(Lp,Lq)-multiplier problems

by

J. F. PRICE

A thesis presented for the degree of Doctor of Philosophy in the Australian National University

STATEMENT

Except where I have credited it elsewhere, the material presented in this thesis is my own work.

ACKNOWLEDGEMENTS

Firstly, I should like to record my sincere gratitude to my supervisor, Dr R.E. Edwards. On the one hand, for introducing me to the pleasures of harmonic analysis in general and multiplier-operators in particular; and on the other, for the deep interest which Dr Edwards has shown in my work throughout. I have highly valued his suggestions, his criticisms and our conversations during the past three years.

Secondly, my thanks go to many others for discussions and help-ful comments related to the ~"ork of this thesis. These people

include Professor C.S. Herz, Drs A.C. Baker, Alessandro Figa-Talamanca and Garth Gaudry, and Messrs Mike Brady, David Tacon and Ian Wright.

CONTENTS

STATEMENT

ACKNm..JLEDGEMENTS CONTENTS

CHAPTER 0 INTRODUCTION AND PRELIMINARIES 0.1 Introduction and background to the

multiplier problem 0.2

CHAPTER 1

1.0 1.1

Definitions, notation and conventions

MULTIPLIERS HITH RANGE IN THE SPACE OF TEMPERATE DISTRIBUTIONS

Introduction

Preliminary results on temperate distributions 1.2 Multipliers with range in the space of

temperate distributions

CHAPTER 2

2.0 2.1 2.2 2.3

MULTIPLIERS BETHEEN SOME NORMED SPACES OF DISTRIBUTIONS

Introduction

The spaces of distributions H~ p

Representation of multipliers of H~ P The limiting case of Corollary 2.2.4

CHAPTER 3 3.0 3.1 3.2 3.3 3.4

CHAPTER 4

4.0 4.1 4.2

CHAPTER 5

5.0

5.1 5.2 5.3

CHAPTER 6 6.0 6.1 6.2 6.3

REPRESENTATIONS OF (p,q)-MULTIPLIERS WHEN G IS COMPACT

Introduction and preliminaries Pseudomeasures over G

Representation theorems The representation spaces Two open questions

IDEMPOTENT MULTIPLIERS AND LACUNARY SUBSETS OF

r

Introduction and definitions Basic lemmas

Lacunary sets and the inclusion when 1

2

p < q < 2

COMPLEMENTED CLOSED IDEALS IN Introduction

Complemented sub-modules

Loo Projections onto invariant subspaces of Extensions of multipliers

MULTIPLIERS WHICH ARE NOT MEASURES Introduction

Multipliers which are not measures

CHAPTER 7 7.0 7.1 7.2

APPENDIX A

APPENDIX B

APPENDIX C

C.O C.l C.2 C.3 C.4 BIBLIOGRAPHY

THE STRICT INCLUSION LP ~ Lq P T q Introduction

Preliminary results The main result

SOME BOUNDEDNESS THEOREM$

(00, q) -MUL TIPL IERS \..JHEN L 00 HAS ITS WEAK TOPOLOGY

A CONSTRUCTIVE APPROACH TO BOUNDEDNESS PRINCIPLES

Introduction

Further preliminaries

The case in which E is complete and semimetrizable

The general case

The uniform boundedness principle

INDEX OF SYMBOLS

CHAPTER 0

INTRODUCTION AND PRELIMINARIES

0.1 Introduction and background to the multiplier problem·

Let A and B denote sets of functions defined on the real line, integrable on [0,2n], with period 2n. In 1922, Fekete

[1] considered six special sets A, one of which was the set of continuous functions, and another the set of functions integrable on [0, 2n ] • Taking B = A to be anyone of the said sets, he asked the follm.;ring question: "What are necessary and sufficient conditions on a sequence

{~n}:=O

such that

(0.1.1)

be the Fourier series of a function in B whenever

(0.1.2)

is the Fourier series of a function in A?" Fekete solved the t

problem for these particular cases, but more importantly, he appears to have been the first to have isolated the "factor-function" problem and given i t a general formulation.

tIn all six cases, the condition that

Fekete [1] proved to be necessary and sufficient Eoo 1 ~ sinnx/n be the Fourier series of a function

n= n

[Earlier references to the idea of multiplying a Fourier series term-by-term by a sequence, and then examining the resulting series, may be found in Young [1] (1913), Steinhaus [1] (1919) and Szidon

[1] (1921).) This problem has become, through its many variations and generalizations, a source and stimulus of much mathematics over the years.

If the complex form of the Fourier series is used, instead of the real form as in (0.1.1) and (0.1.2), a natural generalization is to replace the underlying set [0,2n] by any (Hausdorff)

locally compact abelian group and to consider the Fourier

representations of functions or measures on this group. In this case we say that a function ~ on the character group is an (A,B)-factor function if the Fourier transform g of each g in A or B is defined, if f~ is defined for each f in A, and if f~ is the Fourier transform of a function or measure in B. (In fact, this problem may be formulated with just a locally compact group by using the continuous irreducible unitary represent-ations of functions or measures on this group by bounded linear operators on Hilbert spaces. For convenience, the discussion immediately following will be restricted to the abelian case, even though much of the work in the sequel takes place over groups which are compact, but not necessarily abelian.)

1 2. p, q 2. 2. Clearly each (LP,Lq)-factor function ~ defines an operator T from LP into Lq by

(0.1.3) (Tf)

Moreover, T is continuous, from LP into linear and commutes with translations-operators with these properties are called

The general problem, albeit hopeless, is to characterize the set of (A,B)-multipliers. Wendel

[1]

and Edwards

[1] noticed that in many cases, including

(12. p,q 2. 2), every (A,B)-multiplier is generated by an (A,B)-factor function. This often illusive, sometimes non-existent relation is the main reason for the scope and importance of the theory of multiplier operators in modern harmonic analysis.

This thesis is essentially a study of (LP,Lq)-multipliers; the types of questions studied are largely influenced by the structure of the underlying group in each case. Since the problems considered cover a wide range, it is simpler and more useful to introduce them in the introductions to the appropriate chapters or sections. We only remark here that the underlying group for the work of:

Chapters 1, 2 is

Chapters 3, 4, 5 is any compact group,

Chapter 6 is any locally compact abelian group,

All the material of this thesis, apart from §6.2 and several minor results, has been prepared for publication in Price [1],

[2] and Edwards and Price [1]. However, at each stage of this thesis it has been attempted to relate the work to other published results; the amount of these details and a more expositional approach throughout ~",rill dis tinguish the sequel from the above three papers.

0.2 Definitions, notation and conventions.

The notation, definitions and terminology will almost always agree \",rith that in Edwards [3]. In general, only functional

analytical concepts which occur in this thesis, but not in Edwards, will be defined in the body of the thesis. Also the concepts in Rudin

[2]

and Hewitt and Ross

[1]

will often be used without further reference.

In the following list we collect together the basic definitions and notations which will be standard throughout. Section 0.2.7 is very important.

regular, Hausdorff and normal. (This follmvs from Theorems (8.4) and (8.11) of Hewitt and Ross [1] and an obvious

construction.) Locally compact abelian will be abbreviated to LeA.

0.2.2 HAAR MEASURE. The left Haar measure of G will be denoted by A

G, but fGf(x)dAG(x) will often be written as j f(x)dA(x), jfdA, fGf(x)dx or even <f,A>.

When G is also abelian, the dual group of G will always be denoted by

r

and will possess the typical element

y

.

In

this case, if ~ is a bounded measure and f is A-integrable, we define ~ and f to be the complex-valued function on

r

such that

respectively. The function ~ is called the Fourier-Stieltjes transform of ~ , and f is called the Fourier transform of f. The Haar measures on G and

r

will be assumed to be normalized so that the Parseval formula is valid.

If G is compact, the Haar measure will be normalized so

that the measure of G is 1·

,

and i f G is the n-dimensional Euclidean space, Haar measure will be normalized so that the measure of the unit hypercube is 1. In this case, the Parseval formula requires that we define

A

fey)

=

f

f(x)exp(-2ni<x,y» dx, Rn

0.2.3

FUNCTIONS

ANV

M

EASURES.

If the underlying domain of any set of functions, measures or distributions is clear from the context, then it will be suppressed in discussions and notation. C(G) will denote the space of continuous complex-valued functions over G. and C will denote the subspaces of

c C comprised

of functions which vanish at infinity, and which have compact

support, respectively; C

=

Co

=

C when G is compact. c

Also, Co will be assumed to be equipped with the uniform (or supremum) norm, ,,,rhich makes it into a Banach space. Let lv1(G) and ~d(G) denote the space of complex-valued measures and complex~valued bounded measures respectively over G, and equip

~d with its usual Banach space topology (as the normed dual of

Hhen 1..::. p ..::. 00, LP(G) will denote the usual normed Lebesgue space of equivalence classes of functions; denote its nOlU by

II

lip' The symbols LP and L q will always imply that

p E [1,00] and q E [1,00], respectively. T.fuen 1 < p <00

_,

p' will always satisfy lip

+

lip'

=

1; also l '

=

00 and 00' = 1.

0.2.4

VI STRI BUn ONS .

When G is restricted to be

will also need the standard spaces of distributions discussed by Schwartz. For example: V(~), the inductive limit of Frechet

we

VI and SI will be assumed to be equipped with their strong

topologies. Almost all the necessary facts about these spaces can be found in Schwartz [1]. However, for properties and

definitions of convolutions, our main reference will be the masterful

paper, Shiraishi

[1].

The symbol Dr will always imply that r is an n-dimensional vector of non-negative integers, that is, r

=

_{(rl, .•. ,r n), and}

is defined to be where D. _l denotes differentiation, in the sense of distributions, with respect to the i-th natural

co-ordinate function on Rn. In this case we define

0.2.5

TRANSLATION ANV REFLECTION OPERATORS.

right translation operators T

a C by

T f(x) a

p f (x) a

-1 f(a x),

-1 f(xa ),

and (a E G)

The left and are defined on

and are then extended by continuity to measures or distributions by imbedding the set of locally integrable functions into the space of measures or distributions, respectively. When G is abelian, the

operator P

a

=

Ta will usually be denoted by t a • These operators will generally form a group of bijections on function, measure or

When fEe, define the reflection f of f by

f(x)

=

f(x -1 ),

and extend by continuity to measures and distributions.

0.2.6 CONVOLUTION. Another fundamental concept is that of

convolution. However, ,,,e will follow explicitly Edwards [3] in

this matter. In §4.l9, Edwards defines convolution between

certain pairs of measures, and then between certain pairs of functions,

or functions and measures, by identifying locally integrable functions

with the measures \"hich they generate.

f,g ELl, this results in

For example, when

Also, on occassions we have need of pointwise descriptions

[everywhere, almost everywhere (a.e) and locally almost everywhere

(l.a.e)] of convolutions. A frequently used case is: let

~ E ~d' f E LP ; then ~*f is defined and satisfies

).J*f(x)

f

Gf (y -1 x) d~ (y)

0.2.7 MULTIPLIERS. Let A,B be topological vector spaces

of functions, measures, pseudomeasures or distributions over G

which are invariant under the p [resp. T J, a E G.

a a Then a

continuous linear operator T from A into B which satisfies

Tp

=

p T [resp. TT

=

T TJ for each a E Gis said to be a

a a a a

right (A,B)-multiplier, or simply a multiplier [resp. a left (A,B)-multiplier J. When A

=

B, we will usually replace (A,B) by A and talk about A-multipliers.

As stated in the introduction, the main concern in the sequel is with multipliers from LP into Lq. In this case we will usually

abbreviate the above to (left or right) (p,q)-multipliers.

I f p of

00

,

let Lq [resp.

LqC

.Q.

) -

"

£

"

signifies "left"J denote

p p

the set of multipliers [resp. left (p,q)-multipliersJ from LP

into and i f p 00

,

let Lq [resp.

Lq(nJ

p p denote the set of

restrictions of such operators to [It is important to note

that it has become the custom to speak of (oo,q)-multipliers, when what is really meant is (CO,Lq)-multipliers. A discussion of this point is given in Appendix B.J For each T in

Lq

we define

p

its norm

Lq(T)

as the operator norm of T, recalling that p

the domain of T is Co when p

=

00 • When G is

of f such that

let Lq denote the subset of SI p

II f*l)! II _q < cons t . Ill)! II _p

for all lj! in LP when 1

:5-

p < 00

,

or lj! in _Co when p

=

00,

where the constant depends on f. Let Mq denote {f : f E L q}.

P P

Hl)rmander

[1]

has proved the following fundamental result: "Let

T be a (p,q)-multiplier; then there exists f in Lq such that

p

(0.2.1) Tlj! f*lj!

for all lj! in LP when 1 ~ P < 00, or all lj! in _Co when p Conversely, each f in Lq defines a (p ,q)-multiplier via

p

(0.2.1).

_"

Analogous results to this are contained in Brainerd and

Edwards [1] for LCA groups and p

=

q, in Gaudry [2] for LCA groups and p,q E [1,00] , and in Chapter 3 below for compact groups whenever not both p > 2 and q < 2.

The linear spaces and Mq may be made into Banach

p

spaces by equipping them with the norms f ~Lq(f)

P and

f

~

Mq

(f)

=

L q (f)

p p '

operator in (0.2.1).

where L q (f)

CHAPTER 1

MULTIPLIERS ~nTH RANGE IN THE SPACE OF TEMPERATE DISTRIBUTIONS

1.0 Introduction.

The main purpose of this chapter is to demonstrate that the

multipliers from either

V

or

S

into

S'

may be characterized

completely as convolutions \vi th elements in S'. Both of these

results are introductory in the sense that they give no direct

information about Lq.

p However, the second result is needed in

Chapter 2 and, since they are both of interest in their own right,

they both will be included in this chapter. Also, we use the second

of these results as a vehicle to show something of the generality

and scope of this type of characterization (as convolutions).

1.1 Preliminary results on temperate distributions.

The topology of S may be defined by the sequence of

seminorms

(1.1.1)

for p

=

S when

S (\jJ)

p

0,1,2, .••

.

S is equipped

Let S' denote

p

with the norm

I

r

I

< p, x E R n }

the topological dual of

S (.).

p The natural norm of

S' will be denoted by S' (. ) • The following two lemmas are typical

p p

in the theory of countably-normed spaces and, as such, are almost

1.1.1

LEMMA

.

00

IJ S'

p=l p S' •

PROOF. It is trivial that S' c

S'

for each p, and hence

p

that U 00 S' c S'.

p=l p To prove the opposite inclusion, let

X belong

to S'. Then {~E S : 1 <~,x> 1

<

l} is an open neighbourhood of

o

in

S,

and therefore contains a non-void set of the form

{~ E

S

s'

(X)

p

: S (lj!) P

-1

< k ,

<

kL

In this case X

which completes the proof.

is a member of o

S' P

1.1. 2

LEMMA

.

A set in

S'

is (weakly or strongly) bounded if and only if it is bounded in some S' .

m

with

PROOF. Since

S

is barrelled, every weakly bounded subset

of S' is equicontinuous (Edw"ards [3], Theorem 7.1.1). Then, similarly to above, each weakly bounded subset of

S'

is bounded on some neighbourhood of 0 in

S,

which signifies that it is contained

and bounded in some S'.

m The converse may also be simply proved; for example, by first showing that tpe injection of S' into S'

m

is continuous whether

S'

is equipped with its strong or weak

topology. 0

at each point of

V(

~)

,

where

~

is an open subset of Rn.

Then to each non-void compact subset K of ~ there corresponds an

integer m such that U is equicontinuous from V(K), equipped

with the Vm(K)-norm, into

S'.

PROOF.

Since V(~) is the internal inductive limit of spaces

V(w), w a relatively compact open subset of ~, the restriction

of each u in U to V(K) is continuous. Now V(K) is a

Frechet space whose topology is defined by an increasing sequence

of norms

Irl

.:.

m,

x

E K},

p

=

0,1,2, •. •

.

Let B

=

{f E S' : S' (f) < 1}. Then each

p p p

-B is

p strongly (even weakly) closed in S' . To see this, let

{f

i} be a directed family in B p converging 'veakly in S' to f·

,

that is <1jJ, f>

=

lim. <1jJ, f . >

l l for each 1jJ E

S.

Then, for each

index i, 1<IjJ,f.>1 < S (1jJ) for all IjJ E S, and so

l - P

< S (1jJ) for all 1jJ E

S;

that is fEB.

P

p Adding this

fact to lemmas 1.1.1 and 1.1.2 above shows that the conditions of

Corollary A.2 in Appendix A are satisfied with E

=

V(K), H

= S'

and H

=

S'. From this we conclude that there exist integers m

m m

and p such that U is equicontinuous from V(K), equipped with

the Vm(K)-norm, into S'

We ~"ill now give an application of the above theorem which improves some results of Schwartz by giving weaker sufficient conditions for a distribution to be temperate. We first quote

the appropriate result of Schwartz - see Schwartz

[1],

Chapitre

VII, Theoreme VI, ~ et ~ - and then follow it with the improved

version. See also Remark 1.1.6 below.

1.1. 4

THEORE

M.

Let X be a distribution. Then for

X to be temperate it is sufficient that either

(1) X*~ is a function of polynomial order at infinity

for each ~ E

V.

or

(2) X~ (or, equivalently, X~) is a bounded distribution

on (see Schwartz

[1],

p.200) for each ~

E

S.

1.1. 5

COROLLARY

(to

Theo~em

1.1.3).

Let X be a distribution and ~ some non-void open subset of Rn Then for X to be

temperate it is (necessary and) sufficient that either

(1) X*~ E S' for each ~ E V(~). or

(2) X~ E S I for each ~

E

V(~).

PROOF.

The necessity is obvious in both cases. Let

A

hypothesis will also be satisfied for ljJ E V (h+rI), where

so that without loss of generality we may, and will, assume that

rI is a neighbourhood of zero. Let {oil be an approximate identity sequence in

V(rI);

that is, O. E

V(

rI

) ,

f

O. (x)dx

=

1

l _Rn l

and lim. (supp o . )

=

{OJ. Define operators

l l

u. : ljJ ~ X.)~ljJ [resp. v. ljJ ~ Y.ljJ] where X.

=

X)~o. [resp. Y.

l l l l l l l

Xo.]

l from

V(rI)

into S' • Evidently the u. [resp. v. ]

l l

are continuous linear operators from

V(

rI

)

into

S'

and satisfy the hypotheses of Theorem 1.1.3.

Let K be any compact neighbourhood of 0 in Rn. Then there exists an integer m such that {u.} [resp. {v.}]

l l is

equicontinuous from

Vm(K)

into

S'

(where u. [resp. v.] also

l l

denotes the unique continuous extension of u. [resp. v.] _{l l} to all By a suitable choice of a parametrix ljJO E

Vm(K)

(see Schwartz [1], (VI, 6; 22) or Edwards [3], 5.11.2) we have

where 6k is the iterated Laplacian, 0 is the Dirac measure, and ~ E

V(K).

Thus

(1.1.2)

[resp. Y. l

X. = X.)~O

l l

Y.o l

Now {u,(1jJ)} [resp. {v,(1jJ)}) is bounded in

S'

for each

1. 1.

1jJ E Vm (K) • (Let U be a convex neighbourhood of 0 in

S'

and let 1jJ E Vm(K). Since V(K) is dense in Vm(K) and the u,

1.

[resp. v.] are equicontinuous, there exists 1jJ1 E V(K) such that

1.

u, (1jJ) - u. (1jJ1)

1. 1.

But {u

i (1jJ1) }

exists A > 0

E

U [resp. vi (1jJ) - vi (1jJ1)

E

U] for i

=

1,2,., • •

[resp. {vi (1jJ1)}] is bounded in

S',

whence there

such that {u

i (1jJ1)} C AU [resp. {vi (1jJ1)} C AU],

and so u, (1jJ) E U

+

AU [resp. v. (1jJ) E U + AU] for all i.

1. 1.

Since

U is convex, U +A U

=

(l+A)U, so that {u,(1jJ)} [resp. {v,(1jJ)}]

1. 1.

is absorbed by U, and thus is bounded.) Thus (1.1.2) shows

{x.} [resp. {Y,}] is bounded in

S'

,

is therefore strongly relatively

1. 1.

compact (Edwards,

[3],

8.4.12), and so possesses a limit point

in S'. This limit point must be X, which completes the proof. 0

1.1.6

REMARK.

After having proved Corollary 1.1.5, it was

brought to the attention of the author that a result similar to

Corollary 1.1.5(1) is proved in Yoshinaga and Ogata [1]. Lemma

1 (1) of this paper states that: "For any distribution X

satisfying for any CP,1jJ

E

V,

then XES' "

A surprising fact is that this lemma is equivalent to 1.1.5(1) above

PROOF. Let X be a distribution satisfying X*~ E

S'

for all ¢ E

V

.

Then X"~~'~1jJ f 0 for all ¢, 1jJ E V (see Schwartz

m

[1], pp.243-244), and Lemma 1(1) of Yoshinaga and Ogata [1] shows

that X is temperate. Conversely, let X be a distribution

satisfying for all ¢,1jJ in

V.

Application of

1.1.5(1) above twice, with yields that X must be temperate. 0

The last sentence shows us how to give a stronger version of

Lemma 1(1) of Yoshinaga and Ogata. Namely, a distribution is /

temperate if there exists an integer m > 1 and non-void open

for all 1J.;. E V(~.), i

=

l, ••. • m.

~ 1

1.2 Multipliers with range in the sEace of temperate distributions.

1. 2.1 THEOREM. Let T be a multiplier from

V

into

S'.

Then there exists a unique XES' such that

(1.2.1) T1J.;

=

X*1jJ

for 1J.; E

V.

Conversely. each XES' defines a multiplier from

V

into

S'

via (1.2.1).

PROOF. The converse follows readily from well-known properties

of the convolution operator; for examp1e.see Schwartz [1], Ch. VII,

Let T be a multiplier from

V

into

S'

,

and put

To.

=

X. E

S

'

,

where {o.} is an approximate identity as

l l l

defined in the proof 1.1.5. Since T is also a multiplier

from V into

V'

,

we learn from Edwards [3], 5.11.3 that

also commutes with convolutions. Thus

(1. 2. 2) Tlj; = lim. T (0 . ~'~lj;)

l l lim. X. l l ~'~lj;

T

for each lj;

E

V,

where X. == To .• Let T. lj;

=

X. "'lj;. Since the

l l l l

T.

l are continuous linear operators from

V

into S'

,

and satisfy

the hypotheses of Theorem 1.1.3, we may imitate the proof of

Corollary 1.1.5 to show that {X.} must have a limit point in

l

S', X say. This completes the proof since the limit of

X.*lj; in (1.2.2) can only be X*lj;. 0

l

By letting V replace S everywhere in the statement of the

following theorem, we see that it has a strong resemblance to

Theorem 5.11.3 of Edwards [3], which in turn may be derived from

results of Schwartz - see Schwartz [1], Chapitre Vl, pages 160-164

and page 197. The definition and principal properties of

0'

_c

the space of "rapidly decreasing distributions", may be found

1. 2. 2

THEORE

M

.

Let T be a continuous linear map from

S into S'. Then the following five conditions are equivalent; (1) T commutes with translations.

(2) T commutes with derivations.

(3) T commutes with convolutions by elements of

S.

(4) T commutes \vith convolutions by elements of

0'

_c'

(5) There exists a temperate distribution X on such that

for all tJ; E S.

1. 2.3 REMARK. For our purposes the most important part

of the above result is the equivalence of (1) and (5). That (5)

implies (1) is easily shm.;rn, and the proof that (1) implies (5)

may be deduced from Theorem 1.2.1, as is done in Price [1], 3.2, by

noting that if T satisfies (1) above, then TID is a multiplier

from

D

to

S

'

.

He choose to give a proof of this implication, and

the others necessary for Theorem 1.2.2, independently of Theorem

1.2.1 in order to display more of the techniques which are available.

The proof will also be independent of that of Theorem 5.11.3 of

Edwards [3].

1. 2.4

PROOF

(on

Theo~em 1.2.2l. Standard results in the

of (1), (2), (3) and (4). Also (4) obviously implies (3), and

the proof that (2) implies (1) is sufficiently similar to the proofs

of the analagous results for continuous linear operators from

E'

to

V'

given in Schwartz [1, Ch. VI, Th. X], or from

V

to

V'

given in Edwards [3, Th. 5.11.3], to warrant exclusion. He now

prove that (1) implies (3).

Let T be a continuous linear operator from

S

into

S'

and let ~,¢ t

S,

and hence ~*¢ t

S.

The proof that

T(~*¢)

=

T~*¢ follows readily once it has been established that

~.,.(¢

=

f

t ~.¢(y)dy is the limit in S of finite Riemann sums

Rn y

I.

t ~.¢(y.).m(R.) , where m(R.) is the Lebesque measure of the

1 y. 1 1 1

1

n-dimensional rectangle R. ,

1 and where y. 1 E R 1 ..

Let K (N)

=

{y : \ y . \ < N, i

=

1,..., n } an d 1

-K(N)'

=

Rn'K(N), and recall the definition (1.1.1) of the sequence

of seminorms {S (.)} on

S.

p Given any integer p ~ 0, it

follows from a general property of vector-valued integrals (see

Edwards [3], 8.14.6 and its follmving Remark) that

and therefore that

(1.2.3)

For a fixed N, the uniform continuity of (l+\x\)P Dr~ and

S -norm, and hence in S,

p by sums of the form

L~-l

l - t y.

~.~(y).m(R.)

l

,

l

where M depends only on N. Combining this with (1.2.3) leads

to:

as proposed.

The continuity and linearity of T yield

T(~*~)

which, since T also commutes with translations, is equal to

(1.2.4) limNliTIl. L.t (T~).~(y. ).m(R.). 1'1 l y. l l

l

Now, for any f E S,

<f,

I.

t (T~).~(y.).m(R.»

=

<I. t f.~(y.).m(R.) ,T~>

l y i l l l -y i l l

which, by the preceeding remarks, tends to <U(~, T~> as

M + 00 and N + 00 in that order. But it follows immediately

from Definition 2 and Theorem 2 of Shiraishi [1] that this limit

is precisely <f,T~*~>. Since S is a Montel space, weakly

convergent sequences in S' are strongly convergent, and consequently

The cycle of implications ,,,,ill be completed by proving that

(3) implies (5). Let ~,~ E S and assume (3). Since R

n

is an abelian additive group, convolution is commutative and so

we have T~*~

=

~*T~. By the "exchange formula" (see Schwartz [1, Ch. Vll, Th. XV]) we have

(1.2.5) (T~) • ¢ ~. (T¢) in S'.

Select in S any non-vanishing function ¢a and define

Now Y is a distribution since (T¢a) is a

(temperate) distribution and l/~a is indefinitely differentiable.

Furthermore, from (1.2.5),

(1.2.6) Y.~

for ~ E S. This shows that Y.~ E

S'

whenever ~ E

S,

and so,

by Corollary 1.1.5(2), Y E S' . Introducing X E S', such that

X

=

Y, into (1.2.6) and taking the inverse Schwartz-Fourier

transform results in

CHAPTER 2

MULTIPLIERS BE~vEEN SOME NORMED SPACES OF DISTRIBUTIONS

2.0 Introduction.

By the systematic use of Fourier transforms and weight

functions L.R. Volevich and B.P. Paneyakh brought many classes of

spaces of distributions (including the Sobolev spaces) and their

topological duals under one unifying definition.

1.Je begin the study of (p ,q)-multipliers by relating

In this chapter

L q with the

p

space of multipliers between distribution spaces of the type just

mentioned. The proofs of these results will use Theorem 1.2.2 as a

sort of "blanket theorem". As a corollary, a representation of

multipliers from into

V

will be given.

Lq

2.1 The spaces of distributions

The spaces H).J

P are essentially those defined and studied in

Volevich and Paneyakh

[1],

although for a different class of weight

functions. This is because we will not require H).J to be a module

p

over

V,

but will require that multiplication by the weight

functions and their inverses define continuous isomorphisms from

S onto itself. (See 2.1.3 below for a brief discussion on classes

of weight functions.) Chapter II of Hormander [2] also gives an

independent investigation of the spaces Throughout this

property that each of ]J ,]J -1

indices) is of polynomial order at infinity.

ranging over all

2.1.1 DEFINITION. H\.1

=

{Iji E S' : ¢

=

F-1 (]Ilji) A E LP },

P

where 1 < P ~ 00, \.1 E B, and IjJ

=

Fiji denotes the

Schwartz-Fourier transform of Iji.

2.1. 2 We define a norm on H~ by

11

1J;

II~

=

II

¢

lip' thus making

H]J isometrically isomorphic to LP • Also

p

(2.1.1) S C H]J c S' ,

P

where in both cases the embedded space is dense (provided p

#

00

in the first inclusion) in the enveloping space, and the inclusion

maps are both continuous - see Volevich and Paneyakh [1, p.67].

2.1. 3 RE'MRKS OU ['JEIGHT FUNCTIONS. Volevich and Paneyakh

( [1]) require their class B' of weight functions to satis fy two

p

conditions, namely (a) ]J E B' implies that \.1 and \.1 -1 are

p

pointwise multipliers of

S,

and (b) H\.1 is a module over D

p

with respect to pointwise multiplication. In an attempt to

achieve this they define B'

P

to be the class of functions

]J such that, for constants k and c depending only on p and \.1,

for all x,y E . Rn. , and similarly for )J -1 This certainly ensures

that condition (b) above is satisfied as is shown on pages 67-68 of Volevich and Paneyakh [1]. However, on page 66, Remark 13.1 of this paper, the authors give an invalid proof that condition

(a) is satisfied. [Their proof is invalid since for )J and

-1

)J to be pointwise multipliers of S it is necessary (and

sufficient) that each of )J,1l -1 , D ) J , D I l r r -1 (r ranging over all indices) be of polynomial order at infinity, and not just 11 and

-1

11 .]

When p

=

2, definition (2.1.2) is certainly not sufficient

to ensure condition (a) • For example, when n

=

1 define )J by

11 (x)

=

2

+

exp (i exp x). Then )J satisfies (2.1.2) since 00

M; (Rn) is isometrically isomorphic to L (Rn) (see Brainerd and Edwards [1 ], Theorem 4.4 and §4.5), but no derivatives of )J are

of polynomial order at infinity. For p other than 2 it

appears to be a difficult 1uestion to decide whether or not definition (2.1.2) is sufficient to ensure that )J and ₁₁ -1 are pointwise

multipliers of S. However, judging from a review (}1athematical

Reviews 34 # 3194) of a later paper, Volevich and Paneyakh [2], it appears that the authors have strengthened their definition of weight functions by requiring that condition (a) as well as (2.1.2)

2.1. 4

EXAMPLE: THE S030LEV SPACE

wm

.

p

~-Jhen m is any non-negative integer and 1 .::. p .::. co, the

Sobolev space

wm

p is the space of temperate distributions which,

along with their generalized derivatives of orders not exceeding m,

belong to LP • Assume that ,~ is equipped with its usual norm,

p

namely A theorem of Lizorkin

[1],

refining a result of Mihlin, shows that g E MP for all

P

1 < p <

rl,···,r

D n _{g is continuous on {x x.}

_i:

_{0, i l, ..• ,n} and i f}

1

r rl, •.• ,r

x n D n

g/

n is bounded for all x, where

r.

=

0 or

1 1. These conditions are satisfied for

r -1

x P where

1 < p < co, the fact that Xrp- l E MP

p

equivalent to the statement that

topological vector spaces, with

We will prove that, with

for

/r

/ ..:.

m

is

coincides, in the sense of

PROOF. Calderon

[1,

Theorem 7], for example, uses

r -1 p

/ r / < m, to prove that HP coincides with I~.

x P E 1-{ ,

P P P

Alternatively, assume that uP and

wm

coincide. Let

p P

1jJ E LP . Then F-l (p -l~) E uP

P

I,-Jm _and

p so Dr (F- 1 (p -1~)) E LP

for /r/ < m. Now

where 1 . 2 -1. Applying the "exchange formula" (see Schwartz

5» to the last term shows that it equals [1], p.268, (V11, 8;

(_i)lrIDr(F-1(

p-l~»

_{which, from the above discussion, belongs to}

To prove that x r p -1 E MP

p' or equivalently, that

F-1(x rp-l) E LP , it will suffice to show that the linear operator p

-1 r -1 p P

w: \jJt--;)oF ex p )*\jJ is continuous from L to L - see 0.2.7. Since

F,

F-

1 and multiplication by COO-functions of polynomial

order are continuous linear operators from S' to

S',

it follows

that the graph of w is closed. Banach's original version of the

"closed graph" theorem yields the required continuity of w. 0

As is commonly done, the normed dual of ~, 1 _{::.. p} < 00

,

will

be denoted by _W -m _"

P where lip' + lip

=

1.

follows readily that this dual is precisely

p

When p' H _P "

1 < p < 00

,

where p' (x)

-1

1 12

-m/2

p (x)

=

(1

+

x ) ; see Volevich and Paneyakh [1, p.66].

2.2 Representation of multipliers of H~ •

P

i t

The proof of the main theorem in this section uses the idea of

(S')-convolution introduced by Hirata and Ogata [1]; the

(S')-convolution of f,g E S' exists if and only if a,S E S

implies (P'(a) •

(g)~S)

ELl. The convolution f*g is then a

temperate distribution defined by

« Pg)*a,S>

_f

(f*a) (x) (g*S) (x)dx

n

for any a,S E

S

.

(It is a consequence of Corollary 1.1.5(1),

for example, that f*g must be temperate when defined as above.)

Of course, the existence of the (S')-convolution implies the

existence of the Schwartz convolution and in this case the two products

are identical. (See Shiraishi

[1]

for definitions of the

(S')-convolution and the Schwartz convolution which display their

similarities.) The motivation behind the S'-convolution is that

if the (S')-convolution is defined between f,g E S', then

the exchange formula is satisfied, that is

(2.2.1)

2.2.1

and _~ E LP

I f p

:f

00

,

PROOF.

P ' f E Lq

=

L

P q'

Lg

LEMMA

.

The (S')-convolution between f E Lq P

is defined, where 1 _{..:. p, q ..:.} 00

,

and belongs to

then ~ t----?> f)~ ~ is continuous from LP into Lq.

Let

a,

S

E

S

.

Then f*a E L

P

n L

P'

since

(H~rmander

[1], Theorem 1.3); and ¢*S E LP,

so that

v I 1

(f*a) . (~*S) E LP .LP C L, as required. When

1 ..:. P < 00

,

into Lq

that the operator is continuous from

follows immediately from the definition of

Lq.

When P

=

00, we need only consider q

=

00, since L

q

=

{OJ

P

when P > q: HBrmander [1]. First construct a sequence {~ } C V

00

which is bounded in L and converges to ¢ weakly in

V'

.

(Eg. 'Pn

=

f _n *6 , _n where f _n

=

f for Ixl ~ nand 0 otherwise.)

co co

Since {¢}

n is bounded in L , {f*¢ } n is bounded in L and thus is relatively weakly compact, so that it possesses a

00 L .

00

This limit point in L can be none other than f*¢. 0

2.2.2

THEOREM.

Let T be a multiplier from H)..I

P into

H~, where )..I,V

E

Band

1

~ p,q ~ 00. Then there exists a unique YES' satisfying

(2.2.2)

for t./J E H)..I i f

p

Tt./J

p

:f.

00, or t./J E S i f p = 00.

Y E F-l()..IMq/V)

p defines a multiplier from S eqipped with the p

=

00, into

PROOF

.

Throughout the proof we assume

Conversely, i f

1 ..:. p

p

:f.

00, or

via

< 00·

,

(2.2.2).

the proof for p = 00 follows with minor modification. We begin with the converse. Let

such that t./J " = )..1 -1¢; " f E Lq such that

p Y

t./J E H)..I _p' and let

"

= )..If/v. of the proof of 2.2.1 we have

-1 " " t./J*a

=

F (t./J.a)

then there exists _¢ E LP

Y E F -1 ()..1M q Iv), then there exists p

since and

-1 ~ p'

Y*6

=

f*F (~6/v) E L since

,

L1

LP, • Thus (Y*6). (1jJ*a) E

,

and so the (S' )-convo1ution q

of Y and 1jJ is defined. Furthermore, application of (2.2.1)

to Y,1jJ, then to f,cp, combined \\I"ith Lemma 2.2.1, yields

(2.2.3)

-1 A A

F (f.cp) f*cp E Lq,

that is Y>~1jJ E HV.

q That the operator 1jJ ~Y>'<1jJ is continuous

from H~ into HV follows from (2.2.3) and the continuity of

p q

cp ~ f*cp from LP into Lq·

,

while that it is linear and

commutes with translations follows from the references in the

first paragraph of the proof of Theorem 1.2.1.

Let T be a multiplier from H~

P into From

2.1. 2

we know that TiS is a continuous linear operator from

S

into

S'.

Thus it satisfies hypothesis (1) of Theorem 1.2.2 and so

there exists YES' such that (2.2.2) is valid for 1jJ E

S.

"

It remains to show that Y E ~Mq/v. p

Now T is continuous, and therefore bounded. We note from

Remark 2 of Shiraishi [1] that the conditions of Hirata and Ogata [1]

for the exchange formula to be valid are satisfied in the following

manipulation. Thus we have, with the supremum taken over non-zero

I/TII

(2.2.4)

(since

F,

multiplication by II E B

continuous isomorphisms from S into S)

-1 A A

= sup{IIF (\J/ll.Y.W)II

Ill

wll

}

q p

-1 A

sup{IIF (\J/ll.Y)'~~1

Illwll

}.

q p

A

Mq(\JY/ll)

Since \JY/ll E S', we have p

the definition of the norm of Mq• Thus p

and

=

IITII

A

\JY/ll

-1

II

by

E Mq p

define

(2.2.4) and

and so we

the required Y E llMq/\J. I f P < 00, S is dense in p

Hll by (2.1.1), and (2.2.2) for

W

E Hll is obtained as the

p p

continuous extension of W ~Y*W for W E

S

.

0

have

As our first corollary of the above theorem we prove Theorem

13.1 of Volevich and Paneyakh [1]. If Hll C H\J

P q' the closed

graph theorem and (2.1.1), for example, implies that the inclusion

map is continuous. Thus Hll C H\J if and only if the map

p q

is continuous from Hll

p into where denotes the

Dirac measure. As a consequence, it is easily deduced from Theorem

2.2.3 COROLLARY.

sufficient that v/~ E Mq_p •

For H~ c HV it is necessary and

p q

The following specialization of 2.2.2 to the Sobo1ev spaces

mentioned in 2.1.4 is obvious. For 1 < p .::.. 2,

1 < q ~ 2 and m,k E {O,1,2, ••. } this result was first proved

by J.C. Merlo [1] using a completely different method which

relied on a decomposition of elements in LP. More recently,

I.W. Wright proved the result for 1 < p,q < 00 and

m,k E {O,1,2, ••• } by a method which relied on the description of

vf1

given in 2.1.4 - see Hright [1] .

p

2.2.4 COROLLARY. Let T be a

Wk where 1 < p,q < 00 and m,k are q'

a unique Y E F-1 ((1

+

IxI2)(m-k)/~q) p

and conversely.

2.2.5

REMARK.

When p

=

1 or

multiplier

integers.

such that

00

,

from

Ifl

into p

Then there exists

(2.2.2) is valid;

and m

is a positive integer, a sufficient condition for an operator from

s,

equipped with the 1f1-norm,

p into

vf1

q to be a multiplier is

that it is of the form (2.2.2) with Y E Lq.

P This follows readily

by noting the continuity from S with the LP-norm into Lq of

Drf ~~Y*Drf ~

=

Dr(Y*f) f or eac h r : r I I ~ m. An unpublished

result, Theorem 1.3.13 of Wright [1], shows that this condition

conditions of this nature appear to be known for higher dimensions

when p is 1 or 00

2.3 The limiting case of Corollary 2.2.4.

Following Schwartz [1, p.199], we define V

LP

to be the space of

indefinitely differentiable functions on Rn which, along with all

their derivatives, belong to and equip it with a topology

defined by the sequence of seminorms

{ II

.

II

}

,

m

=

0,1,2,... .

\.·r

_p

When 1

.2.

p < 00, the topological dual of equipped with

its strong topology, will be denoted by

2.3.1

COROLLARY.

Let T be a multiplier from

V where 1 < p,q < 00.

Lq

Then there exists a unique

Y E F-l«l+ \x\2)c/2Mq)

p for some integer c > 0 such that

(2.3.1) T1jJ

for all 1jJ E

V

LP

Conversely, given any integer c > 0 and

Y E F-l«l + \x\2)c/2Mq ),

p

is a multiplier from V

LP

the operator defined by (2.3.1)

into V Lq

PROOF. Let T be a multiplier from into

Then TiS is a multiplier from

S

into

S'

and so, by Theorem

1.2.2, there exists YES' such that (2.3.1) is valid for

lji E S. Since T is continuous from V

LP

into

V

,

Lq

to each

positive integer k, there exists a positive integer m and a

constant K such that

(2.3.2)

for all lji E S. If m < k, inequality (2.3.2) may be replaced by

IIY*ljill k2.. KlIljili k'

w

w

q p

so that, by Corollary 2.2.4, b

=

max«m-k)/2,0).

Conversely, let c >

°

be any integer and suppose that

By Corollary 2.2.4, to each k

=

0,1,2, •• •

there corresponds a constant K such that

for each lji E V

LP

Thus T: lji ~Y'klji is continuous from V LP

into V , and it is clear that it is linear and commutes with

Lq

2.3.2

and

V'

Lq

REMARK.

replacing

Theorem 2.3.1 remains valid 'vi th V I LP

respectively. This may

be seen by examining the adjoints of the operators in Theorem

2.3.1 (cf. Brainerd and Edwards [1], §1.5), and using the fact

that V is reflexive.

CHAPTER 3

REPRESENTATIONS OF (p,q)-MULTIPLIERS WREN G IS COMPACT

3.0 Introduction and preliminaries.

3.0.1 In this chapter we commence the study of (p,q)-multipliers

when the underlying group is compact. Hhen G is compact and

r

contains a certain type of infinite lacunary set, it is readily

proved that there are (p,q)-multipliers, 1 < p

2

00 and

1

2

q < 00, which are not expressible in the form f r-+ 11 )~f for

any measure \1. (For the details of this result see 4.2.2(1)

-it is sufficient for our present purpose to only remark that

r

has this lacunarity property for a large class of infinite compact

groups, including those which are abelian.) Thus, for there to be

any chance at all of giving a general characterization of

(p,q)-multipliers as convolutions, we need a set of "suitable" elements

larger than M.

Kahane [1] introduced the idea of "pseudomeasures" over a

locally compact abelian group. In §3.l, we will show that this

technique is applicable to compact groups; however our treatment

will follO\v more closely that of Brainerd and Edwards [1, §4.l].

This space of pseudomeasures over compact G will certainly be

large enough to characterize (p,p)-multipliers as convolutions,

but it is an open question whether it will do this for all

In §3.2 we prove some basic representation theorems, and in

§3.3 we prove some simple properties of representations of

(p,q)-multipliers.

The motivation for the approach to (p,q)-multipliers

when G is compact given in this chapter is provided by the standard

treatments of this problem when G is a (locally) compact abelian

group, particularly the treatment of mUltipliers on the circle

group contained in Edwards [4, Chapter l6J. For this reason it

would not be surprising if other developments of the basic theory

of multipliers when G is compact, similar to that given in this

chapter, are known. However, at the time of preparing this work,

November and December of 1968, the author could find no suitable

reference. Since then the author has had the privilege of seeing

a preprint of sections 35 and 36 of Hewitt and Ross [2J, which is an

extremely detailed and exhaustive treatment of multipliers when

G is compact, treated from the point of view of multiplier

functions or factor functions on f . As would be expected, there

is some overlap between the results of Hewitt and Ross and the

follow-ing chapter, but the only result of any consequence in this

intersection is Theorem 3.2.4.

Also, the author gratefully acknowledges the fact that much

use was made of a preprint of an early version of sections 27 and 28

3.0.2 Throughout this chapter G will denote a multiplicatively written compact group (not necessarily abelian) with identity e

and equipped with normalized Haar measure AGo

f is uniquely represented by a Fourier series

f ~ ) dey) Tr[f(y)y(.)],

y~r

I f fELl

,

where:

r

is a set of representatives, one selected from each then

equivalence class of continuous, irreducible, unitary representations of G'

,

dey) is the (finite) dimension of the representation y;

Tr denotes the usual trace; and f is the Fourier transform of f, defined by

(3.0.1) fey)

=

fG

f(x)y(x)*dx,

y(x)* denoting the (Hilbert) adjoint of yCx). Let H denote the

y

Hilbert space corresponding to the representation y. There are

several equivalent ways of interpreting (3.0.1). For example, one may use the general notion of vector-valued integration as outlined in Edwards [3, 8.14]. However, by using the fact that

space, a much simpler interpretation is available. A

fCy) to be the endomorphism of

A

H

Y defined by

<f(y)a/S>

f

e

f(x)<y(x)*a/S>dx,

H

Y is a Hilbert He define

for each a,S E H where < / > is.the inner product in H .

Y

The following "exchange formula" will be used often:

(3.0.2)

where

(f*g) (y)

=

g(y)f(y)

1

f,g E L . The proof provides a pleasant introduction

to manipulations of Fourier coefficients of integrable functions .

Let a,S E H •

Y Then

< (f*g) (y)a \,S>

f

f(y)dy

f

g(y-1x )<y(x)*a \S>dx

f

f(y)dy

f

g(x)<y(x)*y(y)i<a \S>dx

(by making the translation x~ yx in the inner integral)

f

g(x)dx

f

f(y) <y(y)*a \y(x)S>dy

A A

<g(y)f(y)a\S>

as asserted. Also the Peter-Wey1 completeness theorem will be

frequently used, and without comment, generally in the form:

If fELl and f(y)

=

0 for each y in

r

,

then f

=

0 almost

everywhere. See Hewitt and Ross [2] .

3.1 Pseudomeasures over G.

such that f.! (y) is an endomorphism of H

Y for each y in

For each y in r we will define two norms on the space of endomorphisms of H

Y

11<1> II (1)

by

and 11<1>11(00)

=

11<1>11

=

1..1/2,

r.

where

A

is the maximum eigenvalue of <1><1>*. Note that 11<1> II is the usual operator norm for an endomorphism <I> of H .

Y The

1/2

positive self-adjoint (p.s.a.) operator (<1><1>*) is often written

It is known that there always exists a unitary transformation,

U say, such that <I>

=

u

i<l>

i;

this is called the polar decomposition

of <I> (see Dixmier [1], Appendice III). The norms defined

above are only two examples from a class of norms defined and

studies in v. Neumann

[1]

.

A special case of formulae (19) and (20) of this paper is the important inequality,

(3.1.1)

where are endomorphisms of H .

Y (For a more direct proof of

(3.1.1), see Dixmier [1, p.l04].)

Let f.!

E F and define

and

in the extended real number field. Then _{11t-t111 .:... 1It-t 11}00·

For p :: 1 or 00

,

denote by E the subspace of F consisting

p

of t-t such that Iit-til < 00 Then E1 C E

.

(E

1 and E are

p 00 00

the two extremal examples from the family of Banach spaces E , 1 < p

p

-

< 00

,

defined and discussed in Hewitt and Ross [2] .)

3.1.1 DEFINITION.Denote by A the set of f in C with f in E

1, and equip A with the norm IlfilA ::

Il f11_{1 ·}

As mentioned on page 510 of Fig~-Ta1amanca and Rider [1], i t may be shown that A

=

L2*L2

,

whence we see that A is the space extensively studied by Eymard [1 ], when G is a

locally compact group. For example Eymard [1, Chapitre 3] proves that A is a Banach algebra under pointwise multiplication and is

dense in C with

(3.1.2)

for f in A. Whenever f E A, inequality (3.1.1) shows that the Fourier series of f is uniformly convergent since

(3.1.3) f(x)

for all x E G, since f is continuous.

The normed dual of A will be denoted by P, and will

be called the space of pseudomeasures over G. Inequality

(3.1

.

2)

shows that corresponding to each measure is a pseudomeasure. In

the sequel we will identify measures with their corresponding

pseudomeasures. Since

A

and El are isometrically isomorphic,

and E is isometrically isomorphic to the normed dual of

00

(see Hewitt and Ross

[2]),

then there is an isometric isomorphism

bet\.;reen P and E.

00 One such isomorphism may be defined via an

extension of the Fourier transform.

3.1. 2

DEFINITION.

The Fourier transform 0 of 0 in P

is a unique element in the normed dual of El defined by

<f ,0> <f,o>

for each f E A.

Since 0 belongs to the normed dual of E

l , then it

corres-ponds to a unique element,

x

say, in E ₀₀ by the relation

~ A ~

..

(3.1.4) <f ,o> _-- _L~ _yEr d(y)Tr [f (y*)x(y)] ₌

Lyer

_d(y)

_{Tr [[}

_(y)XX(y)],

Clnt i

of G defined by y*(x) In the sequel we will

not distinguish between 0 E

Ei

and X E E when they are 00

A

related by (3.1.4). Thus

110

II

₀₀

=

110

II

_p

•

For complete rigour, it is necessary to show that the

definition of Fourier transform for pseudomeasures given above is

an extension of that for integrable functions and measures.

Suppose that ~ E Mbd generates the pseudomeasure o.

Then, for f E A,

<f,o>

=

fG

fd~

=

~*f(e)

=

I

d(y)Tr[f(y*)~ (y)]

by (3.1.3), where f,~ are the usual Fourier transforms of f

and ~ respectively. By comparing with (3.1.4) and letting

f range over A, we see that ~

=

°

on

r

,

as required.

Since the Fourier transform for measures extends that for integrable

functions, there is nothing more to show.

To make P into a Banach algebra under convolution, we

(3.1.5) (0 *0 ) A

1 2

Inequality (3.1.1) shows that A on

r,

where 0

1,°2 E P.

3.2 Representation theorems.

Often a space of multipliers may be represented as operators

defined by convolutions with a set of functions, pseudomeasures,

et cetera. However, in most cases it is difficult to give an

independent description of this set. For example, see the

in 0.2.7. In this case, Lq is defined p

in terms of the space of (p,q)-multipliers and when p

1

1,

or q

1

00, or p and q are not both 2, Lq has no p

satisfactory alternative description. (See Hormander

[1]

for

the other cases, where it is shown that

00

=

Lq

L ,

q (1 < q ::. 00) and L

2

= P.)

2

The first two theorems of this section are representations

of multipliers from

A

into p _{and from} L2

a compact group, in terms of well-known spaces.

into L , 2 when

The following notation and terminology will be needed:

Let A and P denote the linear spaces A and P equipped

w w

G

with a(A,P) and a(P,A) topologies respectively (see Edwards [3, p. 501)). Let ~ be a function in F with finite support.

Then there exists a unique function f in C such that f

=

~ ,

namely

X !---+ f (x)

The set of all such f is closed under linear combinations, and

will be called the linear space of triponometric polynomials.

3.2.3 THEOREM. Let T be a map from A into P.

the following five conditions on T are equivalent.

(1)

(2)

T is a right

T is a right

(A,P)-multiplier.

( A, P)-multip1ier.

w w

Then

(3) T is a linear map from A to P which commutes with

right convolutions by pseudomeasures, that is

for each f in A and 0 in P.

(4) Condition (3), with the set of trigonometric polynomials

replacing the set of pseudomeasures.

(5) There exists a unique ~ in F, satisfying

sup II~(Y)II/d(y) <

00

,

such that

y

(3.2.1) (Tf)

on

r

for each f in A.

PROOF. Ad (1) implies

(2).

This is immediate from a standard

result in duality theory.

Ad (2) imE1ies (3). The proof will be divided into several

parts. Let

measure.

8

=

p 8 where a E G and 8

(a) The set consisting of finite linear combinations from

{o : a E G} is dense in P.

a w Suppose that f E A and

<0 ,f>

=

0 for each a in G.

a Then f

=

0, showing that only

the zero continuous linear functional on P vanishes on w

{o : a E G}.

a An application of the Hahn-Banach theorem completes the proof.

(b) Let g E A and ¢ E P; then the operators and

a

~

¢

*

a

are continuous from P.!..Q A

w w

to wP respectively. This follows from the relations

and <f*¢,a>,

and P

w

where ~,a,¢ E P and f,g E A, which in turn may be derived from Definition 3.1.2 and (3.1.4). For example,

<~,g*a> <~,ag>

A

Ly d (y) Tr [g (y) ~ (y*) a (y ) ]

V A

Ly d(y)Tr[(g) (y*H (y*)a(y)] v A

Ly d(y)Tr[ (~~~g) (y*)a(y)]

<~*g,a>

(c) (2) implies (3) . Assume that T is a continuous

linear operator from A

w into w p which commutes with translations.

Let f E A, and define an operator u : 0 ~ TCf"~o) - Tf*o

from P to P. Application of (b) shows that u is

continuous from P to P.

a E G,

w w

uC6 )

a T (f;~6 a )

T(p f) a

Since p T

=

Tp for

a a

Tf*6

a

p (Tf)

a 0,

so that {6 : a E G} belongs to the kernel of u.

a The proof

that T commutes with convolutions on the right by pseudomeasures

is completed by noting that (a) now implies that u is the

zero operator.

Ad (3) implies (4).

Ad (4) implies (5).

function on G such that

This is obvious.

For fixed y in r

g has support y

let g be any

and let denote

the function x ~ Tr[y(x)]. Hence is a trigonometric

poly-nomial satisfying

otherwise, where I ,

y

when y'

=

y and

is the identity operator on H , .

y

o

By

examining the Fourier coefficients, we see that g

=

d (y)x *g. y Assume (4), that is that T is a linear map from A to P

which commutes with right convolutions by trigonometric polynomials.