS
75-02411-A03
124
(1 •l)
IrFfllq -< Aprlfllp, p : p(iy) ,
where
There tors.
are
The
other
fractional
semi-simple having
q = q(iy)
many
Lie
features
In v i e w
integrals
groups
the c o r r e s p o n d i n g
families
is r e a s o n a b l e
to e x p e c t on the
deed, the
the
ease.
theory The
spaces
to the
analytic
by
its v a l u e s
that
between
spaees).
The
theory
two
given
origin
Thorin
involving
(X,~)
with
that
described such
spaces
such
of
irreducible
analytic
of
D.
families
of l o o k i n g
representations
families
of
of o p e r a t o r s
above.
families
of o p e r a t o r s ,
they
from
family We
of o p e r a -
map
and
is c o m p l e t e l y
shall
see that
at this
it
determined
this
property
and
into,
is,
in-
is in terms of
of o p e r a t o r s . interpolation
Banach
of this
operators
values
each
way
of
ones
linear
boundary
Another
the
known
nature
of
of i n t e r p o l a t i o n
current
and
are well
similar
of the
A S = p i/p/ql/q • P of such a n a l y t i c
and
examples
involves
spaces
theory
acting
(the
is the
on
intermediate
"end-point" celebrated
LP-spaces
are m e a s u r a b l e
the
or
"boundary"
theorem
on a m e a s u r e
functions
of R i e s z space
on a m e a s u r e
space
(Y,v): Suppose having
T
is a l i n e a r
operator
norm
operator
M. < ~, ]
i
l-t
t + -Pl
-
P T
is w e l l
with
P0
defined
operator
on
strip
--
LP(x,b)
if
l-t
=
-
-
and maps
the
into
LqJ(Y,v)
0 S t ~ i,
+
qo
q
Ht This
i
and
LPJ(x,~) Then
-
t -
ql
latter
into
Lq(Y,v)
norm
( i. 2 ) result
considered
mapping
j = 0,I.
has
analytic
{z ~ ~ :
been
extended
families
0 ~ Re
_<
. l-t.t IvI0 ivlI . in m a n y
of l i n e a r
z ~ i}.
He
directions.
operators
showed
that
E.M.
{Tz}
Stein
defined
appropriate
[6]
on the
boundedness
qj assumptions implied Lq(Y,v) an
of
Tj+iy
mapping
the b o u n d e d n e s s (p
and
inequality
q
there
theorem.
The
was
related
an
principal
general
Banach
spaces
of the
operator
T.
was
the
LPJ(x,~)
Tt
corresponding
published
vestigators
of
to to
t
goal
More
Shortly
Lp
explicitly, one:
the
Suppose
T
he
Stein's
into obtained paper
of the
Riesz-Thorin
was
include
as the
problem
j = 0,i,
LP(x,~)
Moreover,
extensions spaces)
(Y,v),
after
of e x t e n s i o n s
of t h e s e
just
following
L from
as above).
(1.2).
"explosion"
(not
into
as an o p e r a t o r
to
domains
posed
by
is d e f i n e d
was
more
and r a n g e s several on two
in-
125
Banach
spaces
spaces
CO
B0
and
B t = [B0,BI] t T
maps
Bt
and CI,
can
and
(like
the
into
an
ators,
above,
8D
all
them boundedly
construct for
into
"intermediate
0 ~ t ~ i,
these
B<,
of a d o m a i n
spaces"
Bz,
defined
results
intention
two
Banach
spaces"
in such
a way
on
D.
z ( D,
The
the
that
construction
dealing
will
with
<
of
to c o n s t r u c t
a way
families
"endpoint" a situation
points
purpose
in such
for a n a l y t i c
two
we are
with
It is our
for each
with
to c o n s i d e r
where
associated
D ccn.
theory"
dealt
here
in the b e g i n n i n g )
spaces,
"interpolation
{T },
then
It is our
of B a n a c h
"intermediate
it m a p s
Ct ?
spaces.
boundary
obtain
one
one we d e s c r i b e d
a continuum the
and
C t = [C0,Cl]t,
As we m e n t i o n e d or " b o u n d a r y "
BI
that
of l i n e a r
be b a s e d
we oper-
on an
Z
extension
of the
This tion
presentation
of the r e s u l t s
Taibleson cerning
§2.
and
this
shall
obtained
of A.P.
CalderSn
of a c o n s i d e r a b l y
in [3].
who m a d e
of o p e r a t o r s
beginning
consider
of the Let
consists
E. W i l s o n
Interpolation
also,
method"
We w i s h
several
the
general
of E.M.
and
operator T z into m e a s u r a b l e
suppose
mapping
(X,~)
functions
assume
that
z ~ ]
(Tzf)g
dv
functions
the n o r m The
simple.
of
Suppose,
on
T
D
that
simple
values
as an o p e r a t o r
z interpolation
result
for
suggestions
in
from
in q u e s t i o n
intermediate
M.
con-
that
[0,i].
We
will
follow
z ( D
there
on a m e a s u r e
space
function
LP(Z)(x,~)
(and,
LP-spaces.
each
functions
I p(z)
spaces
theorem
only
on a m e a s u r e
is an a n a l y t i c further,
with
of
involves
Y monic
descrip-
our c o l l e g u e s
helpful
Riesz-Thorin
which
a linear
g
detailed
LP-spaces
of the
Stein)
be a d o m a i n
on
space
and
more
to t h a n k
very
construction
a generalization
theorem
D c ~
acting
exists
f
[2].
paper.
Before we
"complex
(Y,v).
on
D,
for
We each
and
I q(z)
are h a r -
let
~(z)
denote
into from
Lq(Z)(Y,v). the
following
theorem:
Theorem
(2.1).
If
~(z)
< ~
for
z ( D
then
log ~ ( z )
is s u b h a r -
monic.
Proof: Let whose and
Fix
a(z) real
such
functions
z0 ( D and parts
that f
and
b(z) are
a(z O) and
g
suppose
Sp(z 0) = {z
be the u n i q u e a n a l y t i c 1 i p(z) and i - (q--~) and
b(z O)
such
that
are
real
( ~ :
Iz-zol
S p} c D.
f u n c t i o n s in S (z 0) I P ~ ~ (respectively)
numbers.
Choose
simple
126
I
(2.2)
X
Let
f' = f/Ifl
define
g'
P(Zo
Ill
when
f
similarly.
d~
:
1
I
=
Y
zs not zero and,
r(ZO)
Igl
otherwise,
=
An immediate
If~
b(z)r(z 0) f'
calculation,
(2.3)
and
fz0 = f
and
=
in
1
gz0 = g. F(z)
is analytic
gz
which makes
Ilfzllp(z)
Moreover,
f' : 0;
put
We then put
a(z)p(z 0 ) fz
dr.
=
=
Igl
use of (2.2),
=
yields
Ilgzllr(z).
The function I y (T z f z )gz d~
(it is the sum of terms
Sp(Z 0)
g'.
of the form
P
e ~(z )
where ~ is an analytic function and ME, X F (Tz×E)X F dr, Y of are characteristic functions of measurable subsets of X and Y, ]
finite measure). making
Thus,
loglF(z)l
use of this fact and
log
y
(T
z0
d~
f)g
: <
Now,
taking
the supremum
loglF(z0)
I
127
log
i
we obtain
(2.4)
<
i --
12~
-
27
0
the subharmonicity
Let us now make
several
note that the Riesz-Thorin sult.
Suppose
{z E ~ :
T
remarks
theorem
satisfies
Izl < I}
the Dirichlet
02~ ~( Zo+peiO ) -
P(e18)
=
i Pl
loglF(
)1 de
de.
f
and
g
log
~(Zo+pei9 )
of
log ~(z)
concerning
satisfying
de. and Theorem
this theorem.
preceding i 1 p--~' q(z)
data
if
-7 ~ e ~ ~(l-2t)
if
7(i-2t)
< e <
(2.1). We first
case of this re-
the hypotheses
be the unit disc and
i iP0
27
is a very special
problem with boundary
i
!
0
llfIlp(z0 ) = i = llgllr(z0),
This establishes
<
over all simple
log ~(z 0)
Consequently,
(2.3) we have:
-2~
-
is subharmonic.
(1.2).
Let
the solution
D = of
127
and
!
l
i = q (elS) respectively.
Then
the
clearly
be e x t e n d e d ,
p = i;
thus,
since, the
~
last
i
to
_ l-t P0
+ t Pl
inequality
see t h i s
we can
one n e e d e d
monie
family
for
to the
<
8 <
inequality
situation
(2.4)
where
of
§I, o f f e r s
Let
D
be the
i
~
d8
=
l-t q0
=
(l-t) l o g M 0 + t l o g M I. + t ql '
we
obtain
(1.2)
on
interpolation
immediately same
domain
condition the
D : {z = x + i y
( ~ :
harmonic
is u s e d
Tt,
(2.1).
of a d m i s s i b l e
least
which
of a n a l y t i c
from theorem
In o r d e r 0 s x s i}
srowt h is p r e c i s e l y
majorant
to o b t a i n
0 s t ~ i
families
the
(see
of the bounds
(2.4)
in
semi-disc
-i _< y _< i}
bounded
and
is a b o u n d e d
operator
~(m]
the
result
by the
vertical
semicircle that
of n o r m
= i
~(iz)~
then,
gives
could,
also,
use
sider i ment
Tz
was us
1 and
the
for
z = Re { z_-~}
~D 3~ ~ e s -~-,
line
he has
segment
{z = e i8 :
-7 -~ < 8 _ < ~}
fact
Beckner
The
that
operator
result
D
of
again, isometry
in o r d e r
D
to o b t a i n
the
norm
information
y-axis,
let
result
of
Tie: e boundedness
+ Lq(Z)(dz)
D,
the
subharmonicity
~i _ < x ~ 0}.
Beckner's
: LP(Z)(d~) Z
has
z
Izl ~ i,
property
to o b t a i n T
T
in the
use
.
T. : L I + y 2 ( d ~ ) ~ L ( I + y 2 ) / y 2 ( d ~ ) ly the t r i v i a l fact that
by L. Gross.
{z = x + i y ~
and,
and the
shown
the
the r e f l e c t i o n
of
subhar-
[6]).
T i8: L2(d~) ~ L2(d~) is an i s o m e t r y , then t h e o r e m (2.1) e g i v e s n s the r e s u l t ~ ( z ) ~ i. When z : x is r e a l and in
operators
from
T : H ~ znH , d e s c r i b e d in the b e g i n z n n e x a m p l e s for the a p p l i c a t i o n of t h e o r e m (2.1).
good
Beckner's
that
and
exponentials.
operator
If we a s s u m e
fact
can
z0 = 0
of o p e r a t o r s
ning
{z : iy :
the
His
log ~ ( z ) ,
of the
we gave
theorem
take
~(l-2t)
log ~ ( e 18)
to c o n s t r u c t
function
for the n o r m The
z
by t a k i n g
in [6].
if
e T
and
Stein's
-~ ~ 8 S ~ ( l - 2 t )
case,
can be o b t a i n e d
he c o n s i d e r s the
T
i- 12~ -2~ 0
Similarly, of o p e r a t o r s
argument
< _
log ~ ( 0 )
i__ ql
in this
letting
if
q0
We
about
the
We m e r e l y
p--~
on the
i.
of
: Re { y-axis
L 2 (dz)
~ L 2 (d~),
of the
operators
con}, seg-
128
for
z ( ~.
In p a r t i c u l a r ,
: L l+Ixl (d~)
T
we o b t a i n
L(l+Ix[)/IXI(db)
the
fact
that
is an o p e r a t o r
of n o r m not
exceeding
x
1
for
order
-i ~ x ~ I. to
is o b t a i n e d
§3.
The
from
We h a v e D
the p o i n t s
(2.1).
This
of
seen how
an a p p r o p r i a t e
Dirichlet
type
the
problem
(2.1) case
The
difficulty
l e m is that
where
there
intermediate
the
this
"small",
intersection.
stantial
subspace
of t h e
Lp
of o p e r a t o r s
{T z}
arises
in the t h e o r y
of
two
Banach
spaces
by considering norms. depend since
We
all
shall
B0
and
Banach
see t h a t
on the d i m e n s i o n a large
appropriate
class
limits
of of
and
that
(3.l)
We
avoid
to be
cn
interpolation For most
Banach
spaces
the
estimates
fhat
for e a c h
e ~ Ivl ie e
kl(e)llvll
to be the
domains
e ( [0,2~)
is a m e a s u r a b l e
~
Ivl ie
~
or
problem
between difficulties
with
general
we obtain
this
property
prob-
to c o n s i d -
This
these
endowed
subspaces.
general
this
f o r m a sub-
[B0,BI] t
D
to m o r e
spaces.
for e x a m p l e ,
all
of meth-
non-trivial
domain.
applications has
this
a null,
functions
spaces
shall
n.
us,
a common
a meai p~,
spaces
Banach
have
the d o m a i n
of o u r m e t h o d
cn s u c h t h a t
simple
to
results
to a t t a c k
defined,
spaces
allows
spaces
general
well
the
the
to e x t e n d
dimensional
extension
on
that
like
in
is a s s i g n e d
then
part,
for the m o s t
then,
the
(2)
interpolation
attempts
Banach
having
B I.
in
result
c a n be c o n s t r u c t e d
finite
consider,
We a s s u m e ,
are
one
intermediate
our
case
spaces
data,
now
to h a v e
spaces
er a f a m i l y even
spaces when
hope
fact
Lp
for w h i c h
boundary
The
of
We w o u l d
arises
if t h e
to be the
boundary
spaces
is no r e a l
spaces
result
p r e c i s e l y , if w e are g i v e n i 0 _ < p(~) _ < i, and a solution,
boundary
that
last
if B e c k n e r ' s
More
c a n be o b t a i n e d .
od to the first
with
intermediate
see
spaces"
family
of the b o u n d a r y ~D. l p(~) , ~ ( ~ D ,
and
spaces
"intermediate
function
are
not
intermediate
just
L p(z)
to t r y to use t h i s semi-disc
appears
when
of t h e
the
is t e m p t i n g
{ n the u p p e r
construction
a domain
surable
It
interpolate
do not
is all we n e e d ,
that Also, unit
they we
are
shall
disc.
The
is r o u t i n e . we h a v e
function
a norm
for e a c h
1 1 ie e v E ~n
k2(e)11vlI'
e
( 2 ) S e e [3] f o r f u r t h e r d i s c u s s i o n c o n c e r n i n g (2.1) and B e c k n e r ' s result. The i n e q u a l i t i e s o b t a i n e d by W e i s s l e r [7] c o u l d a l s o s i m i l a r l y s t u d i e d in c o n n e c t i o n w i t h (2.1).
be
129
n [[vi[ = ([k=l
where
log kj(e), family
IVkl
j = 1,2,
of B a n a c h
2 1/2 )
is the u s u a l
is i n t e g r a b l e
spaces
on
B is : (~n,
I [ iS )
e
of the b o u n d a r y family
B z = (cn,
For h (8)
8D.
:
i
z
l + z @ i8
_
Z
(8)
is u s u a l l y
disc.
a
an a p p r o p r i a t e
z ( D.
1
2n l _ z e - ~
the P o i s s o n
therefore,
to the p o i n t s
let l-r 2
+ i _I
2~ i - 2 r cos (~-8) + r 2 :
h
assigned
e
for
6 D
n o r m and
We have,
Our t a s k will be to c o n s t r u c t
I [z )
z = re l~
Euclidean
[0,2~).
Pz(8)
called
~ i - 2r cos ( ~ - 8 ) + r 2
+ Qz(8).
the H e r g l o t z
and c o n j u g a t e
r sin (~-0)
Poisson
kernel
kernels
P (8)
'
and
Z
associated
Qz(~)
are
with the unit
We let 2~
W.(z) 3 for
j : 1,2
and
=
exp {
1
0
z ( D.
The
h (8) log k.(%) z ]
integrability
that the f u n c t i o n s functions the
on
symbol
of
log k.(8) a s s u r e s us ] never vanishing analytic
W.(z) are well d e f i n e d , ] (in fact, log W.(z) ( H P ( D ) for 0 < p < i). We let i8 ] z m e d e n o t e a g e n e r a l n o n - t a n g e n t i a l a p p r o a c h of z ( D D
to the b o u n d a r y Ch.
dO}
point
e i8
From classical
HP-space
theory
(see
[i0],
VII) we have
(3.2)
limi8 IWJ(Z)l
=
k.(e)3 '
z~e
almost
everywhere.
For
of all
{n-valued
analytic
ilFIl~
Thus,
0~r
(by
It f o l l o w s (3.2) we h a v e
H~ = H ~ ( D ; { n ) , j = 1,2, 3 ] on D satisfying
• . llW'(relO)F(reae)llP3 2~-d8)i/P
<
lim ie
H p#
introduce
=
H~ c H E
< -
a.e.).
for
2~ " INF[I[P: ('0[ Lr(e18)Ipel8
because
of
1 IIFIIp
(3.1) <-
and IIIFIIIp
(3.2),
k =
exist We can,
the s p a c e s
{F ( HE: and,
IWj(z)l -I = k . ~1 3
consist
-.
E H~ if and o n l y if Wjf k ( H P ( D ) ] that the n o n - t a n g e n t i a l limits F(e mO)
z~e
therefore,
Thus,
Let
functions
F : ( f l , f 2 , . . . , f n)
1,2,...,n. a .e.
e
p > 0
de i/p ~) < "}"
130
It will
be u s e f u l
z0 E D
we
to i n t r o d u c e
the
following
2w
have
IIIFIII
= (
°
IIIFlllp
lllFJllp,0
Moreover, g E HP(D)
ing this
HE:
for each
p
i/p
and
IIIFIII®
this
dS)
norm
is e q u i v a l e n t
is i n d e p e n d e n t
of
and
it is easy c
ctlgllp
to c h e c k
an a p p r o p r i a t e
inequality
to each
to
z 0 ( D.
'z°
z0 E D
for
.
IF(e~e) I e i s P z 0 ( 8 )
o
P'Z° If
on
let
Ill rlll p,z We then
norms
that
Ig(zo) I
constant
component
of
(m_lzol)l/p
independent for
W.F, ]
of
g.
' Apply-
F E H~, ]
we
F E H~, 3
then
obtain
IIF(z0)ll
(3.3) for each
Lemma
z 0 E D.
(3.4).
exists
~
From
If
K
a constant
cllFIl~/ (l-]z01) 1/plwj(zo)]
this
we
obtain:
is a c o m p a c t C K = c~
such
subset
IIP(z)ll for
all
can
now
Its d e f i n i t i o n shall
For
•
gives
and
there
-< cKIIFII j
v
we
on
impression
is not
seeking
E Cn
the
for the
{n
that
case
associated it d e p e n d s
and
that
intermediate
with
on the
it does space
each
z 0 E D.
index
satisfy
p > 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
<
find, s.
given
By
e > 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
<
c'~.
v : 0.
its d e f i n i t i o n ,
131
II
that
is a norm.
z0,P pendent of p,
we obtain a norm on
{n
(3.6).
Proof:
{F k}
is a Cauehy sequence
lllFk-Fmlllp ,
HP(D;~ n)
it follows that
(the space of
in classical
HP(D)).
{n-valued
in
F ( Hy
H~.
{WIF k}
Then,
of the latter space we are
such that
lim IIF-Fkll~ = 0. k~
a.e.
if necessary,
since
is a Cauchy sequence
functions having each component
From the completeness
assured of the existence of an Relabelling,
Before
we prove the following:
is complete.
Suppose
llFk-Fmll~ ~ in
H~
is inde-
z0,P 0 < p < I.
also when
showing this independence result, however, Lemma
II
Since, as we stated above,
we can assume that
~
.
lim Fk(el0)
= F(e ie)
Thus, using Fatou's lemma, we have:
I11F-FklII PP
1
2~
=
0
" Fk(ei0) P d~ IF(el0) 1e i8
[2~
=
lim IF (e 18) - Fk(eiS)IP d__88 J0 m~ ® m ei8 2~
lim 12n IFm(ei8 ) - Fk (ei8)l p i8 d8 2. m~® 0 e :
IIIPm-Pklfl~
lim m~
But the last expression Thus,
F = lim F k
in
is as small as we wish if H~,
k
is large enough•
showing the desired completeness•
We shall now state and prove the following basic result concerning the norms
1 Iz0,p
we introduced:
Theorem I.
If
IVlz0, p
inf {IIIGIII:
:
is independent
v ( ~n,
of
p.
z0 ( D
and
G E H~, G(z 0) : v, Furthermore,
F = Fz0,v ( {G E H; : G(z 0) = v, IVlz0
IVlz0'P
Proof:
Let
0 < p ~ -, then
IF(eiO)l eiO
= eonst,
a.e.}
there exists an extremal function
IG(eiO)lei8
= const, a.e.}
such that
a.e
[ : ivP'Z0 = {F ( H
(observe that our hypotheses
IG(emO)lei8
: F(z 0) : v}.
We first show
do not imply that the constant
[ ¢
function
132 F(z 0) [ v
belongs
To check that
to
F. ( H~
[Wl(Z)/W2(z) I (because Wl(Z) Wg~
F(e
ie
This shows that
Clearly,
(
P (e) l o g [ k l ( 8 ) / k 2 ( e ) ] z
0
~ 0).
to
It follows
that
@k(Z)
H ' ( D ; { n) c HP(D;¢n). a.e.;
Thus,
=
F E H~.
_<
IW2(Zo)lllvll
a sequence
We also
<
such that
lim k~®
= exp{
(e i8) de
1
2~
P,Z 0
log}Fk(eie) [eie de}.
the notation
loglG (eie )l iePz 0 (e ie ) de}.
0
on
IllFklll
e
mean of [0,27).
G
with respect
If, for some
to the p r o b a b i l i t y
p > 0,
IIIGNI < " P,Z 0
then
(3.7)
IIIGIIIp,Zo
lim
p~0
(see Chapter
6 of [ 5 ]).
IIIGII[ o 'z0
We then have,
IIWI (Z)Kk (z)II
= lw~(z)I l~(z) IllFk(~)ll kl(8) =
More-
-.
{Fk} c [
introduce
is the geometric z0
i
i/p ( i2'~ 0 Pzo (e)iw2(z°)l rElY1 eie/k2(e)]P de)
' z0
P
~
=
IIIGI;I o lllGlll0,z0
de)
consequently,
= exp { 12~ 0 [hz0(e)-hz(e)]
K k = %kFk .
F(z 0) = v.
Wl(Z)F(z)
and let
z0,P
measure
W2(z 0 ) - W2(z ) v.
F(z)
F ( [.
We now choose
Put
exp
W2(z0) --~ v W2(ele)
IIfFIIIp'Zo
IvI
:
belongs
) =
Let
we note that
log[kl(e)/k2(8)]
W2(z 0) v
over,
H~).
lllFkllJ0,z 0
exp
0
Pz (e) log
de} IFk (eie)l
ie e
llFk(Z) II
133
I2T~ exp 0 Pz(e)
_< ]llrkl{I o ,Zo llFk(z)ll
It follows
IIIFkNI0 . ,z 0
lJlFklllp,z0 .
Thus,
<
Ivlz0,p lTt~klllp,Zo
Ivl
~
This shows that
as
llWl(Z)Kk(Z)ll
Since
lllFklll0,z0 -
JKk(eie)lei8 Kk(Z 0) = v <
JllFklli 0,z °
k + -.
<
s
:
lllFklll0,z0
we must have
JllFklllp,z0
It follows
that
z0,P
inf {IIIGIII~ :
z0,P
=
dS}.
that the last term does
Moreover,
K k ( Hi~ c H~.~
Ill~
P (8) l o g l l k l ( e ) F k ( e i e ) l l z
0
HP-theory
Kk ( H ~ ( D ; ~ n ) .
This shows that
But
exp {
easily from classical
not exceed
a.e.
$2~
lllrklll0,z0 IlWl(Z)mk(z)ll /
:
de
log IIFk (eie)ll:t
G ( H~, G(z 0) = v,
ie = const,
IG(eie)l
a.e.}
e
and the first part of theorem I is established. In order to show the last part let us accept the fact be shown in the proof of theorem closed
subspace
of a reflexive
II) that,
Banach
for
space.
F
IIIFIII P,Z 0
of minimal norm
As before, 9(z)
and put
=
K = 9F.
(see pg.
i < p < -, Since the set
convex and closed (2) subset of this reflexive an element
(which will
Banach
H~
is a
[
is a
space it contains
•hus,
244 of [9]).
Iv 1 z 0
:
let
exp { 12~ [hz0 (e)-h ( 8 ) ] l o g F(ei0)II 0 z The argument we just gave
(with
ie dS} e
Fk = F
for all
k)
implies
-<
IVlzo
llIKlllp,z0
:
Thus, we must have equalities probability
space
(M,~),
if
IIir111 o,~o
-<
lilt iilp' ,z °
some
p > 0, we must have
Let us make we just gave (2)Lemmas
IVlzo •
in all the last inequalities. exp
loglf I d~ = ( M
This shows that
:
some observations
Ifl p d~)
a.e.
a.e,
(see Ch.
6 of [ 5 ]].
and theorem i is proved.
concerning
this proof.
The argument
showed:
(3.4) and
for
M
If(x) I = const,
IF(elS)l eie = IVlz 0
But, on a
(.3.8) can be used to show
[
is closed.
134
Corollary
(3.8).
If
F
is an e x t r e m a l
function
in
[ : Ev
•
F ( H;
and
IF(elS)lei8
We a l s o F(z)
<
IVlzo That
showed,
-
IIIFIII.
= IVlz 0
in the
W2(z 0 ) W2(z ) v ( E.
-
But
:
implies
proved
e([0,2~)
that
[ ~ @,
that
that
lW2(z0)1 m2(e )
sup. {
<
Ivlz ° Our n e x t
task
of the
dual
spaces
of the
introduce
some
N
is a n o r m
on
each
w
( Cn ,
dual
will
Ivl
e
<
ie}
lW2(z0)1 llvll•
norm
: v ~
JW2(z0))llvll.
investigate
spaces
original
notation C n,
_
be to
intermediate
we
L
this
that
is,
spaces
the
a.e.
argument
ess.
(3.9)
the
then
p,z 0
boundary
and make
we d e f i n e
where
is well
with
defined
on
question
constructed
of how
N*
is,
the
linear
({n,N)
to
to do this
N(v)
then,
and has
dual
observations.
: sup{ll:
Thus
the
are r e l a t e d
In o r d e r
elementary
vjwj.
N).
the
spaces.
some
N*(w)
n = [j=l
(associated
we h a v e
If
: i}
also
for
a norm,
functional
norm
S N*(w).
By a
W
simple
compactness
argument
one
can
show
that
the n o r m
of
L
is W
actually that
equal
every
fies the (~n,N*).
Lemma
N*(w).
linear
N**(v)
Clearly,
construct,
using
norm
i such
that
N*(w)
: I,
such
By a d i m e n s i o n
functional
identification
(3.10).
Proof:
to
= N(v)
N**(v) the
({n,N) dual,
for
= N(v). L = L
To
one
is of this
(~n,N)*
all
~ N(v).
Hahn-Banach
L(v) that
on
of the
argument
of
can a l s o
form.
This
(~n,N)
with
The
justi-
v ( sn.
show
theorem, Since
the
there
we h a v e
opposite
a linear
N(v)
exists
inequality
functional w
= L(v)
E {n,
= L
W
N**(v)N*(w)
show
L
we of
with
(v)
=
the
defini-
W
= N**(v).
following
lemma
follows
easily
from
(3.10)
and
tions:
Lemma
(3.ii).
k-lllvll
Now
kllvll ~ N(v)
(for all
suppose
v ( {n,
we h a v e
s Kllvll k, K
the
if and
positive
situation
only
if
K-lllvll ~ N*(v)
constants).
we e n c o u n t e r e d
in t h e o r e m
I:
a
135
{I
family of norms
] ie}
is associated
with the points of the bound-
(3.1).
by
e ary of
D
which
satisfies
Then,
(3.11),
the
dual
norms
satisfy
(3.12)
i
~
and the functions
Ilvl/
Ivl*iee
~
<
i
-
km(~y Ilvll
1
log ~
are integrable, by hypothesis. That is, ] we can apply the method of theorem I to the boundary space (¢n } I* , i@ ) e and obtain corresponding intermediate spaces. More generally, D
is a domain
Banaoh spaces
in
let us introduce
¢
and
the following notation.
{B<} = {(¢n, I I~},
for which the construction
We then denote by
[B<] z
space obtained
[B ]z : [B< ] ,
Proof:
For simplicity we give the argument
z : 0;
the modifications
2 L#
=
{f
defined
on
required
aD,
~
snace in general!)
2 L2
aD,
defined on
cn-valued
iif112(2) This last space is a Hilbert be convenient 2~ L2
=
{g
aD,
D
z E D. z ( D.
the unit disc and are obvious.
Let
If
(ei@
2
d@
)1 i e e
F~ )
1/2
< ~}.
let
and measurable:
(
12~
(ei8
0 IIk2(e)f
d8
)H2 ~%- )
1/2
it is self-dual.
< ")"
It will
to identify the dual as the space
¢n-valued
(2),
Ilgll2 via the correspondence
for all
z ( D
Also,
space; hence,
for us, however,
defined on
=
Ii ~
(
(This is not a Hilbert {f
at
and measurable:
IIIflll 2
=
for
for other
{n-valued
is a family of
of theorem I can be performed.
the intermediate
Theorem II (The duality theorem).
< ( aD,
Suppose
g ~+ Lg,
:
(
120~
where
and measurable:
~
I
Lg(f)
g(eZ@)
" 2
=
f
2~
0
easy to check that the norm of the linear functional (2)* equals llgll2 If Clearly, IIIflll2 < Ilfll 2(2)., thus, L 22 c [ #2.
de
2-# )
1/2
dO ~ Lg,
< -} .
It is
llLgll 2*' L2
f ( L#, 2
let
136
fn : 9n f
where
I 9n(8)
Thus,
=
0 _< 9n(8)
_< 9n+i(8) _< n.
to
@
convergence
as
linear that
functional on -'i]iglll2
(3.13)
otherwise
and,
.
consequently, k2f n ( L 2
shows
llfn(ele)k2(8)II
for each
This
shows
that
L2
is dense
is a bounded
_: (
1 2~
linear
" ' 2 de i/2 (ig(ele) l~ie ) ~%-) e
e
where
e
valued
function
n
)
=
L (f) g n
=
: But,
by the monotone
sion tends
to
-<
fILl[.
convergence
functional
since on
lim L~,
l~(e)12
IIIflll2 Thus
: i.
Put
2~ 0 ~n(e)~ (e) Ig (eie) I* e
theorem
lllfn-fllI2 we m u s t
:
of Lebesgue,
and put f n : ~n f'
de 2-%-
d__88 ie 2~ " the last
expres-
12~ ~de0
= 0
have
which satisAlso let
de 2~ = i
12~ 9n (8) " 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 = <~(eie),g(eie)>.
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 <_ IILII. Thus, there exists g ( L 2, 2 2 L 2. We claim that L
its norm.
L = Lg
k2 (e) Ilf(eie) II < n
theorem,
n ~-.
Now suppose denotes
_< 1 This
2 ll[fn-fHl 2 tends
if
n k 2 (8) llf(e 18) II
9n(e)IIk2(8)f(eie)H the d o m i n a t e d
i
and lim n ~
L
L(f
is a c o n t i n u o u s ) = L(f).
n
.
Thus,
On
137
L(f)
I:~(e)ig(eme)l*
:
i8 d__88 2~
:
Taking
the s u p r e m u m
with
I]~II2 = 1
of all these
we obtain
lllglll 2
:
exmressions
)
"
0
(writing
L(f
sum
d8 2~
e
over all
~ (L2(0,2E)
f = f )
~
Th(h)i
sup
IrLII.
:
IrlhHI 2~1 This establishes
For any 12~ 0
(3.13).
f ( [~
l
(el8
d__88 < 2~ -
)>I
1 2~ 0
<_
Again
writing
fn = 9n f
we have,
If
(eiO
IIIflll 2
using
(eie
)[ isIg e
IIIglll 2
<
, d8 )I i8 2~ e
~.
the d o m i n a t e d
convergence
theorem, L(f)
:
Thus,
lim L(f ) n n~ ~
g
=
lim L ( f ) g n n~
=
f
2~ < f ( e i S ) , g (ei8 )> de ~ 0
represents
lows that
NL[I s
lim 12~ ~ n ( 8 ) < f ( e l g•) , g(ei0 )> d_e8 2~ n~ 0
=
L
[#2
on
lllglll~ •
(3.14)
IL(f)l
~ lllflll2 • II]gHl2
~
It fol-
Consequently, [IL/I =
Now let us choose
and
•
IH~III ~ •
v ( Cn
and let
L
be the linear
functional
V
on
H#2
defined
[IIFNI2 Ivl0 boundary
and
values
by
Lv(F)
it follows of the
that
subspace
of
extend
to a linear
Then
ILv(F) I <_ IF(0)101vl ~
IILvll 2), (H#
functions
be a closed L
= .
in
L$.(3)
H$
!Ivl0 "
we can consider
By the H a h n - B a n a c h
functional
L
By c o n s i d e r i n g
on
L~
the latter
theorem
without
the to
we can
increasing
its
V
(3)
The fact that
H~
lishing
gives
[~ =
(3.14)
as the lllfll]2
proof
space
is closed g
that
I we used
of a r e f l e x i v e Banaeh p, i < p < -).
from
us an i d e n t i f i c a t i o n
of all
it follows
of t h e o r e m
follows
space
(3.6).
The argument 2 of the dual, (L#)*,
estabof
such that lllgIll2 < ~. Since 2 is reflexive. Recall that [#
(lllfNl2)* in the
the
subspace
fact
(this
that
argument
H~
is a closed
extends
to other
indices
138
norm. Let g ~ (L~)* be the representative of L. (zk/w2(z))u, for k = O,l .... and u 6 ~n. Then,
Lv(Fk)
=
= I/W2(0) 0
On the other hand,
that the Fourier series of the components of type. By (3.11)
~k2
=
i
k = 0
if
k > 0.
g/W 2
i9)
)11
~
Ig(e
Consequently
(u, ( ~ ) \ is an analytic function in W2(z)/ using the mean value theorem for analytic functions, (u
g(0)~ ' W2(0)/
=
12~ (u ( e ~ ~ 0 ' W2(ele )
It follows
are of power series
(el@
tlg
Fk(Z) =
if
12~ g(eie). 0
Lv(Fk) = i
1 g(ei@) W2(eZe )
Define
de = L 2--~v(F0 )
I*ie'e
H2(D)
_
Thus,
W2(~7 "
Thus, = for all u ( {n., therefore, g(0) = v. If N0(v) denotes the intermediate norm of v obtained, by the method of theorem I ' from the boundary norms I I* ie we then must have N0(v) < 2 = IILvll s Ivl*0 ' That is, e
llYglli"
No(V)
(3.15)
<_ [Vlo.
We shall now show that the reverse inequality also holds. This, then, would establish theorem II. Consider two extremal functions F = F0, w and G = G0,v, the first corresponding to the boundary norms {I Iie}, e
the second corresponding to
{I ]*ie}.
.,.
Thus, by theorem I,
e
N0(v) = IIIGN]2; also WIF and W21G belong to H2(D;Cn). Hence, (WI/W2) ~ HI(D). By the subharmonicity of the log of the modulus of this function we have Wl(0) ~W2
log
But, and
log G
Wl(0) ~
=
are extremal,
-<
12~ 0 log
12~ I Wl(eie) 0 log W2(ele )
Wl(eme) (eie ei W2(eZe ) . de 7~- "
Consequently, since
de 2--~ "
F
139
log
l<w,v>l
12~
s
d8 l o g l < F ( e m.e ) , G ( e i e ) > I ~-~
0 12~ 0
-< That
is,
all
w ( {n
Basic
e
lwl0N0(v),
~
with
Ivl~ ~
ity: §4.
l<w,v>l
. l°glF(emO)l
lwl0 = i
" * d9 ie IG(ele)l ie 2~ e
-
xf we n o w c o n s i d e r
loglwl0N0(v).
the
we t h e n h a v e the d e s i r e d
supremum over reverse
inequal-
N0(v).
properties
Theorems
of the i n t e r m e d i a t e
spaces
I and II h a v e m a n y c o n s e q u e n c e s .
some of t h e m and
illustrate
the i m p o r t a n c e
inequality
(3.9)
We shall n o w p r e s e n t
of the d u a l i t y
result
we
just o b t a i n e d . If we a p p l y mediate
spaces
of the f a m i l y
{I
to the dual n o r m s I*i8}
obtained
as i n t e r -
we o b t a i n
e
< -
Ivl~0 Using
(3.I0)
and
(3.1].), t o g e t h e r
lw~(z0)IJlvll~
Corollary (4.1). and
I lWl(Z0)l
llvll.
with
Ivlz 0
(3.9), <
we o b t a i n
iW2(z0)Irlv[l
for all
z0 ( n
v ( ~n. This tells
us that
h a v e the c o n d i t i o n we n e e d e d
is a d o m a i n w h o s e
corresponding
kernel
associated
It is n a t u r a l and
to
(3.1)
of t h e o r e m
with
can be u s e d to o b t a i n
z ( 6. spaces
~
for the c o n s t r u c t i o n
a Herglotz ( ~,
if
closure
is in
on the b o u n d a r y I.
That
spaces
[B<] z
to i n q u i r e w h a t are the r e l a t i o n s
we
which
is, a s s u m i n g
B< : (~n,l
it, the spaces
intermediate
8~
D
~
has
I<),
for all
between
these
B : Z
Corollary the type
Proof: then,
IVlz 0
(4.2)
(The i t e r a t i o n
just d e s c r i b e d
Suppose
Nz0
by d e f i n i t i o n ,
~
inf {
then
is the n o r m of IF(Z)Iz
<~nSUpIP(<)l<:
inf { ass.
sup
If
theorem.) [B ] z = B z
< _
[
[B<]z0 ,
IIIFIII,
F (H~(D)
c D
for all
is a s u b d o m a i n z ( 6.
< ( 8~.
for all
z ( D.
If
F (H~(D)
Thus,
, F(z o) = v}
IF(<)I< : F ( H ~ ( 2 ) ,
F(z 0) = v}
=
Nz0(V).
of
140
Using
theorem
II and this
last r e s u l t
we t h e n h a v e
IvI*
'
Taking
the dual n o r m s
we t h e n h a v e
Ivl
of both
~ N z0
We h a v e
It turns
sponding Theorem ferred
with
II.
and
exists
= N(v)
Corollary
(4.3).
Proof:
By c o r o l l a r y
and
and
D.
=
and
w
Thus,
(3.8)
must b e l o n g
IVlz 0
(3.10)
is an e x t r e m a l IF(eiS)lei8
fact
function
corre-
is also a c o r o l l a r y
this u n i q u e n e s s
is u s u a l l y
is s m o o t h p r o v i d e d such that
between
func-
= IVlz 0
N*(w)
this n o t i o n
of re-
for each = 1
and
and the u n i f o r m
[4]).
spaces
are
with a point
smooth, then the e x t r e m a l z0
in the d o m a i n
and a
is unique.
F(z 0) = v.
z0
v
see
F then
the e x t r e m a l
w ( Cn
associated
v ( {n
Z 0
B
if
This
B = (~n,N)
If the b o u n d a r y
vector
that,
guaranteeing
(for the c o n n e c t i o n
of the dual of
Fz0,v,
and u s i n g
(4.2).
v ( {n,
is unique.
a unique
convexity
function,
and
in m a n y cases
v
The p r o p e r t y
there
(3.8))
z0 ( D
out that
z0
inequality
(v).
z 0
z0
to as s m o o t h n e s s :
v ( {n
to
of this
This p r o v e s
seen ( C o r o l l a r y
tion a s s o c i a t e d a.e.
sides
(v).
~ N*
z 0
an e x t r e m a l
to
HI,,
Let us c h o o s e Suppose
obtained
G
function
satisfies w ( ~n
associated IVlz0
with a.e.
with
lwl* = i and s a t i s f y i n g z0 f u n c t i o n c o r r e s p o n d i n g to
is an e x t r e m a l
f r o m the dual n o r m s
in p a r t i c u l a r ,
F
IF(eie)le ie=
{I
iG(eie)] , ie = I
l*ie}
a.e.e
on the b o u n d a r y
of
Then the f u n c t i o n
e
~-
IIIFIII®
belongs
IIIGIII ~'~ :
H'(D)
to
Ivl z
).
(since
l1
~ IF(Z)IzlG(z)1 [
Consequently,
0
2~
IVlz 0
1
=
Pz0 (8) d8
0
IrlFIIl~.
o
~-
IIIGIII '~
Since
]l-~
above,
m u s t hold)
Pzo(9) Ivl
z0
=
d8
a.e.,
Ivl
z0
the e q u a l i t y
(which,
2~ I
P 0
implies
Zo
(8)
= 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 °
(
<
i/p
1
ir(w(e))ip
0
for all
p > 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
<
= c,
It has b o u n d a r y
which
It f o l l o w s
clW(eie) I
=
implies
values
ciwI*i8
that
a.e.
e
for all
w.
F r o m this we see that
IF(ei6)I
i8 ~ c.
But
c = IF(z)iz
e
IIIFH,
! c;
thus,
IF(eiS)l
proved,
Let us now s u p p o s e ness c o n d i t i o n Corollary
i8 : c
giving
(4.3)).
our b o u n d a r y
for
and the c o r o l l a r y
z,z 0 E D A(z,z0):
We i n t r o d u c e
tion assciated
and
with
function
however,
it does
some of them.
v E Cn.
cn ~ cn
analytic
of
Banach
us the u n i q u e n e s s
A(z,z0)(v)
ping
a.e.
is
e
and
z E D.
=
That
v.
the s m o o t h -
functions
(see
Fz0,v(Z) is,
for e a c h
A(.,z0)(v)
z0 ( D
A(z,z0)(v)
A ( z , z 0)
properties.
is an i m m e d i a t e
we have
is the e x t r e m a l
In p a r t i c u l a r ,
In g e n e r a l ,
satisfy many basic
The f o l l o w i n g
satisfy
the n o t a t i o n
such that
z0
spaces
of the e x t r e m a l
is not
a mapfunc-
is an
linear;
We shall n o w d e r i v e
consequence
of the
143
definition (4.6)
of extremal
A(z,z)
= I = identity
Let us choose
G(z)
function
for
and
z E D.
associated
: A(Z,Zl)(V)
with
v E {n Then z0
it is the unique
extremal
function
= F(z0).
(The propagator
for all
z ( D.
and put
F(z)
is analytic
=
and is the
A(z0,zl)(V).
function.
associated
Thus,
equation).
F and
is also an extremal
G(z 0) = A(z0,zl)(V) (4.7)
Qperator,
z0,z I E D
A(z,z0)[A(z0,zl)(V)] extremal
functions:
The function
In fact,
with
F(z) ~ G(z).
z
and
This
by
(4.5),
G(z).
But
shows:
A(z,z0)OA(z0,z I) = A(z,z I)
for all
z,z0,z I E D. From these two properties (4.8)
A(z,z 0)
maps
the left and right Let
A(z)
a(z): ally,
of
:
onto
and
A(z0,z)
is both
a(z)
= A(0,z).
We then have,
from
(4.7)
(4.5):
A(x)oa(z0) ,
tells
~n
A(z,z0).
and
property
A(z,z 0)
The last equality
one-to-one
inverse
= A(z,0)
and the constancy (4.9)
{n
we obtain
IVlz
=
la(z)(v)I0.
us that the mapping
B z = ({n i iz) ~ B0 = ({n,l i0 ) is norm preserving (more generthis is true of A(z,z0): Bz0 ~ Bz). Since I Iz is a norm we
have the subadditivity
(4.10)
property:
la(z)(vl+V2)lo
The uniqueness
~
of the extremal
la(z)(vl)lO functions
+ la(z)(v2)lO.
gives us the h o m o g e n e i t y
property: (4.11)
A(z,z0)(Xv)
(4.12)
A(z,z0):
Proof:
Choose
Fk(Z)
{n
= kA(z,z 0) ~
v ( ~n
= A(Z,Zl)(Vk).
{n
for
X E ~,
is continuous and a sequence
Then
IVklz 0 =
z,z 0 ( D
for each pair {v k)
and
z,z 0 E D.
converging
lllFklll2,z0
and
v E {n.
to
v.
Let
144 =
lim IVklz0 [vlz 0 k~We saw in the proof reflexive;
topology 2 F E H#. ges to
of
.
Two
F(z 0) : v
{Fk}
2
in
set which
weakly
is closed
closed
we can assume
2
H# c L# 2 L# is
(3)) that
converges
a weakly
and
z E D.
The
is continuous.
immediate
consequences
and
: k~lim
IF(Z)Iz
in the n o r m
set.
{Fk}
to an
Consequently
itself
conver-
linear
functional
Thus, of this
IFk(Z) Iz : k~lim
mapping
lim < F k ( Z ) , W > = 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
-<
IVlzIA*(z)(wj)l[ / IW2(z)l
Thus, by (a) and (c),
f].(eie) :
a.e.; but, by (b),
:
IVlz/ lW2(z){
-<
i{vI{.
limie f.(z)3 : {I/k2(e) z~>e
' I"8) (wj)>l l
Ivl ielA*( e ie )(w.)l * i8 e ] e
S
:
Ivl i8" e
Hence, (e)
f.(e ] ie) ~ ]vl ie/k2(8) e Let
s Nv{l.
h(e i8) = sup f.(eie).
By the last inequality
h(e ie) ~ llvN.
j Thus, the Poisson integral of [2~ J0
P (8)f z
.(ei 8
) dS.
But
]
is defined and
from (d), we have
j
IVlz _ IW2(z){h(z).
>- 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
is e a s i l y
z E D.
Nz0(V) with
choose
= i : M[0(w). z0,v
associated
checked
We
and
with
the z 0,
that
F(z),G(z)> 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[¢(<(e))l
de
0 2~
1 I
log ~ ( < ( e ) N < ( e ) ( F ( < ( e ) ) M ~ ( e ) ( G ( K ( e ) )
-
2g
-
1 I0 2Tr
-
1 I0 2~
de
0 2rr
log R ( < ( 8 ) )
de N z0
(v)M* (w) z0
2~
log R ( < ( e ) )
Now, taking the s u p r e m u m over all
de.
v, w
N
satisfying
(v) = i = z0
M* (w) z0
we obtain
Theorem
(4.14)
(The I n t e r p o l a t i o n Theorem).
family of linear o p e r a t o r s mapping
Bz
R(z)
then
is the operator norm of
Tz,
into
If
{T z}
Cz
for
log ~ ( z )
is an analytic z ~ D
and
is a subharmonic
function. As is the case in t h e o r e m are given i n f o r m a t i o n about mediate
(2.1) one can apply this result when we
~(<),
< ( ~D,
by c o n s t r u c t i n g the inter-
spaces from the c o r r e s p o n d i n g b o u n d a r y spaces
by the method of t h e o r e m I. that the analytic
family
{B~}
and
{C~}
In such a p p l i c a t i o n s we have to check
[Tz}
satisfies
"admissible growth" condi-
tions of the kind m e n t i o n e d in §2. Before giving some a p p l i c a t i o n s of these results let us make an o b s e r v a t i o n c o n c e r n i n g our c o n s t r u c t i o n of intermediate
spaces.
We
have solved a general version of the Dirichlet p r o b l e m for the region D
when the b o u n d a r y data is not n e c e s s a r i l y a numerical valued func-
tion but is a function whose value at the Banach space
B~ = (C n,
1 I~))-
this b o u n d a r y norm has the form
~ ( ~D If
n = i
is the norm
I I~
(or
then, by h o m o g e n e i t y
Ivl~ = k(~)llvll and the basic
148
assumption grable. IVlz the
(3.1)
Thus,
=
and the
For
{Bz}
of the
it's
norms
reasons
data
families
is the
mean
value
cussion
§5.
Some
second
with
ary" i
LP(<)-spaces
the
by
basic
which
Ivl ie
done
is used
function
logk
of this
solu-
spaces
is d e t e r m i n e d
by the
characterizing
plays
the role
A more
these
that
complete
the dis-
[3].
these
:
a family
They
the
same
I.
a family
HVIIp(e)
:
(
For
such
of
LP-spaces
obtained
problem
spaces
would
simplicity
LP(8)-norms
n [ k=l
IVkl
p(6)
and
from
for the
that
of
0 < mk v = (Vl,V 2 ,-..,v n ) ( ~n, Then, such that i ~ p(e) ~ ®. k2
were
the D i r i c h l e t
are
of t h e o r e m
given
D.
function and
see that
intermediate
functions.
in
e
kI
of
(which
is inte-
the E u c l i d e a n
exponential
property
we c o n s i d e r e d
solving
that
and we are
(S.l) where
the b o u n d a r y
family
(4.2)
of a d o m a i n
by the m e t h o d
disc
is g i v e n
we
we h a v e
by m u l t i p l y i n g
family
The
Theorem
section
points
We c l a i m
obtained
the
(4.1)
what
log k
applications
In the
unit
with
by the
for h a r m o n i c
notions
iated
p(~)
value)
where
from
that
obtained
~ (8D).
does
and,
problem
are
we call
Iteration
of t h e s e
~ W(z)
a log-harmonic
{B~},
theorem
k I = k 2 = k,
therefore,
absolute
we c o n s t r u c t e d
boundary
see,
Dirichlet
the
these
with
= W2(z) We
intermediate
(here,
tion.
Wl(Z)
IW(z0)lllvll.
0 solution
norm
can be made
function have
suppose on
assoc"bound-
been is the
D
cn:
i/p(6) mk)
'
p(8)
clearly,
is a m e a s u r a b l e there
exist
constants
that
klllVM
(5.2)
Ivl i e
~
~
k211vll;
e
consequently, and
(4.5)
condition
(3.1)
it is r e a s o n a b l e
before
(2.3)
z0 ( D
the
is c e r t a i n l y
to e x p e c t
to be an e x t r e m a l function
of
z,
the
function.
satisfied.
function That
B(z,z0)(v) ,
is,
whose
In v i e w
defined if
k th
of
(2.3)
i~ediately
v ( ~n
and
coordinate
is
)[a(z)-~(z0)]P(Z0), vk llvlIp(Zo ) <'vkl
where
~(z)
=
h z
d8 p(el8)
be an e x t r e m a l
function
0 world, out
that,
at least
(8)
in this
,
should,
associated
case,
there
if t h e r e
were
justice
with
and
z 0.
v
is justice.
First,
in this
It t u r n s observe
that
149
B(z,z0)(v) Izl S i
is a n a l y t i c
we have,
in
z
and
analogously
(5.3)
to
B(z0,z0)(v)
z 6 D
:
that
if
Ilvllp(Zo )
(this is a s t r a i g h t f o r w a r d
see i m m e d i a t e l y
Moreover,
(2.3),
IIB(z,zo)(v)llp(z)
for all
: v.
B(',zO)(v)
( H:
computation).
and,
F r o m this we
thus
ff
We shall n o w A(z)
by
show that the r e v e r s e
be an e x t r e m a l
1
Ilvllp(Zo ).
IVrz ° <_
(5.4)
i = I - p(z)
Choose
~(z,z0)(w)
(instead
of
function
and
associated
w ( Cn
the f u n c t i o n
p(z))
w
~
with
just d e f i n e d
(instead
of
is a l s o true.
z0
and
v.
IIWHq(z 0 ) = 1
such that
I B ( z ' z 0 ) ( w ) I *z ~ llB(z'z0)(w)llq(z)
lr
inequality
above
v).
in terms
Then,
= llWllq(z0)"
by
Let
With and d e n o t e
of
q(z)
(5.4)
and
(5.2),
Thus
IA(Z)Iz]B(z,z0)(w)l* z
Ivl
:
z0
l~(z,z0)(w)l~
IVlz0rlWNq(z0) : Ivlz0. Consequently
%(z)
=
belongs
to
H'(D)
and,
there-
fore,
:
~(z 0)
:
2~
1
P
0
(8)%
(eie
) de.
z0
Hence,
ll
~
12~ P (e)llA(eie)ll (eie llB(ei0,z0)(w)ll . de 0 z0 p ) q(e 18 )
:
( 12~ o P o(e)
If we n o w take the
supremum
llVNp(ZO) < _
see t h a t
IVlz 0 .
d e ) l VlzollWllq(Zo) over all This
IVlz° for all
z 0 6 D.
therefore,
The c l a i m made
w
shows
=
IVlzo.
satisfying (because
of
IlWlIq(z0) = i (5.4))
that
IlVllp(zo) at the b e g i n n i n g
of this
section
established.
Let us n o w t u r n to a n o t h e r spaces obtained
when
we
interesting
e a c h of the b o u n d a r y
class
norms
of i n t e r m e d i a t e
is a H i l b e r t
space
is,
150
norm: and,
for each for
v ( £n
measurable denote
e ( [0,2~)
operator
Ivl
the c o n d i t i o n
IIF(8)-III
are
Ilvll
whenever
z0 E D
extremal
function
:
invertible
We a s s u m e
matrix
e ~ F(8)
llr(e)ll
and
is a
IIF(8)-III
is s a t i s f i e d
v ( ~n. H#2
F(z 0) = 0}
endowed
and
P
we
can
of a p r o j e c t i o n
with
be the
llF(e)ll
and
F(z 0) = v}
situation
in terms
log
I we h a v e
2 F ( H#,
:
In this
A(z,z0)(v) space
if we a s s u m e
By t h e o r e m
inf { IIIFNI 2,Zo
and
IFr(o)lIIIvll.
< e ie
(3.1)
=
Hilbert
H 0 = {F ( H
n× n
letting
Ivl
<-
integrable.
Ivlz °
ing on the
be an
= llF(8)vIl.
Then,eclearly,
1 rlr(o)-l[l
Then,
ie
F(e)
norms,
(s.5)
log
let
function.
let
the n o r m projection
express
the
operator
actLet
III ll12,z0 H#2
of
onto
H ±0.
2 Gv - W2(z0) - v b e l o n g s to H# (see the b e g i n n i n g of z0 W2(z) the p r o o f of t h e o r e m I) Then, c l e a r l y the set [ = [v intro. , 2,z 0 d u c e d in the proof of t h e o r e m I is the c o s e t GVz + H0" 0 The
function
Lemma
(5.6).
Proof:
Since
extremal we n e e d PG~0 with fore,
(PG~0)(z)
the
= A(z,z0)(v)
boundary
function
are
with
z0
among
all m e m b e r s
2,z 0
that,
smooth and
v.
we k n o w v
by
(4.3)
is unique.
=
]llh+h Olll ~2,z 0
:
IIIPG 1112
v
proves
the
The m a p p i n g operator
on
2 H#.
+
lllh ±111
of the
that
Thus,
the
all
Gv + H 0, z0 has m i n i m a l n o r m ( o b v i o u s l y G v (z 0) = v). Suppose Gv = h + h± z0 z0 I h ( H 0 and h i E H 0. The g e n e r a l e l e m e n t of Gv + H0, therez0 has the f o r m (h+h 0) + h I w i t h h 0 ( H 0. Consequently,
III (h+h 0) + h i rll 2
This
z0,
spaces
associated
to do is c h e c k
for all
,z 0
coset
>_
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)
-"
<
I
log det P(0) dS, 0
then
can be factored as
P(e)
(bjk(eie))
is such that
1,2,...,n,
and each
P(8)
bjk(ei0)
bjk(eiS)
: b(eiS)*b(eiS), belongs
to
has a Fourier
where
L2(0,2~),
b(e ie) =
j,k =
series of power
series
type. That
(5.8)
the largest
implies
this theorem
proper value of
P(8)
of the smallest proper value of with the integrability logllP(8)II
and
of the
logllP(8)-iN.
is immediate:
and P(e),
p~k'S,j
Nb(eie)ll
pjk'S
spaces are separable
One then obtains established students,
theorem
(5.9). Thus,
gives us the square
It is not hard to extend theorem boundary
(5.11),
Consequently,
s llP(e)llI/2.
the extension
by Devinatz.
S. Bloom,
is
condition
together of
(5.8) can be
Since
b(z)
is analytic,
series of power series type on the boundary.
(5.10) we obtain tion on the
IIP(8)II
is the reciprocal
imply the integrability
applied and we obtain the factorization it has a Fourier
Since
IIP(O)-III
From
the integrability
integrability
assump-
of the bjk'S.
(5.8) to the case where the
infinite dimensional
Hilbert
of the W i e n e r - M a s a n i
spaces.
theorem that was
This extension has been done by one of our
and will appear elsewhere.
Other applications
and observations
concerning
these results
can
be found in [3]. The m o t i v a t i o n lectures
for the choice of the material
at the University
As we stated before, dimensional
we considered
spaces.
the boundary
in order to avoid considerable
finding an appropriately Moreover,
presented
of Maryland was of a pedagogical
large common
in these nature.
spaces to be finite
technical
difficulties
of
subspace of these boundary
this choice also simplified
all questions
concerning
153
"duality results."
Another aspect of the theory we have not discussed
involves interpolation of nonlineam fact, it is the analyticity of T z) that is of basic importance
operators on Banach spaces.
ples of nonlinear analytic operators arise frequently for example,
in mathematics;
it can be shown that the functions arising in the Riemann
mappin Z theorem vary analytically with the domain rect" parametrizations).
(if we have the "cor-
Various regularity results of these functions
can be proved by using our interpolation theory. to publish,
In
(T F)(z) (and not the linearity of z in our interpolation theorem. Exam-
It is our intention
in the near future, a paper containing the general theory
and more applications.
REFERENCES [i]
Beckner, W. Inequalities (1975) , pp. 159-182.
[2]
Calder~n, A.P. Intermediate S ~ c e s and Interpolation, plex Method, Studia Math. (1964)~ ppT ~13---~.
[3]
Coifman, R.,Cwikel, M., Rochberg, R.,Sagher, Y.,and Weiss, G. Complex Interpolation for Families of Banach Spaces, Proceedings of Symposia in Pure Mathematics, vol. 35, Part 2, A.M.S. publication (1979), pp. 269-282.
[4]
Dunford, N. and Schwartz, J.T. Publishers, iNew York (1958).
[5]
Hardy, G.H., Littlewood, J.E. and P61ya, G. Cambridge Univ. Press, London (1934).
[6]
Stein, E.M. Interpolation of Linear Operators, Math. Soc., vol. 83, No. 2 ~-1956--~, pp. 482-492.
[7]
Weissler, F.B. Hypercontractive Estimates for Semigroups, Proceedings of Symposia in Pure Math., vol. 35, Part i, A.M.S. publication (1979), pp. 159-162.
[8]
Wiener, N. and Akutowicz, E.J. A Factorization of Positive Hermitian Matrices, J. Math. and Me~h. 8(1959), pp. 111-120.
[9]
Wilansky, (1964).
[i0]
A.
in Fourier Analysis,
Linear Operators,
Functional Analysis,
Zygmund, A. Trigonometric Cambridge (1959).
Ann. of Math.
102
the Com-
Interscience
Inequalities, Trans. Amer.
Blaisdell Publ. Co., New York
Series, Cambridge Univ. Press,
M A X I M A L FUNCTIONS:
A P R O B L E M OF A. ZYGMUND
A. C~rdoba Princeton University In 1910 H. Lebesgue extended the f u n d a m e n t a l theorem of calculus in his w e l l - k n o w n paper, (Ann. Ec. Norm.
27):
Sur l ' i n t 6 g r a t i o n des fonetions d i s c o n t i n u e s ,
Let
f
be a locally integrable
function on
~n.
Then i lim r÷0 ~[B(x;r)]
where
~
r f(y)d~(y) JB(x;r)
denotes Lebesgue m e a s u r e
The q u a n t i t a t i v e
f(x),
a.e. x,
~n.
i n t e r p r e t a t i o n of this result was obtained by
Hardy and Littlewood in 1930 applications,
in
:
Acta Math.
(A maximal theorem with f u n c t i o n - t h e o r e t i c
54).
Given a locally integrable
function
f
let us define
Mr(x)
i
:
sup ~[B(x;r)] r>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<~.
any r e s t r i c t i o n ,
LP(~3),
i ~ P ~ ~.
then
t h e n a.e.
H e r e we can
s e t t l e the c a s e y
because
:
an e a s y c o m p u t a t i o n y
(Yl
0 ) ~ 0
0
Y2
shows
that
=
in a s u i t a b l e
= Suplu(X Y
+ iY) I
where
~ 0
is m a j o r i z e d ,
Mr(X)
sense,
Y2
by
M~f(x)
with
~(s,t)
= (s-t) I/2.
157
Therefore,
we have a.e.
convergence
for the class
L(I + io} L)(]R3).
Proofs. [A]
Proof of the covering
lemma.
We can assume that the given family satisfies B{UR } < ~,
otherwise
can also assume that tained
{R }
Therefore
is finite and no one of its members
we is con-
in the union of the others.
Let us choose cal side. element of among the
R1
Assuming {R }
to be an element
that we have chosen
such that its vertical
of
{R }
with biggest
RI,..., Rj_ 1 ,
let
R].
vertibe an
side is the biggest possible
~'s that satisfy 1
exp(
R
The subfamily,
j :l
(x))dx
~
1 + e
k
{Rj}j=I,...,M
exp( ~ XR. ( x ) ) d x M URj
the condition
there is nothing to be proved.
obtained
=
exp(
~
H-1 URj -R M
j:l
in this way,
M-I [ XR.(X))dx j=l ]
satisfies I
+ e
exp( RM
M-I [ XR.(X))dx -< j=l ]
j:l f -<
exp( H-1 UR. 3
M-I [ XR (x))dx + (l+e)~{R M} j :l j
j:l (I+e)ED(Rj)
Next,
given
R ( {R } - {Rj},
s
E'
is extended
cal d i m e n s i o n
than
~(URj).
we have
p(E XR.(X))dx 3 where
(l+e)(e-l) e-l-e- I
~
i + e
over all parallelepipeds
R. 3
with bigger verti-
R.
Let us rearrange
the rectangles
appearing
in
E'
in the following
way
E'XR " ]
=
XRI +-''+
XR
p
+ XR
p+l
+-..+
XR
q
158
where the
R.'s,
and those sion of
R.'s 3
j = l,...,p with
have
s-dimension ~ s-dimension of
j = p + l,...,q,
have
t-dimension
R
{ t-dimen-
R.
Then i + e-i
-R
z{--~T < i I exp(Z'XR')dx
P =
--
r , s [: 0r!
!
--
s!
kl'''" 'kr=l £i'''"
p
1 1
r, :0 r! s!
If
q
~
!
~
£s =p+I
q
~{~ln.
kl,...,kr= I £1,...,£s=p+ I
~{Rkln'''NRkrNR£1n'''nR£snR]
# 0
i
IRXRkl
... x~
• .n~nR,ln.
the intersection
then the intersection must be of
of the R,
to
s" =
R.'s whose t-dimension is bigger than the 3 and analogously for the block 6. 8
Therefore, ~{Rkln...nR £ nR} s
~{R}
Given
• • nNsnR}
~{R}
the form shown in the figure, where "the block corresponding t-dimension of
dx
s
P = {x0' Y0' z0) ( R,
sx6 sxt
consider
Ip I
:
{(x, Y0' z0) ( R},
Ip 2
:
{(x0, y, z 0) ( R}.
"
159
It happens, again by the monotonieity, I
that
Rkz n • • • nR k r n I P l j lip 1 I
s
and
'
iRwin .....NR~snlp2 I
lip21
=
~
•
Therefore, i + e -I
~ i
IRexp( ~'XRj )
p
q
i i [ r! s! r,S:0 kl...kr:l £1...~s:P+l
IR~ln..-nRk nzpll llpll
R~IR...nR~s nIP2
lip21
E ,x ], I
Ip
I
Here, I 1 denotes 1-dimensional Lebesgue measure. Therefore if M x, M denote the extensions to R 3 of the one-dimensional Hardy-LittleY wood maximal function in the x and y directions respectively, we have: R c {Mx(eXp(EXR.) ) • My(eXp(EXR.) ) ] ] c {Mx(eXp(EXR.) ~ 71 + e -1} ]
U
i + e -1} {My(eXp(~XR.)) ~ 71 + e-l}. ]
Thus, 2
I
URj
exp(ZXR ") ]
~
C~(UR.). 3 Q.E.D.
160 [B]
Proof of the theorem. It is a well-known
and the implication of the proof satisfies maximal
fact that statements
is as follows:
the hypothesis
find such a family Let
B
B
with
B,
lemma and if
then
M
satisfies
with the property that
M~f ~ CMf
B
denotes
the
(b); next we for some uni-
C.
satisfy the hypotheses
of the covering
Sup ~
=
lemma,
and define
If(y) Ida(y),
x(R* R~B R*
The strategy
M
Mf(x)
where
(b) are equivalent,
First we observe that if the family
of the covering
function associated
versal constant
(a) and
(b) ~ (a) is the easier of the two.
R
is the result of dilating
R
by a factor of 3 with respect
to its center. Claim.
There exists a constant
p{Mf(x)
> ~}
_<
C<~
such that
C fIR 3 [f(x)Ik { i
+ log _______21) If(x) d~(x).
Proof of the claim. The set
{Mf(x)
> i}
is a union of parallelepipeds
{Rc}
such
that i | If(y)Ida(y) n:
_>
k.
We can apply the covering
lemma to show the existence
{Rj}
properties.
with the prescribed
* ~{URj}
~
27 ~ ( R j )
~
I
27
If(x)l k
ZXR.
UR,
Next,
observe that if
that
u • v ~ u log u + exp(v-l).
exists a constant
C
u < ~
and
v
of the subfamily
We have
(x) d~(x)
"
3
are positive real numbers,
it follows
Furthermore,
such that
for every e > 0 there + u • v ~ C u(l + log u) + exp(sv-l).
In particular, ~{UR.* } 3 But
_<
I
C s
If(x)I X URj
{i + log TI f ( x ) l } d ~ ( x ) +
e -I I URj
exp(aZXR (x))dB(x). ]
161
where
C
> 0
is the constant appearing in the covering len~a. so that
-
Finally we must
2
show how to get the family
all we can reduce the values of {2+n}n6Z;
then we define
is clear that if
B
in such a way that z = 0,
s
and
$(2k,2 £) : 2m
t
B
R
of
B#
p{R} < 8~{R}.
is c o n t a i n e d
Clearly
M#f(x)
obtained
in an element
It
in this R
of
Next, we c o n s i d e r in the h o r i z o n t a l and to each one of
having the r e c t a n g l e as
h o r i z o n t a l base and w i t h v e r t i c a l d i m e n s i o n given by
integer m u l t i p l e s
First of
2m-I < ~(2k,2 ~) _< 2m.
the family of dyadic rectangles,
in the v e r t i c a l direction,
B#.
so that they are in the set if
these r e c t a n g l e s we attach a p a r a l l e l e p i p e d ,
late,
from
is the family of p a r a l l e l e p i p e d s
way then each element
plane
To finish, we just choose
C~e -I < i
~.
Then we trans-
each one of these p a r a l l e l e p i p e d s by
of its v e r t i c a l length.
The family obtained
is
B.
_< 8Mr(x). Q.E.D.
References [1]
[2]
A. C6rdoba, On the Vitali covering properties basis, Studia Math. 57 (1976), 91-95. ,
sXtX~(s,t),
Mittag-Leffler
of a d i f f e r e n t i a t i o n
Institute report
9, 1978.
[3]
, and R. Fefferman, A geometric proof of the strong m a x i m a l theorem, Annals of Math. 102, 1975.
[4]
B. Jessen, J. M a r c i n k i e w i c z and A. Zygmund, Note on the d i f f e r e n tiation of m u l t i p l e integrals, Fund. Math. 25, 1935.
MULTIPLIERS
OF
F(L p)
A. C 6 r d o b a Princeton University I.
The d i s c
multiplier
I would which
are
Fourier sions
related
Series.
Some
I will
a more
Consider
restrict
the
1 A > [
what
happens
The
known
outside
theory
where
Analysis
of m u l t i p l e to h i g h e r
dimen-
it is p o s s i b l e
now
defined
formula
I > 0,
to
KA
by the
f (S(m2).
is an i n t e g r a b l e
ease,
and
we s h a l l
Kk
fails
to be
and
disc).
kernel,
concentrate L1,
in
Now C. H e r z
(T O
so the
only
on
is the
[7] o b s e r v e d
that
the r a n g e 2n n+l+2A
=
of
means
be e x t e n d e d
multipliers
where
to the u n i t
p(A)
~2
in F o u r i e r
Cesaro
can
A^ f({),
in this
k s ~1
where
to
of F o u r i e r
= Kl*f
easy
associated
is u n b o u n d e d
myself
~2
results
spherical
results
(1-I~]2)+
Tlf
is v e r y
multiplier TX
=
three
in
description.
family
then
LP-theory
today
to the
of t h e s e
complete
A Tlf({) If
related~roblems
to p r e s e n t
closely
but
present
like
and
T1
in
2n < p < n-l-2 A
~2
can be
-
p'(1).
summarized
in the
following
theorems. Theorem
p(X)
(A)
(a)
TO
(b)
If
is o n l y b o u n d e d on L2(~2), (C. F e f f e r m a n [5]). i 7 ~ k > 0, then TX is b o u n d e d on LP(~2),
< p < p'(k), Given
(arbitrary
(L.
Carleson
N { i, c o n s i d e r direction)
and
Mf(x)
P.
associated
Sup
(B).
There
exists
[i]). of e c c e n t r i c i t y
maximal
S N}
function
£" I If(Y) IdY.
x(R(BN~<~;
Theorem
Sj~lin
B N = {rectangles
the
:
and
a constant
R C,
independent
of
N,
such
that
llfll2 ~{Mf(x)
for e v e r y The
> ~ > 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
satisfies the i n e q u a l i t y I
llTkfIl4 -< Ck~llfll4, Vf E S(m 2) because i n t e r p o l a t i o n with the obvious estimate k
llTkfll~ _< C~711fll~
184
yields t
IITkflIp i p series
t 4
If
~
Ck
4 4 < P < 1-21'
and if
X2
2
k(l-t) 2
then
i -klk4
2
Ilflip.
i - t < 21,
and t h e r e f o r e the
k(l-t) 2
k=l
converges,
proving the C a r l e s o n - S j 6 1 i n
theorem.
Statement and Proofs of Some Results (a)
Suppose that
[-i, +i], 8 > 0
~:~
+~
and c o n s i d e r the family of Fourier m u l t i p l i e r s
for r a p i d l y d e c r e a s i n g T h e o r e m i.
:
where
for every
~(8-i( j~ I-l)).f(~),
smooth functions
There exists a constant
IISsfN4
f.
C,
independent of
8,
such that
i ¥ C (log(i/6))llfll 4
!
f (S(~2).
(b) of
S 6,
in
is small, defined by the formula A $8f(6)
B
is a smooth function supported
Given real numbers
rectangles
in
a r b i t r a r y direction.
N ~ i
R 2
and
a > 0,
with d i m e n s i o n s
a
consider the family and
Na,
For a l o c a l l y integrable function
f
but with let us define
the m a x i m a l f u n c t i o n Mf(x)
where
~
:
sup ~ x(R(B
If(x) Idb(x) R
denotes Lebesgue m e a s u r e in the plane.
Theorem 2.
There exists a constant
C,
independent of
a
and
N,
such t h a t i jlMflj2
(e)
~
C(log 2N)~llfIl2,
for every
Given two p o s i t i v e real numbers
the family of intervals in the
y-axis,
NI
f (L2(~2).
and
{lj}_~<j<+=
N2
let us consider whose
length is
165
equal to vals
NI
and such that the distance
is equal to
N 2.
whose projection we may define
Denote
by
E~j
onto the y-axis
the
between
two consecutive
the horizontal
is the interval
strip
I.. ]
in
inter~2
Given
f (S(~2),
g-function i
g(f)(x)
:
(ZIPjf(x)12) 7,
where
P.f(~)
j
Theorem
3.
For every
p ~ 2
Hg(f)IIp
(d)
Corollary.
there
~<
exists
The operator
sion to
decreasing
Lp (~2)
XE.(~)f(~). ]
a constant
!
T&,
Cp
such that
(S(]R2).
Vf
CpHfHp,
A
for rapidly
=
]
2 >- & > 0,
defined
by
2
and smooth functions
f,
has a bounded
exten-
if and only if 4 4 3+2X < p < 1-21"
Proofs. (a)
Using
smooth cut off functions
we may decompose
3 =
j !0 ~ J'
_<
arg(z)
where
j~
~
supp(9.)2 c {z:-~ + --2 Since we may consider and since the Fourier
~j,
Next,
following
{¢j}j:l,...,[6-1/2]
j : i, 2, 3,
transform
to prove that the multiplier
j~
< ~ + 2}"
commutes
40
with rotations,
satisfies
[2], we consider
as a rotation
the estimate
a smooth partition
in the unit circle,
of
~0'
it is enough of Theorem of unity,
such that:
i ~j(9)
where
~
bounds
independent
=
--
2~j
~(6 2(8
6_112
is a smooth function of
d.
E
supported
Therefore
]
))
'
on the unit
interval,
with
i.
166
~o(~) If we define
: mo([[leie)
if([) Tj
In2 I~Tjf(x)]4dx
:
: m.(~)f(~), ]
Z~o(¢)~j(e)
Zm.(£).
j~
then we have
:
Z Tjf(X)Tkf(x) 12dx In2 I j,k
=
f ~2
:
f 1 Z iTjf*Tkf(~)12d~ j,k
C ~ f j,k ~2
<
/'~f Tjf*Tkf(~)12d~.
This last inequality holds because no point belongs to more than 16 sets of the family
Supp(T~)
+ Supp(Tkf).
H~Tjf]I 4 where
s
C is independent of Next, we split the sum
Therefore, i CH(Z lTjf(x) 12)TII4
j
6.
i
(z[Tjf(x)12) 7 3
i
<
I
(Z]T2jf(x)12) ~ + (zlT2j+if(x) 12)7 3
J
and we estimate each of the two sums separately. Since the supports of the multipliers m. are basically perpendicular to the direction of the x-axis, we may find a configuration, {E2j}, horizontal strips, as in Theorem 3, with the property that
__CT_L
XE2j([)
" m2j([)
I Figure i Therefore, 1
(ZlT2jf(x) 12)7 where
i
:
(zlT2~P2jf(x)I2) 7
=
m2j([) .
167
A P2jh([)
:
XE2 j ({)fi([)"
An elementary computation (integration by parts) shows that the kernel K 0 of T O satisfies the following: For every pair of integers p,q ¢ 0 there exists a finite oonstant Cp,q, independent of 6, such that IK0(x,y) I Therefore the operator kernel is given by
TO
C
R 0 : {(x,y):
is majorized by the positive operator whose
2-n 0
where
~
3 i Cp,q6~16xl-P]67yl -q
i p(R )
Ixl s 2n6-i,
x ,y
,
n
IYl S 2n6-i/2}
for a suitable con-
n
stant C, independent of 6 > 0. By an appropriate rotation we may get an analogous majorization for every operator T.. ] Due to the exponential decay factor 2 -n it is enough to show that, for each n, the L4-norm of the function i (Z I i ,P2jf(x)12)[ J ~(R~ j) XR2Jn i is dominated by C(log 6)$IIfI14, in L 2(IR 2) we have
JE
fm2 IP(R! j )
for every
f (L4(~2).
Given
w ~ 0
XR2J*P2jf(x) 1 2 w ( x ) d X n
Ej I ~2 IP2J f(y)I2 P(R~ i j--~ XR2j*w(y)dYn i
Z f j where
M
n
m2
IP2jf(y)12MnW(y)dy
2 lieIP2jf12)2-114 ]
is the maximal function of Theorem 2, i N = 6 2 and a : 2n"
with
" IIMnWll2,
168
Therefore i
2 I 2.~,, ) jl4
II(ZIT2jf
_<
I
Sup
ZlT2jP2jf(x)12w(x)dx
Ilwll 2<-m
j
I
-< CZ2-n(l°g n by Theorems
i
i
1/5)TH(Z]P2j
fl217112 4
-<
C(log(i/6) )~IIfll24,
2 and 3.
Q.E.n.
An analogous argument works for the odd sum.
This Theorem,
with the bigger power
[2] for the first time.
of T h e o r e m 3, yields the best exponent, (b) proved, f(x)
T h e o r e m 2 was proved as the case
= 0,
if
a = i,
Ixl > N
(log 1/5) 5/4,
was proved
The proof presented here, using the
in [2].
f(x)
in
g-function
1/4. The exponent 1/2
= (l+Ixl) -I,
if
cannot be im-
Ix I _ < N
and
shows.
Proof of T h e o r e m 2. First of all, whose d i r e c t i o n s
it is enough to prove the estimate for r e c t a n g l e s
lie in the interval
d i l a t i o n we may also assume that cal and h o r i z o n t a l lines, tor
M
[0,~/4].
a = i.
By using a c o n v e n i e n t
We divide the plane by verti-
into a grid of squares of side
acts "independently"
we can simplify the problem by c o n s i d e r i n g only functions on one of the squares of the grid.
So, let
Q
p a r a l l e l to the c o o r d i n a t e axes, and length = N, f (L2(Q).
We d e c o m p o s e the square
=
Q
0
Qip'
Therefore,
(+)Q*
supported
and suppose that
x ( Q .
9N 2 lines.
one can find a r e c t a n g l e
[0,~/4],
Qip n Rip
if
into
by v e r t i c a l and h o r i z o n t a l
interval
f
be a square with sides
(+)
,
= i,
The opera-
Then
Mf(x)
square
N.
on the squares of the grid, and t h e r e f o r e
#
and d i m e n s i o n s
~
and
if for a fixed
Mf(x)
f
_<
small squares
of side
The point is that for every Rip
i × N)
2 ~
{Qip} ,
(with d i r e c t i o n in the such that
IR. If(Y) IdY ip
XQip(X)"
we define the linear operator:
denotes the square e x p a n d e d by the factor
3.
169
Tf(g)(x)
we have
=
if
[ i,p
IRip I
Mf(x) ~ 2Tf(Ifl)(x).
R. g(y)dy Ip
" XQip (x),
In order to prove the theorem,
it is
enough to prove that i llTf(g)N2 where
C
~
is independent
C(log 3NlTllgll2Vg ( L 2 ( Q ~ ) ,
of
f
Thus we have linearized Tf
of
Tf,
Now, given hi
of width
N. and we can consider the a d j o i n
which is given by: Tf(h)(x)
where
and
the problem,
h ( L2(Q *) = hiE . 1
i.
is
=
~ i ,p
( ip
Qip
h(y)dy)xR" (x). lp
we have the decomposition
the
restriction
of
h
to
the
h = h I + ..+ h3N, vertical
strip
Ei
Then in order to prove that i IITf(h) Ii2
~
C(log 3N)[Nhll 2
it is enough to show that i lITf(hi)JI 2 *
i
-< CN--2(log 3N) 711hill2 ,
i : i ..... 3N,
since this implies that i , IImf(h)ll 2
_<
, Z llTf(hi)ll 2
i
i
l
Suppose that the function
h
compose E i into 3N squares also we decompose the function * Tf(h)(x)
i
_< CN -7(log 3N)gZIlhirl2. _< C(log 3N)211hl12.
=
is defined on the strip
pETf(hp) (x)
=
PE IRipl
Qiphp(y)dyXRip (x),
which implies that
IT (h)(x)l Therefore,
E i.
We de-
{Qip}p=l,...,3N' each of side i, and h = Zhp where hp = h lQip. We have
-<
3N X flhpllZ×R. (x). Ni p:l ip
170
I ITf(h)(x)12dx * and an e a s y c o m p u t a t i o n
This
implies
1 p !qllhpll N--~-
~
shows that
IIhpll211h 112 l+lP_ql
3N
i although
In w h a t
follows
the
we s k e t c h
loss of g e n e r a l i t y , is c e n t e r e d Let
latter theorem the p r o o f
that
N I = 2,
at the origin:
~
be a s m o o t h
the
lj
of T h e o r e m
= (wj-l,
of T h e o r e m
basically
3, a s s u m i n g
wj+l),
where
~i
on
3 to p r o v e
one-dimensional.
and the first
such that
Q.E.D.
without
interval
wj I0
I0
= j(N+I). and
~e0
interval
(
-i
and let
version
is
N2 = N
function
2
CN -I log 3N • Ilhll 2
-<
We h a v e used a two d i m e n s i o n a l
Theorem
outside
N
n Riql _< Ci-¢lP_ql
IRip
that
IIT~(h)ll~ _< C~1 p , qI= l (c)
21]hqll2[Rip n Riql,
9j(t)
_ N,
and
: 9(t-wj)
i
Sjf(~)
+
N)
: 9j(~)f(<).
nemI~a. i
ll(zlsjf(x)12)711p _< CpHfllp, for e v e r y Proof.
p ~ 2.
For e a c h
8, 0 ~ 8 ~ 2~, A Tsf({)
and o b a e m v e uniformly
in
that
its k e r n e l
8.
:
~8
Therefore,
consider
the m u l t i p l i e r
EeX0J@j(~)f({)
is a m e a s u r e
for e v e r y
8
of f i n i t e
total variation,
we h a v e
i
Integrate
p
I
with respect > -
__ 2n
to
8, (
0
and o b s e r v e IT~f(x)IPdx)d0
that
if
p h 2,
then
171
; p (z isjf(x) 12)2dx. J Q.E.D.
P 2Tr
The proof of Theorem (d)
3 is now easy to obtain.
Let us discuss
Theorem
pattern as that of Theorem Theorem. on
iTef(x)12dO)2-ax
If
f ~ LP(~2),
l<J = i,
and we have
C,
whose proof follows the same
A. i ~ p < ~,
If(~)]qda(~)
q
then
~
f
exists almost
everywhere
Ap,qllfllp
J~I=l ] q : ~-p'
where
i P 3 p-l"
It is enough to show that
Proof.
lim 6-~0
(f 8 -I
p', q'
_< Ap,qllZllp
as those of the theorem.
it is enough to prove the following
8-lllEaj~jHp,
where
q
1-7-- I~ I-
under the same conditions By duality
<
if(<) Jqd
8<
are conjugate
_<
estimate:
i i C(ZIaj jq'62) q'
exponents
to
p
and
q
respectively,
the
a.'s are complex numbers and the ~j's are characteristic functions of ] the "blocks" (61/2-in the tangential direction, 8-in the normal direction)
of the canonical
serve that
p' > 4.
~-7
partition
L'
A A J ~ i2 d j ,k=l a]ak@3~k
Where joint,"
2 = i. ri + p' and
of the annulus
I-8/2~J
0b-
Therefore,
£ p'
. ! 2
6-7 -<
Since the supports
I ~ ajak~j*~k Ird j ,k:l
of
{~j*~k )
b{supp(~j*~k) ) _< 63/2(Jj-kl+l),
yl . i 7
are "almost dis-
it follows that
172
3
67 lie j * ~ k l l ~ Hence
-<
ij_kl+l
"
we h a v e
HZaj~j IJp, A fractional
Ilzaj~jNp,
-< C6T(Z+r) ~ Z.
laJ
laklr
,k -l
integration argument yields
I )I
3( rl_ c64 i+ )
<
2r z laJ 13-r
3-
2r <
i i c6(zlajl q ,52) q T.
Q.E.D.
Observe that the only property used of the unit circle is its positive curvature.
Therefore,
we have in fact,
proved a more genera~ theorem.
References [i]
Carleson, L. and Sj~lin, P., Oscillatory integrals and a multiplier problem for the disc, Studia Math. 1972.
[2]
C6rdoba, A., The Kakeya maximal function and the spherical tion multipliers, Amer. J. of Math. 1977.
[3] [4]
,
A note on B o c h n e r - R i e s z
Fefferman, C., of Math. 1973.
[5]
,
operators,
Duke.
Math.
A note on spherical summation multipliers,
The m u l t i p l i e r problem
summa-
J. 1979. Isr. J.
for the ball, Annals of Math.
1972. [6]
, Inequalities tors, Acta. Math. 1970.
for strongly singular c o n v o l u t i o n opera-
[7]
Herz, C., On the mean inversion of Fourier and Hankel transforms, P.N.A.S. 1954.
[8]
Zygmund, A., On Fourier coefficients and transforms of two variables~ Studia Math. 1974.
II.
Further Results.
of functions
Let us begin by e d n s i d e r i n g a sharper version of the CarlesonSj~lin theorem. polygon of
Let
N sides in
Xp
denote the c h a r a c t e r i s t i c ~2
function of a regular
and consider the Fourier m u l t i p l i e r defined
173
by
Tf(~) *"
Theorem C
P
xp(~)9 (~).
=
i.
and
For
a(p),
each
p
such
independent
of
IITflIp for e v e r y
theorem
N,
<
p _ 4,
such
there
exist
constants
that
-< Cp(log N)a(P)llfll p
is a c o n s e q u e n c e
of the
following
maximal
function
Define
Mf(x)
where
<
~-_
f E S(~2).
This result:
4
that
the
parallel
Theorem
"Sup" to one
2.
=
is t a k e n of the
There
over
sides
exists
Sup i xER ~-
[ if(Y)idY JR
all r e c t a n g l e s
of the
polygon.
a constant
C,
in
~2
having
sides
Then:
independent
of
N,
such
that
Iifi122 ~{x:
These and
results S
Theorem polygonal the
L p,
p # 2?
sharp,
operator
D,
and
!
they
by
C(log
explain
the
2
the d i f f e r e n t
sides A Tf(~)
have
following
the r e g i o n
sequence PG
many
of the
e f -' . . . . . . . . . . .
4
on some
of angles, figure.
8
2
of
Is t h e r e
directions,
is b o u n d e d
P
i
behavior
question:
infinitely
= XD(~)f(~)
be a d e c r e a s i n g
consider
N)
SO
A.
naturally
whose
given
01 > 8 2 > ...
0 < 8±• < ~ ,
and
in T h e o r e m
i suggests
region
that
Let
are
X > 0
Mf(x)>e}
I
8
with
a such
174
The m u l t i p l i e r
Tel(I) and the maximal
= Xpe(~)~([),
function
H0f(x)
:
Sup
1 I ]f(y)Kdy,
x(R6Bs~ where
B@ : {rectangles of a r b i t r a r y e c c e n t r i c i t y oriented in one of
the d i r e c t i o n s Claim.
O.}. 1
Boundedness
very precise (A).
R
properties
We
shall
Lq(R2),
H8
and
of the operator
T@.
T~
are equivalent
in a
sense.
The b o u n d e d n e s s
operator on
of
show
now
under
L(P/2)'(~2),
with
the
that
assumption
Te
that
M@
is a bounded
is also a bounded o p e r a t o r on
q ~ (p',p).
One of the main ideas here is to invoke the inequality
f where
A w(x) s mal function.
If(x)ILw(x)dx
S
Cs
f
*
denotes the H a r d y - L i t t l e w o o d maxi-
I
= [(wS):'(x)] s,
and
Ek = { ( x , y )
Suppose that
( ~2,
If(x)12AsW(X)dx'
2 k ~ x _< 2 k + l } ,
(see [8])
and l e t
us c o n -
sider the operators
f Skf([)
=
XEk(~)f([)-
Then we can use the L i t t l e w o o d - P a l e y theory to obtain: i
IITfllp However, plane
if
Fk
Hk
~
is the m u l t i p l i e r operator c o r r e s p o n d i n g to the half-
tangent to
P@
along its
SkTf Therefore,
Ir (z I SkTf ] 2)~[Ip,
=
k th
HkTf.
side,
we have
175
P 2 2
I[Z [HkSkf [ [l2
Sup
=
L@)
C Sup E |ISkfl2w*(x)dx ~J
E
W
where
f [HkSkf(x)]2w(x)dx ,~i k
llwll
2
function
C Ilfll~ nP P
w* = Sup[mk(wl+g)]i/l+g k
maximal
S
k
and
in the direction
wk
denotes
the Hardy-Littlewood
Ok .
Q.E.D. (B)
We shall now show that if
is of weak type First LP(~2),
we
((p/2)', show
Te
is bounded
(p/2)'),
under
modulo
the
we have the following
L p,
p > 2
some tauberian
assumption Meyer
on
that
T8
then
Me
condition.
is bounded
on
Lemma:
i
i
ll(ZlHkf[2)711p~ Cpll(~lfk[2)711p, k where
Hk
represents
To see this, functions, support.
the Hilbert
transform
in the direction
0 k.
it is enough to work with finite collection
fl,...,fN
and we may also assume
Then we look for estimates
that each
with constants
~j
Cp,
of smooth has compact
independent
of these assumptions. Let us expand
Pe
by a convenient
Sup(diam(Supp
of
j
Then,
for each
m Hkfj(~)
j = I,...,N
=
Xp~(~+wj)
=
e
~w.-x
where of
P ~.
T~
]
A • fj(~)
f.))
]
exists
! wj
p
so that
p/2. so that I -iw.x
=
Xp (~+wj)le
\
] fj)(6+wj)
-iw.-x T~(e u
is the multiplier
3
f j),
associated
with the characteristic
function
Therefore, i
II(ZlHkfkl2)~-Ilp = II(ZlT~(e k
there
factor
j
-iw. -x
I
i
~ f~l12)711p -~ CpIl(zlfjl2)Tiip. j
Q.E.D.
176
Suppose now that
{R k}
is a c o l l e c t i o n of r e c t a n g l e s
in
B8
having
the f o l l o w i n g property:
VR,
(P)
@ U R I
]Rk
<
j
~lRkl
-
estimate must be true 2 [vi]
_< CpIURk IP .
NZXRkllE 2
q . .
Proof.
Consider
Ek :
Rk - U R i ;
IEkl ~ 9lRkl,
then
by hypothesis
J
j
w h i c h implies that
IHikkEk (x) I where
->
1
I00
on
R
k(1)
(see figure),
denotes the Hilbert t r a n s f o r m in the d i r e c t i o n of
H. zk
If we d e n o t e by
H~ zk
R k-
the Hilbert t r a n s f o r m in the p e r p e n d i c u l a r diree-
tion, we obtain,
T h e r e f o r e we can invoke Heyer's Lemma to conclude the proof of the c o v e r i n g lenm~a [vi]. Suppose, estimate:
for example, that we know that M e verifies the f o l l o w i n g i I{M~XE(X) > T}I ~ CIEI for every open set E. Then p r o p e r t y
[vi] will imply that
Me
is of weak type
(p/2)'
automatically.
It is now an i n t e r e s t i n g q u e s t i o n to decide for w h i c h families of
{e k}
spaces.
the operators
M8
or
Te
are bounded on some range of
Is there any geometric c h a r a c t e r i z a t i o n of good sets
directions? ing examples:
B a s i c a l l y our present k n o w l e d g e (i °)
If the sequence
{e k}
is contained
LP
{O k }
of
in the follow-
is l a c u n a r y then
T e is
177
bounded on every
LP
space, w i t h
i < p < ~,
i < p.
(2 ° )
If
Ok ~ k -n,
L2(~2)
and
M9
is bounded only on
Kakeya set. [C]
(See [I],
Further remarks.
Theorems
(A) and
[2],
(n = 1,2,...) h ~
and
then
( R 2)
.
M0
TO
on those with
is bounded only on
Here
,
the enemy is the
[4], and [7].)
Recently, A. Ruiz
[5] has obtained v e r s i o n s
(C) for more general types of curves
in
him to give a n e g a t i v e answer to a problem of N. Rivihre: mental
~n,
of
enabling
Is the funda-
solution
m(x,y)
=
i 2
x -y+i of the SehrSdinger operator a Fourier m u l t i p l i e r
of
LP(~2),
p # 2?
This result has been obtained i n d e p e n d e n t l y by C. Kenig and P. Tomas by a slightly d i f f e r e n t method
[3].
References [i]
C~rdoba, A. and Fefferman, R., On the e q u i v a l e n c e b e t w e e n the boundedness of c e r t a i n classes of m a x i m a l and m u l t i p l i e r operators in Fourier analysis, P.N.A.S., USA, 1977.
[2]
,
On d i f f e r e n t i a t i o n of integrals, P.,
F.N.A.S.,
USA,
1977.
[3]
Kenig, C. and Tomas,
to appear.
[4]
Nagel, A., Stein, E., and Wainger, S., D i f f e r e n t i a t i o n along lacunary directions, P.N.A.S., USA, 1978.
[5]
Ruiz, A.,
[6]
C6rdoba, A., Math. 1977.
[7]
Stromberg, J., Maximal functions for r e c t a n g l e s with given directions, M i t t a g - L e f f l e r Inst. (1976).
[8]
C6rdoba, A. and Fefferman, C., A w e i g h t e d n o r m i n e q u a l i t y for singular integrals, Studia Math. 57 (1976).
Thesis, U n i v e r s i t y of Madrid. The m u l t i p l i e r problem for the polygon, Annals of
