An abstract version of Herz' imbedding theorem

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

S TEFANO M EDA

R ^ITA P ^INI

An abstract version of Herz’ imbedding theorem

Rendiconti del Seminario Matematico della Università di Padova, tome 86 (1991), p. 37-46

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An Abstract Version of Herz’ Imbedding ^Theorem

STEFANO MEDA - RITA

PINI (*)

The aim of this paper is ^to

generalize

^{to a}

quite

^abstract

setting

^a

well-known result

by

^{C. Herz}

([6],

^Theorem

⁵⁾

^{which can be stated as}

follows.

THEOREM.

Suppose

^{that and}

is

continuously

embedded in whenever

The reader is referred to the notation below for _every

unexplained terminology

^or

symbol.

^Herz’

proof

^{is based on an} inversion formula for standard

higher

order difference

operators

^on

R n,

^{which cannot be use-}

fully generalised

^{to other contexts.} Our contribution is to

provide

^{a dif-}

ferent, perhaps simpler, proof

of Herz’ theorem in the Euclidean context which can be

adapted

^{to a} very

general

situation. We assume that G is a

locally compact

unimodular

topological

group with o-fmite Haar mea- sure.

Furthermore,

^we suppose that we can find an

approximate

^iden-

tity (~~ )~ ~ o ,

^which possesses the

following properties

where

p’

denotes the index

conjugate

to p. It is well-known that _ap-

proximate

identities

satisfying

^{the above}

requirements

^{can be found on}

many groups. For

instance,

^{let G be a} connected unimodular Lie _group and let be the

family

of heat kernels associated to the heat semi- group

generated by

^a

(suitable)

^invariant

(sub-)laplacian.

Then it is not hard _{to prove} that satisfies

(i)

^and

(ii)

above

(see,

^{for in-}

stance

[4]).

Conditions similar to

(1) (i), (ii)

^have been considered

by

^L.

(*) ^Indirizzo

degli

AA.: S. MEDA:

Dipartimento

^di

Matematica,

Universita di

Trento,

^{38050 Povo}

(TN), Italia;

R. PINI: Istituto di

Matematica,

Università di

Verona,

^via

dell’Artigliere

19, ³⁷¹²⁹ Verona, ^Italia.

De Michele and I. R.

Inglis ^[2]

^{on spaces of}

homogeneous type (in

^the

sense of R. R. Coifman and G.

Weiss).

Our method

applies

to contects different from

locally compact

groups and

gives «multiplier

^{results» re-}

lated to Theorem 2.2 in

[7].

^{We shall} examine these

implications

elsewhere.

Our _paper is

organised

^as follows. We fix some notation in Section 1.

In Section 2 we define the

Lipschitz

spaces

AJ§, q

associated to an approx- imate

identity satisfying ⁽¹⁾

^and

give

^a sufficient condition

assuring

that

Lipschitz

spaces

arising

from different

approximate

identities

give

rise to

equivalent

spaces.

Finally

^we prove ^our main

result,

^which gen- eralises Herz’ theorem.

1. Notation.

Let M be a a-finite measure space and let

E ~

denote the ^measure of the measurable subset E of M. We denote

by f *

^the

noncreasing

^rear-

rangement

^{of the} measurable function

f (see [8]).

^If

f

^is

integrable

^on

the sets of finite measure we may also definite its

averaged

nonincreas-

ing rearrangement f ** by

the formula

The Lorentz _space

Lr, q ,

^{1 r 00,} the Banach _space of all measurable functions

f

^{such that}

is finite. It is easy ^to prove that

Lr, p is continuously

^{imbedded in}

Lr, q

whenever We refer the reader to

[8]

for more on the Lorentz _spaces.

2. The main result.

Assume that G is a

locally compact

unimodular group with a-finite.

Haar measure and is an

approximate identity satisfying ^(1).

^For

every

integer n,

^{we set}

39

For _every

positive

real number ^a let

8;, q,

^{1 ~ p 00,}

1 ~ q ~

00, denote the set of all

compactly supported

continuous functions

f

^{on G such}

that

if finite. Set It is not hard to show that Z =

(01.

^{We shall show that}

where

Qn = ~2-n ~ ~2-n+1 ~

^the convergence

being

in the uniform sense.

Suppose

^{now that}

= 0;

a. e. ,

whence

f =

⁰

by (3).

It remains to show that

(3)

^{holds. A}

straightfor-

ward

computation gives

^that

by (1) (i).

^{Since =}

_~2-N

^*

_~2-N

is an

approximate identity

^{on G as}

and

f

^{is a}

compactly supported

continuous

function,

^{we have that}

f * f

in the uniform

topology. Furthermore,

for every p ^{00 we}

get

^that

~ 0 as

thereby proving ^(3).

The above cal- culations show that is a norm on

~p, q .

^The

Lipschitz

space is defined as the

completion

^of

8J§,q

^with

^respect

^{to the norm}

Lipschitz

spaces have been considered

by

^{several authors. The}

reader is referred to

[11]

and

[15]

for a rather

complete

treatement of the

properties

of such spaces. For ^{a more} coincise introduction to the

subject,

^see

^[13].

REMARK.

(a)

^The

Lipschitz

space

depends

^on the choice of the

approximate identity (~s)s&#x3E;o. Let,

^for

instance,

^{G be} consider the canonical

family ^(see [1],

p.

139)

and the associated

Lipschitz

space

~i~, q . Let f be

the inverse Fourier transform of the characteristic function of the interval

[1, 2].

Then it is _{easy to} check

that f is

ⁱⁿ

for

every a &#x3E; 0. Indeed

if k ~

0, l,

^while

if

k = 0, 1.

Let

(~8)8 &#x3E;

^{0 be the}

family

^defined

by

^{the rule}

~ denoting 1/2

the characteristic function of the interval

[- 1, ^1].

Then

We claim

that f is

^{not in}

Ã2,

^{1 if a ~ 2.}

Indeed,

^{let n be a}

large positive

^in-

teger.

^Then

(C independent

^on

n),

whence the claim.

It is clear that it is not hard to

produce

^{functions which are in}

but not in for _any admissible _{a, p, q.} We omit details.

Although

^the

Lipschitz

space

depends

^on the choice of the ap-

proximate identity,

^{we do} not stress this

dependence,

^to avoid comber-

some notation.

(b)

^For suitable choices of the

approximate identity 0

^{1 the}

Lipschitz

spaces defined above ^are

equivalent

^to

Lipschitz

^spaces

defined

by higher

^order difference

operators

^{in a}

variety

^{of con-}

texts.

41

If G is Rn the

equivalence

^{is a} classical result

(see,

^for

example [1],

Theorem

6.3.1).

If G is

R n,

^but

nonisotropic diagonal

^{dilations are}

considered,

^the

result is

implicit

ⁱⁿ

^[12].

If G is the

Heisenberg

group Hn the result is

essentially

^contained

in

[9],

^Remark

(b),

p. 44. A

generalization

^{of his}

argument

^{covers the}

case of stratified groups ^as well

(see

^also

[5]).

If G is a

compact

^connected

semisimple

^Lie group the

equivalence

^is

proved

ⁱⁿ

[10],

^{Theorem 1.9}

(see

also

[4]).

It is

interesting

^to find sufficient conditions which ensure that

Lips-

chitz _spaces, defined

by

^different

approximate identities, give

^{rise to}

equivalent

spaces. Let

(~8)8 &#x3E;

⁰ be another

approximate identity

^{and de-}

fine

Pn , Ng, q

^and

consitently

^{with the}

previous

^notation.

PROPOSITION 1.

Suppose

^that

for

^{some Y)} &#x3E; ex there exists a

positive

constant A such that

Then

continuously

embedded in

PROOF.

Clearly

it suffices _{to prove} that there exists a

positive

^con-

stant C such that

Np, q ( f ) ~ CN~, q ( f )

^{for all}

compactly supported

^con-

tinuous functions

f.

^We

give

^details

only

^{in the case} q ^{= 00.} The case

is, perhaps,

^{easier. Set}

Qn

⁼

~2-n

⁺

~2-n+~ .

^{Then a routine} argu-

ment

(see (3))

^{shows that .}

the convergence

being

in the uniform sense. Therefore

Since

Pj ^{11, IIQn 111} 1 IPj ^11,

^C

(independent of j

^and n; C ⁼ 4 does

work),

^{we have that}

by (4),

^whence

Choose c such that 0 ^{e a.}

Then, by

^H61der’s

inequality

and Fubini’s theorem we have that

Choose now E such that: ^« E q.

Then, again by

^Holder’s

inequality

and Fubini’s

theorem,

^{we have that}

Collecting

^{the above}

estimate,

^{we obtain} the desired result.

COROLLARY 1.

Suppose

^that

for

^{some Y)} &#x3E; a there exists a

positive

constant C such that

Then and

~1~, q

^are

equivalent

spaces.

PROOF. It is an immediate consequence of

Proposition

^1.

We are now

ready

to prove ^our main result. Let dx denote a two- sided Haar measure of G.

43

THEOREM 1.

Suppose

^{that acnd}

a &#x3E;

0;

^then

for

^all q, 1 q _ ^00,

Aap,q

^is

continuously

^{embedded in}

Lr, q ^(G).

PROOF. We

give

^details

only

^{in the case} q ^00.

Slight

modifications

give

^{the case} q ^{= 00 as} well. It is a well-known fact that

if f is integrable

over sets of finite _measure, then the

following

^{formula holds}

(see,

^{for instance}

[14],

p.

202).

^Set

Let f be

ⁱⁿ

~p, q . Then, owing

^to

(3),

^{we have that}

Choose an

integer

^{m such that}

2m -1 ~ t -1~Y 2m.

Then

Choose e such that 0 ^e

lip - «/y. By

^H61der’s

inequality

^and

^{(1) (ii)}

we have that

By

^H61der’s

inequality

^and

^{(1) (ii)}

^{we also have that}

Collecting

the above estimate we

get

^that

By

^Holder’s

inequality

^{and Fubini’s theorem}

Choose 6 such that 0 e «. Then Holder’s

inequality

^{and Fubini’s the-}

orem

imply

^that

45

A

density argument completes

^the

proof.

COROLLARY 2. Assume that ^a &#x3E; 0 and let k be in

A’, q .

^{Then the}

map k

* f

^{is bounded}

from Lpo , qo

^to

Lpl,q1 ^provided

^that

and

PROOF. It follows

immediately

from Theorem 1 and the convolu- tion theorem for Lorentz _spaces

([8],

^Theorem

4.10).

REFERENCES

[1] J. BERGH - J.

LÖFSTRÖM, Interpolation Spaces, Springer Verlag,

^Berlin-

Heidelberg-New York,

^1976.

[2] ^{L. DE} MICHELE - I. R.

INGLIS,

^{Lp estimates}

for strongly singular integrals

on spaces

of homogeneous

type, J. Funct.

Anal.,

³⁹

(1980),

pp. 1-15.

[3] ^{G. I.} GAUDRY - R.

PINI,

Bernstein’s theorem

for

compact ^{connected Lie}

groups, Math. Proc. Camb. Phil.

Soc.,

⁹⁹

(1986),

pp. 297-305.

[4] ^{G. I.} GAUDRY - S. MEDA - R.

PINI,

^{A heat}

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of Bernstein’s

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Math.,

¹¹⁰

(1990),

pp. 101-114.

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GIULINI, Approximation

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stratified

groups, Proc.

Amer. Math.

Soc.,

⁹⁶ (1986), pp. 569-578.

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Lipschitz

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Mech.,

¹⁸

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HÖRMANDER,

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algebra of

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Heisenberg

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PINI, Lipschitz

spaces ^on compact ^Lie groups, Mh.

Math.,

¹⁰⁵

(1988),

pp. 177-191.

[11] ^J.

PEETRE,

^New

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^{on Besov}

Spaces,

Duke Univ.

Press,

^1976.

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SOARDI, On nonisotropic Lipschitz

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WEISS,

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Manoscritto pervenuto ⁱⁿ redazione il 16

luglio

^1990.