Oct 27, 1986 - 1. Richard Rochberg. The basic theory of Toeplitz and Hankel operators acting on the Paley-Weiner space is developed. This includes criteria.
Integral Equations and O p e r a t o r T h e o r y Vol. I0 (1987)
0378-620X/87/020187-4951.50+0.20/0 9 B i r k h ~ u s e r Verlag, Basel
TOEPLITZ AND HANKEL OPERATORS ON THE P A L E Y - W I E N E R
Richard Rochberg
SPACE
1
The b a s i c t h e o r y of T o e p l i t z and Hankel o p e r a t o r s acting on the P a l e y - W e i n e r space is developed. This includes c r i t e r i a for b o u n d e d n e s s , compactness, b e i n g of finite rank, and m e m b e r s h i p in the S c h a t t e n - v o n N e u m a n n ideals, similar questions are c o n s i d e r e d for the r e l a t e d o p e r a t o r s formed b y c o m m u t i n g the d i s c r e t e H i l b e r t t r a n s f o r m w i t h a m u l t i p l i c a t i o n operator.
i_~. I N T R O D U C T I O N : of ~, H 2 the a s s o c i a t e d projection
Let L 2 = L2(~)
H a r d y space,
from L 2 to H 2.
be the L e b e s g u e
and P+ the o r t h o g o n a l
For a f u n c t i o n ~ d e f i n e d on ~ let M
be the m u l t i p l i c a t i o n
o p e r a t o r a c t i n g on f u n c t i o n s
= ~(x) f(x).
and Hankel
Toeplitz
the r e s t r i c t i o n
of M
is the o p e r a t o r T
to H 2.
(f) (x)
operators
are the two p a r t s of
The T o e p l i t z
o p e r a t o r w i t h symbol
T (f) = P M for the moment,
P_ = I-P+.
on ~; M
w i t h d o m a i n H 2 g i v e n by
(T) where,
space
The Hankel
P = P+.
(f) = P(~f) Let P_ be the p r o j e c t i o n
onto H 2,
o p e r a t o r w i t h symbol ~ is the o p e r a t o r H
defined by (H)
H
(f) =
( I - P ) M (f) = (I-P) (~f).
(If P = P+ t h e n I-P = P_. our c h o i c e of P.) conditions
Thus, M
Foundation.
= T
in a m o m e n t we will change
+ H
Of course,
on ~ this is only a formal calculation.
t h e o r y of t h e s e o p e r a t o r s
iSupported
However,
consists
without Part of the
of g i v i n g c o n d i t i o n s
in p a r t b y a g r a n t from the N a t i o n a l
science
on
188
Rochberg
w h i c h a l l o w such c o m p u t a t i o n s interpretation.
to be g i v e n r e a s o n a b l e
More generally,
the t h e o r y of T
on the r e l a t i o n b e t w e e n the p r o p e r t i e s smoothness,
range of values,
properties essential
w i n d i n g number,
of the c o r r e s p o n d i n g spectrum,
introduction by Douglas
operators
s i n g u l a r numbers,
etc.)
(norm,
index,
to the theory of Toeplitz
operators
include
[PH],
Before g o i n g further, that g i v e n by
[Po i], and
is conjugate
is o b t a i n e d
[Po 2].
More
of H
from
(f) = P(~f).
w i t h e i t h e r one.
linear,
not linear.
(H') has the a d v a n t a g e
d e f i n e d by
uses special p r o p e r t i e s
(H) and by
one
of d e f i n i n g an o p e r a t o r
(Warning:
The fact that
(H') are r o u g h l y e q u i v a l e n t
of H 2 and L 2.
to rather d i f f e r e n t
Unfortunately,
However,
H e n c e for issues of size w e can w o r k
w h o s e d o m a i n and range are the same. the o p e r a t o r s
(H); in fact this
from the other by c o m p o s i t i o n w i t h a
linear isometry.
generalize
on H 2 is g i v e n
we change the d e f i n i t i o n
H (f) = PM
conjugate
A good
(H) to the o p e r a t o r d e f i n e d by
(H')
operator
and the
[R I].
This is n o t the same as the o p e r a t o r d e f i n e d by new o p e r a t o r
centers (size,
spectrum,
etc.)
[D] and to Hankel operators by Powers
recent papers
and H
of the symbol ~
The two d e f i n i t i o n s
operators
in other contexts.
both sets of g e n e r a l i z a t i o n s
are called
"Hankel
operators".) Certain properties Toeplitz
and Hankel o p e r a t o r s
fact that functions (on ~ ) .
of H 2 are v e r y useful
in
on H 2.
First,
kernels,
of course,
H 2 can be r e g a r d e d as a n a l y t i c
A s s o c i a t e d w i t h this d e s c r i p t i o n
reproducing
in the t h e o r i e s
of e l e m e n t s
is the functions of H 2 are
that is, for each p o i n t z in ~+2 there
f u n c t i o n ~z such that if f is in H 2 then f(z) = . H 2 has an a l g e b r a i c product
structure;
group of a u t o m o r p h i s m s
is a
Second,
if f and g are in H 2 and the
fg is in L 2 then fg is in H 2.
Third,
there is a large
w h i c h act on ~2+ (translations
of
and
Rochberg
189
dilations)
and induce nice actions on H 2.
Finally,
H 2 has the
p r o p e r t y that if f is in H 2 then the c o m p l e x conjugate, orthogonal
to H 2.
The b a s i c s of the t h e o r y of T o e p l i t z H 2 are n o w fairly well understood.
and Hankel
spaces of a n a l y t i c just listed.
on o t h e r H i l b e r t
of one c o m p l e x variable,
h a v e b e e n m a d e to o p e r a t o r s
on subspaces
spaces
[McD S],
[Bu], the Fock space
space)
[BC],
[L],
[JPR],
[R 2],
Besov spaces,
[F],
[S],
functions
[JPR],
and H p spaces
L2(~)
consisting
and Hankel
functions
[Un] and spaces of
operators
PW is a s u b s p a c e of L 2 =
of o p e r a t o r s
(H') give e s s e n t i a l l y
on H 2 fail in PW.
to f(ax)
m a p s PW out of itself if
coincide.
Second,
is a c o n t r a c t i o n lal > i.
g r o u p of u n i t a r y o p e r a t o r s on PW.
Third,
at
although PW is
H e n c e the d e f i n i t i o n s
the same families of o p e r a t o r s
and Hankel t h e o r i e s
contractions
First,
functions,
of ~ induce u n i t a r y maps of PW but the d i l a t i o n s
natural
to the
A l t h o u g h this space has m a n y
in PW can be r e g a r d e d as a n a l y t i c
The m a p of f(x)
on PW, the
of the p r o p e r t i e s w h i c h w e r e useful
closed u n d e r c o m p l e x conjugation. Toeplitz
complex
functions w h i c h s a t i s f y a g r o w t h c o n d i t i o n
to H 2, several
in the a n a l y s i s
[Pel 2].
of functions that are the r e s t r i c t i o n
(in the c o m p l e x plane).
similarities
[JPS]
of several
[Up],
space of entire functions.
real axis of e n t i r e infinity
[Pe 2],
(=Segal-Bargmann
[Pel i].
H e r e we look at T o e p l i t z Paley-Weiner
extensions
of H 2 [AR], the B e r g m a n
T h e r e are also result for spaces of functions vector valued
on
functions w h i c h have some of the p r o p e r t i e s
For functions
[BC],
operators
R e c e n t l y w o r k has b e e n done
e x t e n d i n g p a r t s of these t h e o r i e s to o p e r a t o r s
variables
f is
Thus,
(T) and
and the
the t r a n s l a t i o n s of ~ do not.
on PW if
la] ~ 1 but
in c o n t r a s t to the
on H 2, there is a s e m i g r o u p
of
if f and g are in PW and fg is in L 2,
it need not be true that fg is in PW.
T h e s e facts,
m o s t of all, p r e v e n t m a n y of the formal c o m p u t a t i o n s
the t h i r d of
[G],
190
Rochberg
[JPR]
and
[R i] from b e i n g v a l i d
investigate hold
the e x t e n t
to w h i c h
in t h i s d i f f e r e n t
setting.
In t h e n e x t about
spaces
and o p e r a t o r s
H
operators
to be of f i n i t e
5.
and T
.
similar
4.
3 presents
rank.
Boundedness
The Schatten
of H a n k e l
essentially results
into t h r e e pieces.
Hankel
f r o m H 2.
reproducing
realization
see t h a t the o p e r a t o r s
operators
techniques
w i t h the P l a n c h e r e l - P o l y a we o b t a i n r e s u l t s Hankel
operators
looked
and,
using
[R 2] t o g e t h e r
W h e n the p i e c e s
Presumably
to T o e p l i t z
are combined, in
these operators
operators
[R 2] for
also have
on H 2 b u t w e h a v n ' t
into that.
In t h e p r o c e s s these
theorem.
in
(H')
are
p a r t can be s t u d i e d to t h o s e
(T) or
using the
q u i t e s i m i l a r to t h o s e d e s c r i b e d on H 2.
some r e s e m b l a n c e s
Two of t h e p i e c e s
similar
here
on PW.
on PW by
on H 2 and are s t u d i e d
The remaining
kernel
Hilbert
on ~2 a r i s e s
operators
defined
are
is in S e c t i o n
of the d i s c r e t e
as p a r t of a m a t r i x
of
for the
and compactness
ideal t h e o r y
6 w e s t u d y the c o m m u t a t o r
on H 2
of c l a s s e s
the criterion
This operator
can b e b r o k e n
to t h o s e
facts a n d f o r m u l a s
transform with multiplication.
We will
Here we
and g i v e a few e x a m p l e s
Section
in S e c t i o n
In S e c t i o n
results
section we collect basic
operators
considered
in t h i s context.
operators curiously,
smoothness
of s t u d y i n g
conditions
w e are led to c o n s i d e r on ~.
spaces
These
(Besov,
spaces
BMO,
CMO)
s i m i l a r to t h e w a y the c l a s s i c a l
on t h e s y m b o l s
new
function
are a n a l o g o u s
spaces
More explicitly,
the classical
homogenous
defined
of s m o o t h n e s s
in t h e l a r g e
in t e r m s
t h e small.
The c o m m o n l y
are d e f i n e d
in t e r m s
control in t e r m s control function
in t h e large. of d e g r e e s at small
studied
are r e l a t e d
(i.e.
scales.
spaces
are d e f i n e d
in the large and a p r i o r i
This combination
spaces we consider
are
and in
and a priori
The s p a c e s w h i c h we e n c o u n t e r
of s m o o t h n e s s
to H 2.
spaces
at ~)
smoothness
in the small
on ~,
to PW in w a y s
smoothness
inhomogenous
of s m o o t h n e s s
spaces
to c l a s s i c a l
b u t are r e l a t e d
Besov
of
consist
arises because
of f u n c t i o n s
with
the
compactly
Rochberg
191
supported
F o u r i e r transforms.
in the small,
although
Thus the f u n c t i o n s
oscillation
The last s e c t i o n c o n t a i n s
at i n f i n i t y
are all s m o o t h
is possible.
some f u r t h e r q u e s t i o n s
and
comments. The l e t t e r c will be u s e d for v a r i o u s M y t h a n k s to B j o r n Jawerth, Taibleson
for m a n y p a t i e n t
constants.
S t e p h e n Semmes,
explainations
and M i t c h
and h e l p f u l
suggestions.
2__~. DEFINITIONS AND BASICS: A.
T h e s p a c e PW:
Let I = [-~,~]. For a f u n c t i o n
f d e f i n e d on I we d e n o t e by ~ f the f u n c t i o n
on ~ w h i c h agrees w i t h f on I and v a n i s h e s Fourier transform transform
on R\I.
D e n o t e the
of g by ~ g = g^ and the inverse F o u r i e r
of f b y ~ - I f = fv: gg(t)
= g^(t)
= c f g(x)
e -ixt dx
u %
and ~-lf(x)
The P a l e y - W i e n e r u n d e r ~-i.
= fV(x)
space,
PW,
= c
f(t)
e itx
dt.
is d e f i n e d to be the image of L2(I)
That is, PW = { ,-l(~f):
f 6 L2(I)
}.
We n o r m PW b e r e q u i r i n g ~ to be an isometry.
(A one p a r a m e t e r
family of such spaces can be o b t a i n e d by r e p l a c i n g of I.
That e x t r a g e n e r a l i t y
in our context.) description
the r e s t r i c t i o n exponential < ~.
gives n o t h i n g e s s e n t i a l l y
The P a l e y - W e i n e r
of f u n c t i o n s
in PW.
t h e o r e m gives an a l t e r n a t i v e
to the real axis of an entire
an e n t i r e
function
function
llfll =
of
(f If(x) ]2dx) I/2
f is said to be of e x p o n e n t i a l
type at m o s t ~ if for each p o s i t i v e that
different
f is in PW if and only if it is
type at m o s t ~ w h i c h s a t i s f i e s
(Recall:
I by d i l a t e s
e t h e r e is a c o n s t a n t A so
If(z) I < A exp((~ + 6)Izl).).
L2(I) has a c o n j u g a t e , d e n o t e " ", d e f i n e d b y
linear i s o m e t r i c
involution,
w h i c h we
192
Rochberg
(2.1) Thus
h for f in PW,
L2(~),
(f)^ =
PW is p r e s e r v e d
restriction
(t) = h(-t)
(f^)*
In particular,
when taking
complex
to the axis of the e n t i r e
as a s u b s p a c e
conjugates.
function
g(z)
of
(f is the
= f(~).)
For any z in ~ and any t in ~ set e z = ez(t ) = K(t)
e
izt
and set izt fz = fz (t) = ~2I (t) e ~-l(ez) (x) = c and w e
(sin~(x+z))/~(x+z).
introduce
the c o m m o n
functions
s h o w up often
abbreviation sin ~ x ~x
sinc (x) = ~(2~)-i/2ek;
These
k 6 Z~ is an o r t h o n o r m a l
basis
of L2(I).
Hence
J
k
{sinc(x-k)}
is an o r t h o n o r m a l
basis
the f u n c t i o n s
sinc(x-y)
PW.
if f is in PW t h e n
That
is,
of PW.
are r e p r o d u c i n g
It is a l s o t r u e t h a t
kernels
for e l e m e n t s
of
X
(2.2) Thus
f(y)
=
the F o u r i e r
given
by the
expansion
functions
= [ f(x)
sinc(x-y)
of f in PW w i t h
sinc(x-k)
takes
respect
dx. to the basis
a particularly
elegant
form: (2.3) This
f(x) is the so c a l l e d
of f w i t h
interpolation
formula
the p r e a s s i g n e d
discussion
of t h e s e
For the rest projection
f(n)
ne~ cardinal
the e x p a n s i o n
taking
= ~
series
respect
which
of f.
{f(n)}
about
It is s i m u l t a n e o u s l y
to an o r t h o n o r m a l
constructs
values
facts
sinc(x-n).
the u n i q u e
at the
basis
integers.
PW and m u c h m o r e
of this p a p e r we let P d e n o t e
and an
element
of PW
A
is g i v e n
in
[HI.
the o r t h o g o n a l
of L 2 onto PW.
B__~. S p a c e s of the B e s o v
of s m o o t h
functions:
and o s c i l l a t i o n
spaces
We recall
the d e f i n i t i o n s
w h i c h we will m e e t
later.
Rochberg
For
193
l = ( - 1 ) j ยง of ej * e k gives
(ej * ej)(t) = c ( 2 ~ and in the off-diagonal case (ej * ek)(t) of ~.
Itl) ~2i(t)
= c (-I) j+k sgn(t)
The ajk can be expresses transforms
. fj(t),
(fk(t)-fj(t))/(k-j).
most conveniently
in terms of two
Define A~ by
(2.7) (A~)^(t) = ( 2 ~ - Itl) ~2i(t) ~^(t). (The name of the operator is suggested by the shape of the multiplier. ) symbol
of H
Let K~ denote the Hilbert ;
(K~)^(t)
By the previous
= i sgn(t)
computations,
transform
N2i(t)
of the standard
~^(t).
the diagonal
matrix
elements
196
Rochberg
are g i v e n by ajj = c. J supported
However,
on 2I, = gV(_j).
for any f u n c t i o n g
Thus,
ajj = c A ~ ( - j ) . Similarly,
the ajk can be e x p r e s s e d
simply in terms of the v a l u e s
of K~; K~(-j) - K~(-k) ajk = c j - k T h e s e m a t r i x entries are the m a t r i x e n t r i e s of the o p e r a t o r on 82(Z)
which
the s e q u e n c e
is the c o m m u t a t o r
{K~(-n)}
transform which k(n)=
i/n
Hankel
is g i v e n by c o n v o l u t i o n
operators
(on Z) w i t h the f u n c t i o n
This is a v e r y i n t e r e s t i n g
on H 2.
r e l a t e d to the o p e r a t o r s
Hankel o p e r a t o r s on L2(~)
analogy with
on H 2 are c l o s e l y
o b t a i n e d by c o m m u t i n g p o i n t w i s e
and the H i l b e r t t r a n s f o r m
[R 2].
T h a t is, Hankel
operators
on H 2 are c l o s e l y r e l a t e d to the s i n g u l a r
operators
on L2(~)
intergal
w i t h kernel h(x) a(x,y)
for some f u n c t i o n h. off-diagonal
by
and the s o - c a l l e d d i s c r e t e H i l b e r t
(k(0) = 0).
multiplication
of p o i n t w i s e m u l t i p l i c a t i o n
- h(y)
=
x - y This kernel is the c o n t i n u o u s
part of the m a t r i x
a n a l o g i e s b e t w e e n results
for H .
a n a l o g of the
In S e c t i o n
for the two classes
6 we examine
of operators.
F__~. E x a m p l e s :
I.
Finite dimensional
suppose ~ = fz.
operators:
For f in PW, x in ~, cH
P i c k z in C and (f)(x)
= =
= = = = e z. z o p e r a t o r w h o s e symbol is the r e p r o d u c i n g
kernel
for the space PW b a s e d on t h e interval 2I are n o r m a l i z e d one-dimensional spaces
operators.
On the Hardy,
Bergman,
and F o c k
it is the choice of ~ equal to the r e p r o d u c i n g
kernel
for
=
Rochberg
197
the space itself which produces the rank one Hankel operators. Here we use the reproducing kernel for the different space, based on 2I.
PW
This is because the product fex need not be in PW
even if it is in L 2. Of course finite sums ~ = ~ cif~. give finite rank 1 operators. We will see in the next section that those are essentially the only finite rank H .
In Section 5 we will see
that infinite sums of that sort with e p conditions of the coefficients give all of the H 2__~. A n a l y t i c symbols: operators T
in the Schatten ideals S p. If we restrict attention to
for which ~^ is supported in ~+ we get operators
(unitarily equivalent to operators) Sarason
[S], Frankfurt and Rovnyak
which have been studied by [FR] and others.
Let 8=8(x) be the function on ~ defined by 8(x)=e l~x. M u l t i p l i c a t i o n by 8 carries PW into H 2.
In fact it establishes a
unitary map between PW and K, the orthogonal complement of 82H 2 in H 2.
Let PK be the orthogonal projection of H 2 onto K.
Thus
T (f) = 8PK(~Sf). Let R
denote the Toeplitz operator on H 2 with symbol ~. T
We have
is unitarily equivalent to P ~ ] K "
(The bar denotes restriction of the domain.)
K is the orthogonal
complement of a subspace of H 2 which is invariant under m u l t i p l i c a t i o n by analytic functions.
Sarason
IS] developed a
systematic theory of the operators obtained when Toeplitz operators
(on H 2) with
analytic symbol are applied to functions
in such a K and the result projected back to K.
These
compressions of Toeplitz operators are unitarily equivalent to Hankel operators. 82p_(82g)
This follows from the identity PK(g ) =
which is valid for all g in H 2.
the Hankel operator on H 2 defined by we have
Thus,
letting J denote
(H) and having symbol ~ 2
,
198
Rochberg
PK Note, K)
however,
that 82H 2
is in t h e k e r n e l
direct
s u m of T
operators
(which is the o r t h o g o n a l
of J.
T h u s J is u n i t a r i l y
equivalence
ideal c r i t e r i a
Theorem
Suppose
I.
T
complement
equivalent
a n d the k n o w n
results
one can n o w r e a d off b o u n d e d n e s s ,
and Schatten 2.1:
9 of
to a
and a zero operator 9
using this unitary Hankel
IK = e2Jl
~ is analytic 9
is b o u n d e d
II T it ~
for s u c h T
about
compactness,
.
Set @ = p_(~2
if a n d o n l y
if @ is in BMO.
if
if
). In t h a t
case
II @ II,
2.
T
is
compact
3.
T
is
in
the
and o n l y
Schatten
if ~ is in Bp 9 result
ideal
~ is Sp ,
CM@9
1 ~ p < ~,
if
IITIIsp ~ II~IIBp.
In t h a t c a s e
extends
in
to p > 0 w h e n
B
P
and o n l y (This
is d e f i n e d
appropriately 9 Proof:
See
[R i] a n d t h e r e f e r e n c e s
Or c o u r s e conjugate
there
is a c o m p l e t e l y
analytic 9
(29
T
there. analogous
result
Also,
at l e a s t
formally,
= Tp+(~)
+ Tp_(~)
= T+ + T_.
of T
then we could use the previous
t h e o r e m to s e t t l e t h e q u e s t i o n s
subsection, modify
this
However,
splitting
the splitting
w h i c h c a n be s t u d i e d
so t h a t
chosen basis
used
in
it p r o d u c e s
using Theorem
[R 2] a n d
set of r e p r o d u c i n g for the u n d e r l y i n g
diagonalize similar subclass
Hankel
result
is t r u e
of the H a n k e l
Given H
acting
t h r e e terms,
2.H w e
two of
A general
is t h a t an a p p r o p r i a t e l y a l m o s t be an o r t h o n o r m a l
space and w i l l
We will
almost
see in S e c t i o n
For n o w w e n o t e t h a t
operators
on PW m u c h m o r e
5 that a
for a
is true.
on PW we can r e g a r d ~^ as a f u n c t i o n
As s u c h it h a s a F o u r i e r
of
2 9149
kernels will
in PW.
In S e c t i o n
almost-diaqonalization:
[JPR]
Hilbert
operators.
b y T+ and T_
as w e see in t h e n e x t
isn't t h a t nice.
3__~. T h e D h i l o s o D h V o f philosophy
inherited
with
If w e k n e w t h a t t h e p r o p e r t i e s
the n e x t two sections 9
were
for T
development
with respect
on 2I.
to t h e set
Rochberg
199
.{fn/2; n in ~}.j half-integer
(Because 21 has length 4~ we must include the
frequencies.)
~^ = ~ Cn/2 fn/2 " Thus, by the computations of Section 2.F.I, v
H (') = c ~ Cn/2en/2. We can use the Fourier inversion formula to evaluate the c
n
This gives, H (.) = c ~ ~(n/2)en/2. Although this is a simple looking formula the set (en/2) an orthonormal basis
However,
Thus if only integer
show up in the expansion of ~^ then the operator
is in diagonal
H (-) = c ~ ~(n)e n. form and we can read off whatever
For example the eigenvalues
information we
of the operator are exactly
the values of the standard symbol at the points of Z. observation
is not
the set {en}ne Z is an orthonormal
(no almost in this case!).
frequencies
want.
basis.
This
is in line with the results of Widom in [W] that for
certain finite convolution
operators the eigenvalues
are closely
related to the values of the symbol at regularly spaced points on the axis.
This is in contrast to the case of an analytic ~.
that case the eigenvalues
are related to the asymptotics
In
of ~ at
im (Theorem 1 of [FR]). Similar comments apply if ~ only contains form n + 1/2, n 6 Z. operators
However the basis that diagonalizes
of the those
is different.
In general ~^ can be split into two parts, frequencies
and one with frequencies
corresponding way.
frequencies
operators
one with integer
n + 1/2 and each of the two
can be diagonalized
in a very natural
But, we will see as a corollary of Theorem 5.2 that this
splitting of the operator into two pieces
is not continuous
in
operator norm or Schatten ideal norms. The splitting of the operator induced by splitting the symbol into analytic and conjugate analytic parts, also does not respect the boundedness see that using the fn'S.
Suppose
~^ = ~
of the operators.
+ ~[, We can
200
Rochberg
~N
~N
~^ = c
to T
(for all N). ~
inx e
X2I" 1 in the p r e v i o u s
1 of d i a g o n a l i z a t i o n
The d i s c u s s i o n applies
fn =
and we c o n c l u d e
that T
~+ = ~ [ 0 , 2 ~ ] "
By d i r e c t
= ~^/2
+ , = ~^/2
paragraph
is has o p e r a t o r
n o r m one
computation
+ c ~ b(2m+l)/2
f(2m+l)/2
with N b(2m+l)/2 Again,
Hence,
and t h e i r
by s u b t r a c t i o n
bounded.
Thus
the s p l i t t i n g Schatten
are not u n i f o r m l y
is c o n t i n u o u s
norm.
with
respect
by the s p l i t t i n g
of
We see in S e c t i o n to the norms
5 that
of the
T
symbols:
defined
by
If ~ is any b o u n d e d
(T) is a b o u n d e d
function
operator
then
and liT II
Thus lIT I[ ~ inf ~ll~ll : T
[S] S a r a s o n
attained bounded
in
shows
(2.9)
analytic
that
and
bounded
~.
extension
Although
our m o r e g e n e r a l
However, Fourier simplest
choice
interpret 192]
choice
we w i l l
emphasize
Although
the s p l i t t i n g
that there of ~ w i t h
(2.8).
that even
In that
this m a k e s
In fact,
In this
case
our a t t e n t i o n
to
in
all b o u n d e d
for b o u n d e d
T
good
sense
Sarason
analytic
The
the point
as a
clear points
~ such that
will ~^ be a measure.
q the
some care.
case ~ = 60,
it is no longer
with
is has a
~.
m u s t be done w i t h
transform,
are even T
T@ = T
@ which
such a n o r m e q u a l i t y
see t h a t we o b t a i n to b o u n d e d
for ~ is ~ ~ i. Fourier
if we r e s t r i c t
not o b t a i n
calculations
at the origin.
distributional
even
attention
we should
transform
then equality
for a f u n c t i o n
to the u p p e r h a l f plane. T
we will
case,
if we r e s t r i c t
= T~.
if ~ is a n a l y t i c
is a t t a i n e d
we get all of the b o u n d e d
p.
~+
of T induced
in the o p e r a t o r
(2.9)
mass
T
ibn! ~ log N.
classes.
the o p e r a t o r
even
are g i v e n by sup
the o p e r a t o r s
4__. B o u n d e d
II~II.
norms
the s p l i t t i n g
is not c o n t i n u o u s
In
~ log(l+N/m). 1 in the p r e v i o u s paragraph, the o p e r a t o r s
by the d i s c u s s i o n
T@ are d i a g o n a l
1 2m+l-2n
= c ~
h o w to out
IS,
for no
Rochberg
201
G. T r a n s l a t i o n s translation non-zero Suppose
and dilations:
acting on functions
on ~, Ut(f) (x) = f(x-t).
t, let D t be the dilation ~ is a b o u n d e d
the Toeplitz
operators
respectively.
function
For t in ~ let U t be operator
on ~.
Let R , ~ t ~ ,
=
and RDt ~ be
on H 2 with symbols ~, Ut~ , and Dt~
Using the fact that the p r o j e c t i o n
is given by PH2(f)
For
Dt(f) (x) = f(tx).
of L 2 onto H 2
(~(0,~)f^) v, it is easy to check that
(2.10)
RUt ~ = UtR U_t
and (2.11)
RDt ~ = DtR DI/t.
The operators unitary,
U t are all u n i t a r y
the n o r m a l i z e d
are u n i t a r y
on L 2.
versions,
Although
the D t are not
D~, defined by D~f =
ItI-i/2Dt f
and we also have
(2.12)
RDt ~ = DIR D~/t.
(2.11) all the U t.
is a c o n s e q u e n c e
of the fact that P+ commutes
with
The same is true for P and hence
(2.13)
TUt ~ = UtT U_t.
The p r o j e c t i o n
P does not commute with the operators
D t.
However
if Itl > 1 then Di/t maps PW into itself and we have a partial analog of (2.12); (2.13)
TDt ~ = P DtT DI/t I I = P DtT DI/t
H. version
Splitting
the symbol:
of the splitting
Itl ~ i.
We now introduce
~ = ~+ + ~_.
Pick and fix ~L in C O which has supp(~L) ~L(X)
a more gentle
= 1 for x in [-3~,-~].
Set ~R(X)
be 0 off 2I and ~C = I-@R-~L on I. left, right and center portions to introduce
C
= ~L(-X)
[-4~,-~/2],
and define ~C to
(The subscripts
of 2I.)
and
refer to the
It is also convenient
,(t) = (2~ - Itl)-l~2I(t)~c(t). c Define ~L by ~L =
(@L ~^)v
and set TL=T L.
Because @L is in Co,
202
Rochberg
the p a s s a g e function Hence,
from ~ to ~ L is g i v e n by c o n v o l u t i o n w i t h a smooth
in LI(R).
by
Thus ~ L is a w e i g h t e d
(2.13), T L is a w e i g h t e d
sum of t r a n s l a t e s
of 9.
sum of b o u n d e d operators.
Thus IITLII ~ c IITII. U s i n g the fact that the S c h a t t e n classes are n o r m e d
ideals
if p
i, we also find IITLIIsp ~ c IITIIsp. (For p < 1 the S p are only quasi-normed.
In that case it is not
true t h a t II ~- llsp < ~ ll.llsp. Thus the p r e v i o u s doubt and w i t h it m u c h of the a n a l y s i s Similar definitions
inequality
is in
of the next two sections.)
and c o n c l u s i o n s
a p p l y for ~R' ~C'
and
for ~ , as w e l l as for TR, TC, and T ,. We h a v e ~ = ~ L + ~C + ~R C C and T = T L + T c +T RNote that A~ , = ~C" (A was d e f i n e d in (2.5).) A l s o note C that the p a s s a g e s from ~C to ~ , and b a c k are g i v e n b y C c o n v l o u t i o n w i t h L 1 functions. for the S p norms passage
from T
for p ~ i.
Hence
IITcIi ~ lit ,II and s i m i l a r l y C
The crucial p o i n t is that the
to TA~ and b a c k is a tame one if the s u p p o r t of
stays away from the edges of 2I. 3_~. K R O N E C K E R i S
Let
H
operator P_(~). operator
THEOREM
ON FINITE
RANKHANK~L
be a Hankel o p e r a t o r on H 2 as d e f i n e d by is c o m p l e t e l y Kronecker's
d e t e r m i n e d by P_(~)
t h e o r e m states that H
(H).
is a finite rank
in the u p p e r half plane.
to R of) a r a t i o n a l In that case the
rank of the o p e r a t o r equals the n u m b e r of poles. is of finite rank e x a c t l y reproducing
kernels
derivatives
of r e p r o d u c i n g
formulation, general
The
so we a s s u m e that ~ =
if and only if ~ is (the r e s t r i c t i o n
function with poles
OPERATORS:
Equivalently
if ~ is a finite linear c o m b i n a t i o n
(or a limiting
of
form; a sum i n v o l v i n g
kernels).
With essentially
the result is also true for Hankel
B e r g m a n and Fock spaces
H
[JPR].
this
operators
The p r o o f s
on v e r y
in t h o s e
Rochberg
203
contexts space.
u s e the a l g e b r a i c (In t h o s e
cases
structure
of the u n d e r l y i n g
the set of p o l y n o m i a l s
the o p e r a t o r
forms
doesn't
the same type of a l g e b r a i c
have
a Kronecker integral
an ideal
theorem
operators
using with
Hilbert
in the kernel
in the r i n g of p o l y n o m i a l s . ) structure.
the r e a l i z a t i o n
a kernel
H e r e we prove
of H a n k e l
that depends
of
PW
operators
as
on the sum of the
arguments. In S e c t i o n one operator.
2.F.I we saw t h a t
if ~^ = fz t h e n H
For k = 0,1,2,...,
let
~k f~k) (t) = _ _ fz(t) 8z k
= ctke izt.
It is s t r a i g h t f o r w a r d
to c h e c k t h a t the c h o i c e
a r a n k k+l operator.
We now p r o v e
Theorem
H
3.1:
are p o i n t s that
Suppose
~^ = f(k) z
produces
the converse.
is a Hankel
zj in C, n o n - n e g a t i v e
is a r a n k
operator
integers
of r a n k n.
There
k~,j and s c a l a r s
aj so
for t in 2I (kj)
(3.1)
~^(t)
= ~ aj f
(t)
zj
and (3.2)
~ kj+l = n.
Proof: Hence such
The o p e r a t o r there
that
integral
K = ~ H 3 -1 is a r a n k n o p e r a t o r
are 2n f u n c t i o n s
K(f)
= ~hi.
representation
in L2(I),
g l , . . . , g n and h l , . . - , h n
W h e n we c o m p a r e
in S e c t i o n
on L2(I).
this w i t h the
2.D we c o n c l u d e
that
for s,t
in I (3.3) We w i s h
~^(s+t) to s h o w that
straightforward to t h a t
1,2,...,n
we c o n v o l v e
in C a w h i c h O
For p o s i t i v e
(3.1).
with
is positive,
16
= h.
commutes
1
* F .
with
6
This
involved
a smooth
is
are smooth.
function.
To get
Let F be a
supported
e let Fe(t ) = eF(t/6).
let h.
translation
= ~ gi(s)hi(t).
implies
if all the f u n c t i o n s
situation
function
(3.3)
in I and has F F = i. J Let ~e = ~^*Fe and for i =
We n o w use
convolution.
(3.3)
and the
For small
e and
fact that for fixed
204
Rochberg
s and v a r i a b l e
t w i t h b o t h s and t in the interval
(i-36)I,
~6(s+t) = (~^*F6) (s+t) = (~^(s+-)*F6)(t) = ~ gi(s) h i 6 ( t ) . For fixed s this e q u a t i o n involves only smooth functions of t. Thus t h e r e is no p r o b l e m d i f f e r e n t i a t i n g .
~(T6k) (s+t) = ~ n (s) h!k) (t). igi 16
(3.4)
Now fix s and regard t as variable. 0s~6(s+t ) .
Thus by
first n d e r i v a t i v e s gi"
z.
zero.
of ~e,t"
S o l u t i o n s to c o n s t a n t c o e f f i c i e n t
Thus,
(3.5)
and its
at m o s t n, h e n c e t h e r e is a linear
and t h e i r v a r i o u s derivatives.
J space c o n t a i n i n g
= ~e(S+t)
are all in the linear space s p a n n e d by the
This space has d i m e n s i o n
we want.
Note that 8 t ~ e ( S + t ) =
(3.4) the function ~e,t(s)
r e l a t i o n b e t w e e n these d e r i v a t i v e s f
We o b t a i n
these derivatives
for all s u f f i c i e n t l y ~e,t(s)
This is almost what 0DE's are e x a c t l y the
Also note t h a t the linear d o e s n ' t change as e tends to small e,
(k i) = ~ ai(e,t ) fzi (s).
For almost every t, ~6,t(s)
converges to ~^(s+t)
s.
As e tends to zero, the left hand
Pick and fix such a t.
side of
(3.5) tends to a limit for almost all s.
side stays
inside the same n - d i m e n s i o n a l
exponentials function
times polynomials.
is also in the space.
Hence,
as required,
(Reason:
convergence
at lots of points
the t r i a n g l e
of T = T . inequality
the limit
the space is finite
so any two r e a s o n a b l e t o p o l o g i e s
boundedness
The r i g h t h a n d
v e c t o r space spanned by
dimensional
agree,
implies c o n v e r g e n c e
ai(6,t) .) 4_~. B O U N D E D N E S S A N D COMPACTNESS:
for almost all
thus
of the
We now c o n s i d e r the
From the d i s c u s s i o n
in S e c t i o n
2.H and
for o p e r a t o r norms it follows that
tlTll ~ iITLll + IITRII + IITclI. IITLII + iiTRll + liT ,II. C The first two terms can be dealt with u s i n g T h e o r e m 2.1 and it's analog for c o n j u g a t e a n a l y t i c
symbols.
We now look at T ,. C If T , is b o u n d e d then the q u a n t i t i e s Il, x6~ C C m u s t be u n i f o r m l y bounded. T h e s e are almost the d i a g o n a l entries
Rochberg
205
of t h e m a t r i x because
in S e c t i o n
we have moved
operators).
Using
2.E.
(The n e w m i n u s
from Hankel
that
operators
computation
s i g n shows
up
to T o e p l i t z
and the d e f i n i t i o n
~ ,, we C
have (4.1)
~ = cA~ ,(x) = C ~ c ( X ) . C C Thus a n e c e s s a r y c o n d i t i o n for T , to be b o u n d e d is that ~C be C b o u n d e d on ~. T h a t c o n d i t i o n is also sufficient. To see t h a t note that the Fourier fn/2'
neZ
is g i v e n
development
= ~ ~C (n/2)
n6Z sum into two pieces,
the
even.
By the d i s c u s s i o n
bounded
Toeplitz
Similarly, compact compact
functions
fn/2"
according
is S e c t i o n
to w h e t h e r
2.F.3
each part
n is odd or generates
a
operator. a necessary
is t h a t
of the
by ~C
Break
of ~C in terms
and s u f f i c i e n t
each of the s u m m a n d s
if and only
if T , is. C c o n d i t i o n for T , to be c o m p a c t C tends to infinity.
condition
be compact.
A necessary
for T
Also
T
to be is
C and s u f f i c i e n t
is t h a t ~c(X)
t e n d to zero as
ixl
T h u s we h a v e Theorem
4.1:
i.
lIT II ~ IIP_(82~R) II, + IIP+(O2~L) II. + ll~Cfl~.
2.
T
is c o m p a c t
CMO and lim
if and only ~c(X)
If ~ is the s t a n d a r d not n e c e s s a r y a conjugate comparable fact
a n d we h a v e
analytic
for a n a l y t i c
This might
like
symbol,
functions
description in terms
the p r o j e c t i o n
operators
lIT II ~ IIg2~RII . + 1182~LII. + II~CII m. for i n s t a n c e
in L~/H ~.
lit It ~ inf
P + ( 8 2 ~ L ) are
in
= 0.
function,
to the n o r m
if P _ ( 8 2 ~ R ) a n d
Using
we find
this
~2~R,
are
For
the BMO n o r m
is
and the a n a l o g o u s
for any
{ Xlhll ; h ^ = ~^ on 2I}.
of the b o u n d e d
of the symbol
T
is not as e x p l i c i t
~ and the L i t t l e w o o d - P a l e y
as one
206
Rochberg
techniques remains for T
of the next
a problem in terms
standard
section
to give
s e e m to be of l i t t l e
a more
explicit
of @; e s p e c i a l l y
help.
boundedness
It
criterion
in the c a s e w h e n ~ is not a
symbol.
By definition, the o p e r a t o r
T
is c o m p a c t
n o r m of finite
exactly
if it is the
r a n k operators.
However,
limit
in
a bit m o r e
is true. Corollary finite to T
4.2:
is c o m p a c t
rank Toeplitz in o p e r a t o r
Proof:
taking
to o b t a i n
The r e q u i r e d
the p a r t i a l
Toeplitz
operators,
operator
and h e n c e
symbol
theorem
Hn,
n
,
is a s e q u e n c e
1,2,3,
the c o n c l u s i o n
approximation
analytic
symbol.
= 8 H(Sf)
where
P_(82~R ) .
is the n o r m
operators,
T n
sums of the F o u r i e r with
for f in PW, T(f)
on H 2 w i t h
then there
which
o . .
of converge
norm.
It s u f f i c e s
separately.
that
If T
limit
on H 2.
of A d a m j a y n - A r o v - K r e i n
and T L
expansion
T R is a
of ~C"
W e saw in S e c t i o n H is the H a n k e l
of a s e q u e n c e follows,
TR,
by
H is a c o m p a c t
(This
for TC,
to T C is o b t a i n e d
Hankel
operator
operator
of finite
on H 2
rank Hankel
for instance,
[Po 2].)
2.F.2
from the
The o p e r a t o r s
T n defined
by T n ( f ) = 8 Hn(Sf ) are e a s i l y operators
seen to m a p PW to itself which
the F o u r i e r operators
on PW.
similarly,
The crucial
CLASSES:
that these
fact
rank
calculation
operators
of
are H a n k e l
is t h a t u n d e r
will
avoid
conditions
in mind,
and
Suppose
1 < p < m.
t h a t ~ i_ss a s t a n d a r d
convenience
this
side shows
finite
A direct
T L is h a n d l e d
and we are done.
and t h a t
know what
and to g i v e
to T in norm.
by 8 goes to translation.
SCNATTEN
operator section
converge
transform
multiplication
5.
f in PW
T = T
We assume symbol.
in a n a l o g y
on H 2 are
in S p e x a c t l y
"standard
symbol"
for the rest
(This
lots of p r o j e c t i o n
on ~ c o r r e s p o n d w i t h the
if P+(~)
for a Hankel
is a c o m p a c t
of the
is a n o t a t i o n a l
operators.)
to T's b e i n g
operator
(P+(~)
We w a n t
in S p.
fact t h a t H a n k e l
is in Bp,
Hankel
to
With
operators
is the
on H 2) w e d e f i n e ~ p to be
Rochberg
207
the s p a c e
of all
standard
symbols
I1~11~
= IITII p
As noting That
in the p r e v i o u s that
< ~. Sp
section
is s u f f i c e s
~ for w h i c h
we start by s p l i t t i n g
to c o n s i d e r
the s u m m a n d s
T and
separately.
is
IITLII sp + IITRII S p + IITCIIsp
IITIIsp
IITLIIsp + IITRJIsp + IITc.IIsp. AS before,
the
its analog.
first
two t e r m s
are s t u d i e d
using
Theorem
2.1 and
W e n o w l o o k at T C
en;
neZ
is an o r t h o n o r m a l
in S p t h e n b y
(2.4),
w e can r e w r i t e
this
Exactly
the same
{en+i/2)
basis
of PW.
I] v p ~ c
Hence,
c IITc,IIsp.
if T , is C Using
(4.1)
as
l~c(n) Ip ~ c liT ,lip . C Sp c o n s i d e r a t i o n s a p p l y to the o r t h o n o r m a l
basis
and t h u s we also h a v e l~c(n+i/2 ) ip < c liT ,IIp . C Sp
These
are n e c e s s a r y
hand,
b y the d i s c u s s i o n
show that
Also,
comparable
S p norms.
We n o w w a n t {~c(n+I/2)}
T C is the s u m of two d i a g o n a l
has p - t h p o w e r
is in S p.
Theorem
for T , to be in S p. On the other C in S e c t i o n 2.F.3, t h e s e two e s t i m a t e s
the o p e r a t o r
e a c h of w h i c h
and P o l y a
conditions
we noted
in S e c t i o n
to r e l a t e
to the LP(~) states
summable
(Plancherel-Polya):
operators
entries.
Hence
2.H that T c and T , h a v e C
the 8P norms
n o r m of ~C"
that these
diagonal
of
{~c(n)}
The t h e o r e m
two q u a n t i t i e s
and of of P l a n c h e r e l
are comparable.
For any a b e t w e e n
0 and 1
I~c(~+n) l p ~ ~ I~c(x) lPdx. Proof: entire
This
is a v e r y
functions
whose
special
c a s e of g e n e r a l
Fourier
transforms
results
are s u p p o r t e d
about on
Tc
208
Rochberg
compact
sets.
For a discussion
theory of function the t h e o r e m
5.1:
Corollary:
This v e r s i o n
of
(4.1).
lIT lisp = II~II~ ~ II82~RIIB + II82~LIIBp + II~cll P P L p (~) For lO
a similar
set 9
construction
= {(l+e)m(i+en);
for
n6Z,
P m6~,
(l+e)m>e},
~
6
{en;
n6Z},
~
= {z;
z6~
6
the p o i n t s
is d i f f e r e n t differences
set d o n ' t
in the d i l a t i o n of f u n c t i o n s
to ~ h a v e
= 9 6
U Z 6
accumulate
from the s i t u a t i o n
We are t h i n k i n g restriction
}, and A 6
of t h i s
= 6
.
in S
on the real
noted
as e n t i r e
P decompositions
Note
that
6
on H 2 and is r e l a t e d
structures
spectral
U ~ 6
in the
axis.
This
to the introduction.
functions
whose
containing
no h i g h
212
Rochberg
frequencies.
The h a r m o n i c
extension
the u p p e r and lower half planes function) the p o i n t s
in ~.
T h e o r e m 5.3:
off ~ to
(recapturing the o r i g i n a l
will not o s c i l l a t e m u c h near ~ and thus
p o s s i b l e to u n d e r s t a n d
if e 0 such that a i so that
aifzi(t) z.6A 1 e
and ll{ai)llep ~ c llbll~p Conversely,
if {ai} e ~P and b is g i v e n by
(5.6) then b is in
P
and
llbils
~ c ll(ai}ll . p ep It is a bit m o r e c o n v e n i e n t to w o r k w i t h the Hankel
Proof:
operators
t h a n the Toeplitz
that if all of the z i in by the o r t h o n o r m a l that fact.
We saw in S e c t i o n 2.F.3 t h e n H b is d i g o n a l i z e d
{ez. ) . We now need a p e r t u r b a t i o n of 1 (This is one of the p l a c e s w h e r e w e w o r k w i t h sets
that are "almost" Lemma
operators.
(5.6) are in Z
basis
an o r t h o n o r m a l
(Kadec's 1/4 Theorem):
basis.)
If z k are real n u m b e r s
c < 1/4 t h e n the map of e k to e z
and
IZk-k I
extends by l i n e a r i t y to a k
bounded
i n v e r t i b l e map of L2(I)
Proof:
A p r o o f of this result u s i n g Hankel o p e r a t o r s
to itself. is g i v e n in
IN]. If the n u m b e r s the ideas of S e c t i o n ll{Ck)li p. as O(i/e)
z k satisfy the h y p o t h e s e s
2.F.3 extend and we h a v e ll~Ckfzkll~p
The p a r t of the sum
(5.6) w i t h z k in ~6 can be split
sums to w h i c h this a r g u m e n t
the p a r t of the sum
(5.6)
applies.
involving terms
k(t)
= ~ akelzkt ~ I t ) . Zk6~ e
This shows that
in Z 6 is in ~p.
W e n o w c o n s i d e r the terms c o r r e s p o n d i n g (5.7)
of the lemma then
to z i in ~e"
Set
Rochberg
213
By T h e o r e m
1 of
[R 2] the o p e r a t o r on L2(0, ~) g i v e n by
(5.8)
Rf(t)
= J0k(s+t)f(s)
ds
is in S p and has n o r m d o m i n a t e d by ll{ak)ii p. tranform picture
of a Hankel o p e r a t o r on H2.)
o p e r a t o r of m u l t i p l i c a t i o n
by K[0,2~ ].
the o p e r a t o r M R M is also is S p. k(t)
(R is the F o u r i e r
Hence,
by k(t)~[0,4~].
=
B e c a u s e S p is an ideal
M R M is u n c h a n g e d
if we r e p l a c e
if we i d e n t i f y L2(0,2~)
~[0,2~]L2(0, ~) then the map of L2(0,2~) Kf(t)
Let M be the
with
to itself g i v e n by
(s+t)~[0,4~](s+t)f(s)
ds
~0 is in S p.
This is e s s e n t i a l l y
described
in S e c t i o n
construction
is the e x p l i c i t
realization
e q u i v a l e n c e w h i c h was d e s c r i b e d in S e c t i o n
2.F.2.)
the Hankel
Also,
by 2~.
(This
of the u n i t a r y
in a b s t r a c t H i l b e r t
space terms
the symbol of this Hankel o p e r a t o r on
PW is e x a c t l y the part of the sum in 9 .
o p e r a t o r on PW, as
2.D, c o n j u g a t e d by t r a n s l a t i o n
(5.6) c o r r e s p o n d i n g
H e n c e that part of the sum g e n e r a t e s
to the z.
an o p e r a t o r
in S P.
6
The part of the sum c o r r e s p o n d i n g
to ~6 is d e a l t w i t h a n a l o g o u s l y
and h e n c e the o p e r a t o r w i t h symbol g i v e n by satisfies
(5.6)
is in S p and
the r e q u i r e d estimates.
N o w s u p p o s e we are g i v e n b in ~ shown in
(5.6).
and w i s h to split it as P The first step is to w r i t e b = b R + b c + b L.
We
will split t h o s e terms u s i n g the z i in ~e' ~e' and ~6 respectively. set
The F o u r i e r e x p a n s i o n
of b C w i t h r e s p e c t to the
(fn/2)n6 Z is the r e q u i r e d e x p a n s i o n
of b c and we saw in the
p r o o f of T h e o r e m 5.1 that the c o e f f i c i e n t s
of that e x p a n s i o n
s a t i s f t y the r e q u i r e d estimate. We k n o w from T h e o r e m 5.1 that bR(2~+- ) is in Bp. decomposition
theorem
for Bp gives almost w h a t we want.
{(l+e)m(i+en) ; n,m e Z}. 61
From
The W
Let ~e =
[CR] we k n o w that for some small
214
Rochberg
(5.9)
b R ( 2 ~ + t ) = ~ ,akeizkt K ~ t ) . Zk6~ 6
with
(5.1o)
ll{ak}II p < c ilbR(2~+-)ll B . P 1 of [R i].)
(See T h e o r e m
Since b R ( 2 ~ + t ) is s u p p o r t e d on [0,4~] we can m u l t i p l y b o t h sides of
(5.9) b y ~[0,4~]
and still h a v e b R on the left.
If w e
t h e n t r a n s l a t e b a c k by 2~ we have e x a c t l y the r e q u i r e d decomposition
of b R w i t h the r e q u i r e d e s t i m a t e s
fact that we w o u l d h a v e u s e d 9
6
instead of 9
except
for the
W e h a v e to show
6
that we can get by w i t h o u t the extra terms and still h a v e an estimate
such as
(5.10).
c o m p a c t support. Lemma:
Here we use the fact that b R has
We collect w h a t we need as a
G i v e n p, l~p
