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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