TYPE AND COTYPE OF SOME BANACH SPACES

TYPE AND COTYPE OF SOME BANACH SPACES

MIECZYSLAWMASTYLO Instituteof Mathematics

A.

MickiewiczUniversity

Matejki 48/49 60-769

Poznafi

Poland

(Received

^on

June

11,1990 and in revised form

May

6,

1991)

ABSTRACT. Type

andcotypearecomputed^forBanach

spaces

generated bysomepositivesublinear operatorsandBanachfunction

spaces.

Applicationsof the resultsyield thatundercertainassumptions Clarkson’s inequalitieshold in these

spaces.

KEY WORDS AND PHRASES. Type,

cotype,Clarkson’sinequalities.

1991

AMS SUBJECT CLASSIFICATION CODE.

Primary,46B20.

1.

INTRODUCTION.

Given aBanachspace

X,

welet foranyⁿ !1, p^s2

q

<^ooand ss<o,

Ko’.)(X)

and

Kt,.s)(X

^be

x C

X,

where

{r" }’.

denotes the

sequence

ofRademacherfunctionsdefinedby forevery^choiceof

i}i.l

r,

(t)

sign sin2"tr for0s 1. Iftheleft

(resp.

theright) inequalityin

(1,1)

^holds,

X

^isof cotype

(q, s) (resp.

type

(p,s)).

Ifs 1,we

say

that

X

isof cotype

q (resp.

type

p) (see [6]).

The notions oftypeandcotypehaveappearedin various

problems

involvingtheanalysisofvector valued functions or random variables.

One

of thegreatadvantagesof the classification ofBanachspaces in terms oftypeandcotype isthe existenceof a rathersatisfactory geometriccharacterizationof these notions.

For

_example

Maurey

and Pisier

[8]

showed that aBanach space

X

isof type pfor some

p

>

(resp.

cotype

q

for some

q

^<

oo)

iffX does not contain

l"s ^(resp. g’s)

uniformly.

Note

that if

X

isoftype

(p,p’)

^with

K"P’)(X)

1, <

p

^s2

(resp.

cotype

(p, p’)

^with

K0,.,,)(X

^1,

2 p<

oo)

^and1/p +

1/p’

1,then

X

verifiesClarkson’s inequalities, i.e.,forevery x,

y X,

wehave

(11

^x

II"

^/

^{y (1.2)}

llx ^{Yll’+ IIx +yll"’}

(1 )l/v ^(1.3)

resp. x y + x +

Yll" ⁽¹¹ xll ’

⁺

^Y I1"’)’’

Clearly (1.2), (1.3)

impliesthat

X

isuniformlyconvex.

thesmallestconstantsforwhich

The well-knownexamplesofBanach

spaces

for which the aboveinequalitieshold are

L,-spaces (see [2]), ^p-Schatten

^idealsofcompact operatorsonHilbert

spaces (see [9]),

provided 1<p <

In [10]

Milmanshowed, using interpolation techniquesthat if C:

R"

is adomain withminimally smoothboundary,then theinequality

(1.2)

appliestoSobolev spaces

W()

^{for <}p 2.Further Cohos 3

],

usingthe above observation,provedthattheinequalities

(1.2)

^and

(1.3)

hold in

W()

^{foreverydomain}

C

R"

and <p<o.

In

the samewayCobos and Edmundsin

[4]

showed that some

Besov

spacesand Triebel-Sobolev

spaces

verify Clarkson’s inequalities.

In

this

paper

wecomputethetypeandcotype for

spaces

of

large

classofBanach

spaces

generated by ^some positive sublinearoperators and Banach function

spaces.

This class includes for

example:

interpolation

spaces

determinedbythe real method ofinterpolation,

Besov spaces,

Triebel-Sobolev

spaces (see ],[ 11],[ 12]) H’-spaces,

anapproximation

space, L’(Ix, X)-spaces

and theother

(see

^for

example [5]).

We

also show thatunder some conditionsClarkson’s inequalitieshold in these

spaces.

2.

PRELIMINARIES.

Let (,gt)

be a complete o-finite measure

space.

If

X

is a Banach

space,

we denote by

L (X) -L (f,

ix,

X)the F-space [i.e.,

completeand metrizabletopologicalvectorspaceof allequivalence classes ofall

Ix-Bochner

^{measurableX-valued}functions on

.

^IfX

^R,

then we write

L LO(,l.t).

A

Banach

space E

C

L

is calledaBanach

function ^space

^if

Ix[ Y[ -a.e.

^on

,

^x

_ ^L

^and

^{y IE E}

implythatx

tEE

and

[lll

Recall thataBanach functionspace

E

iscalledp-convex

(resp. p-concave),

p<^ooif thereexists aconstant

M

sothatfor all

x,...,x,, E,

wehave

The smallestpossible^valueof

M

isdenotedby

M’)(E) (resp. M0,E)).

In

what follows let

X

be an

F-space

and let

S

be apositivesublinearoperatordefined on

.X’

taking values in

L* -L*(, Ix);

that isfor

every x, y tEX

andanyscalar

.

^{thefollowinghold:}

(i) ^Sx

^0,

(ii) S(),x) )q Sx, (iii) S(x

+

y) Sx

+

Sy.

For

agivenBanachfunctionspace

E

CL*andinjective operator

S:X.- L*,

wedefine

De(S)- {x X ^:Sx E).

If

E -(L,, "11,),

we write in short

D,

^insteadof

^De(S),

^where

Ilxll, -(f= Il’d)’

^for

Throughout

the

paper,

weassume that

De(S)

^isa Banaeh

space

withthenorm definedby

Ilxll o- Sxll .

We say

that apair

(E,S)

is admissibleprovidedthatfor any

A

with

I.t(A)

<=0,wehave

xSx,O

in

E

forevery

sequence {x, }

^C

De(S)

^{such that}

x,

0 in

X. _{Here Xa}

isacharacteristic function ofA.

3.

RFULTS.

Let (T,v)

^and

(, I.t)

be measurespaces.

In

thesequelforany

x x. ^tE.X"

^and

f ,f. @L(T,v),

we write

f ^{(R)xk(t )- A(t)xk}

^for

^ttET.

-1 k-I

An

easy

proof

of thefollowinglemmamaybe omitted.

LEMMA

3.1.

Let ft, ...,f, L(T,v)

Then thefollowinghold:

(i) Forallx x. Xandforanyo2 S .lf(R)x (co)tEL*(T,v).

(iii) lf

^{a measure}

^{space (T,v)}

^{isfinite,then}

for

^all

x x. ^X

^and

fl f. ^L p,

THEOREM 3.1. Assumethat

(T,v)

isa

finite

^measure

^space.

^Let

^E

be a Banach

function

^space

andlet

f

(i) IfE

isp-convex, then

for allxl,...,x X-De(S),

^wehave

(ii) If ^E

^{isp-concave, then}

for

all xl

xn ^X,

^wehave

PROOF.

Firstof all,supposethat

,

^{k 1,...,narestepNnctions.}

^We

^{cancerinlyassume}

^at

cx,, ^withA

^C

r

measurable, paiise disjointand

r _A. en

^for

C -Me(E),

wehave

i-I i-1

(by

p

convexity)

Now

assume that in thesquence

{f}k-, f

^{with k 2 ,n arestep}functions. Takeany sequence

{gi}-

ⁱⁿ

^L’(T,v)

^{f step}functions suchthat_gi

"-

^v-a.e,on

^T,

^andgi

^--*f

ⁱⁿ

L’(T,v). From

this, we easily getthat

and

aj

S

_{gj (R)}

x

⁺

2

^(R)

^x,

^dv

Since a Cb by previously proved inequality,we obtain desiredinequality. Thus, by iteratingthe

proof

of

(i)

^iscomplete. Theproofof

(ii)

^issimilar.

Let

usdefineon a

F-spaceX

afamilyof semi-norm

"1

^by

^{Ix[ Sx(co)}

^foreveryco

. ^Now

^the

maintheorem of thepaperisimmediate.

THEOREM

3.2. Assumethat <p^<doand s<oo.

Let E

be aBanach

function

^spaceandlet

K=D(S).

(i) If ^E

^{isp-convex, <p 2,ands-concave,and}

for

all xi

,x, GX

ri(t)x C llXil

^l-a.e.,

^(3.1)

k-1

then

X

is

of

^type

^(p,

^{s with}

Kte’s )(X) C,(p

s

C M" ^{)(E)Mts)(E ).}

(ii) If ^E

^is

^p-concave,

²

^p

^<

,

ands-convex,and

for

^all

x, x, X

Ix

^a.e.,

(3.2)

thenX is ofcotype

(p,s)

with

Ko,.,

COROLLARY

3.1.

1.f

^{theconditions}

o.fTheorem

^3.2are

satisfd

^{with s}

^-p’

^and

^Ct(p,s)- (resp. C2(p,s)- 1),

thenClarkson’s inequality

(1.2) (resp. (1.3)) holds.for De(S).

PROPOSITION

3.1.

Let (Le,S)

^beanadmissible pair,

,:p

<oo.

Assume

that

D

_e

CX

withcon- tinuousinclusion and that

D

_eisanon-closed subspace in

3C.

Then

D

_e^{is not}typer

(resp.

cotype

r).for any

r>p

(resp..for

anyr<

p).

PROOF.

The aboveassumptions implythat for

any

^e>0,

D

_econtains

(1

+e)-isomorphic

copy

of_e

(see [7]).

Sincetypeandcotypeis inheritedby subspaces,then theproof^isfinished.

In

thetheoryoftypeandcotypethetypeandcotypeindicesof a Banachspace

B

whichare defined asfollows

p(B) sup{p: B

is oftype

p}, q(B)= inf{q: B

^is of

cotype p}

areimportant

(see [8]

^for

details).

COROLLARY

3.2. Assumethattheassumptions

of

Proposition 3.1are

satisfied. ^{Let X- D}

e.

(i) If

^<

^p

^2and

for

all xl

,x, X

r,(t)x C Ylx, l"

^x-a.e.,

then

p(X)

p.

(ii) If

^{2 p<ooand}

for

^all

x ^{x, EX}

i.,Ixil ^c i.ri(t)xl _todt

^Ix-a.e.

then

q(X) p.

PROOF.

Since

L

_eis

ponvex

andponcave Banachnction

space,

^wehave

X

^isoftype

, ^p)

^for

<p 2

(resp.

cotype

,p)

for2 p<

)

by

eorem

3.2. ordertofinishtheproof^itsufficestoapply aresult ofhane

(see [6, eorem 1.e.13])

andProposition 3.1.

4.

EMPL.

We

give

o

general examples injectiveand

sitive

^{sublinearoperatorstisfyingthe} inequalities givenin

eorem

^3.2.

t (,)

^beameasure

space

and

letX

an

F-space.

Fix <q< and aume that

{T,}.

^{is a}

sequence

ofinjectivelinearoperators,

Tt:X L(,),

^suchSat

Thenobviously

e

opetor

S: L

isinjecfive,

sifive

^sublinear.

^For

^{isoperator,wehave}

(ii) If

²

^q

^{p <}

, en for ^aH x ^{,x, X}

PROOF.

Wehave

.e,l)-

1 for all 1<q 2 andby ^aduality argument

’e’)(lt)-

for all 2 q<

(see 10]).

Now

aumethat1<p

q

2.

en

by

q’ p’,

itfollowsthat

for allxt,...,xn

X.

Theproofof

(ii)

is similar.

Let X

^beaBanachspaceand

letX L(f2,1,X).

^{Define an}injective, positivesublinearoperatorS:

X-- L(f,tt)

by

I1 (,o)11 ,

^,o

Then the Banach

space De(S)

^is well-known and is denoted by

E(X). Clearly

the inequality

(3.1) (resp. (3.2))

^isequivalenttothe fact that

X

isoftype

(p,s) (resp.

cotype

(p,s)).

REFERENCES

1.

LOFSTRM, J.

and

BERGH, J.,

Interpolation

spaces (Springer-Verlag,

Berlin-Heidelberg-New York,

1976).

2.

CLARKSON, J. A.,

Uniformly convexspaces,

Trans. Amer.

Math,$0f, 40

(1936),

^396-414.

3.

COBOS, F.,

Clarkson’s inequalitiesfor Sobolev

spaces,

Math.Japonica 31

(1986),

^17-22.

4.

COBOS F.

and

EDMUNDS, D. E.,

Clarkson’s inequalities,

Besov spaces

and Triebel-Sobolev

spaces, Zeitschr. r

^Anal.

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COWMAN, R. R., MEYER, Y.

and

STEIN, E. M., Some

new function

spaces

and theirapplications toharmonicanalysis,

J. Funct.

Anal. 62

(1985),

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6.

LINDENSTRAUSS, J.

and

TLAFRIRI, L.,

ClassicalBanach

spaces II:

Function soaces

(Sprin- ger-Verlag,

Berlin-Heidelberg-New York,

1979).

7.

MASTYLO, M.,

Banach

spaces

via sublinearoperators,to

appear.

8.

MAURE, B.

and

PISIER, G.,

S&ies de variables alatoires vectoriellesindpendanteset

pro-

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espaces

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9.

McCARTHY, C. A., c,,

^Israel

^J.

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10.

MILMAN, M., Complex

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(198),

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11.

TRIEBEL, H.,

Interpolation theory,function^soaces,differential oerators

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Verlag

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Wissenschaften

^Berlin,

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12.

TRIEBEL, H.,

Theory_ offunction

;paces (Birkhuser, 1983).