Multipliers and tensor products of L(p, q) Lorentz spaces

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MULTIPLIERS AND TENSOR PRODUCTS OF L ( p , q ) LORENTZ SPACES* Halcan Avca A . %ran Gurkanla Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayas University, 55199, Kurupelit, Samsun, T U R K E Y E-mail: hakanavaomu. edu. tr; [email protected]. t r

Abstract Let G be a locally compact abelian group. The main purpose of this article is to find the space of multipliers from the Lorentz space-L(pl,ql)(G)to L ( p ; , qb)(G).For this reason, the authors define the space Ag:;if(G), discuss its properties and prove that the space of multipliers from L ( p l , q l ) ( G ) to L(pa, q;)(G) is isometrically isomorphic to the dual of A;;;;; (G). Key words Lorentz space, Banach module, multiplier 2000 MR Subject Classification

1

43A15

Introduction

Let G be a locally compact abelian group with Haar measure dx and Cc(G) denote the space of complex-valued continuous functions on G with compact support. The left (right) translation operators Ls (R,) are given by L , f (y) = f (y - s) ( R ,f (y) = f (s y)) for s, y E G. Let f be a measurable function defined on a measure space ( G , p ) . Let f be finite almost everywhere and for y > 0, assume

+

The distribution function of

f is defined by

and the (nonnegative) rearrangement of f is defined by

f*(t)= inf {y > 0 : X,(y) 5 t } = sup{y > 0 : Xf(y) > t } , t > 0. Also the average function of f is defined by

f**(t) =

-1

*Received June 24, 2004; revised March 31, 2005

l

t o

t

f*(z)dz.

(1.3)

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It is easy to see Xf, f*, f** are nonincreasing and right continuous functions on (0, 00). The Lorentz space L ( p , q)(G,p ) (shortly L ( p , q ) ) is defined to be the vector space of all (equivalence classes) of measurable functions f such that 11 f l/;p,q) < 00 where

It is known that

F OPP, == and so L (p,)= Lp and if 0 < q1< q21< , 0 < p<

then

f (p,q2)< f (p,q1holds an hence L (P, q1) L (p,q2)[7]. Alos L (p,q)(Gu,is a nomed space with the norm

Yap proved in [14] that if 1 < p measure space ( G, p ) , then

< 00,0 < q I 00 and f

is a measurable function defined on a

where C ( p ,q ) is a constant depending on p and q . Let A be a Banach algebra. If Vand W are left (right) Banach A-modules, then a multiplier from V t o W is a bounded linear operator T from V to W , which commutes with module multiplication, that is, T (su) = s T ( u ) for all s E A, u E V. We denote by HomA(v, W ) or M(V, W ) the space of multipliers from V to W . Let V and W be left and right Banach A-modules, respectively, and V @ ? W be the projective tensor product of V and W, [2, 131. Assume that K is the closed linear subspace of V B7 W ,which is spanned by all elements of the form au@w-u@aw, a E A, u E V, w E W . Then the A-module tensor product V @ A W is defined to be the quotient Banach space (V @7 W ) / K . Every element t of (V @7 W ) / K has the form

00

where

C llvill IIwilI < 00 (Rieffel [lo]). It is known that HomA(V, W " )

( V@ AW ) *where W *

i=l

is the dual of W (Rieffel [9]). Then the linear functional on HomA(V, W * ) which , corresponds to t E V @ AW takes the value M

at T E HOmA(V, W * ) . It is clear that the topology on HomA(v, W * )defined by the linear functionals of this form corresponds to the weak*-topology (V@AW)*.This topology is called ultraweak*-operator topology (Rieffel [lo]).

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The Space A PPlz yA q1 a ( G ) and Some Properties Throughout this article, let G be a locally compact abelian group and 1 < p1,pz

< co,1 5

q1,qz I rn orp1,pz = 1 = q 1 , q 2 , p1,pz = 00 = q 1 , q 2 . We need the following theorem for the definition of the spaces A;;::; (G). One can find the proof of this theorem in O’Neil [8].

Theorem 2.1 If T is a convolution operator h = T ( f ,g) = f * g for f E L(pl,ql)(G), > 1, then h E L(r,s ) ( G ) ,where Pl 2- + 2 - 1 = and s 2 1 is any 9 E L(p2,qz)(G)with PZ number such that f 2 $. Moreover

&+ &

I1hll

(T,,)

5 3r II f II (PI ,ql1 II9 I I (pz,q2).

.m

Let = f(-x), then II~lI(pl,ql) = Ilfll(pl,ql) and so .f E L(Pl,ql)(G)for every L(p1,q1)(G) (Hunt [7]). Hence by Theorem 2.1 we may write

f

E

IIJ*gll(T,s)5 ~ l fl (p, q,)1 ~1 (p2,q,)~ By Theorem 2.1, we can define a bilinear operator b by

such that b(f,g) = f * g for f E L ( m ,qi)(G)and g E L ( p z , q2)(G).It is easy to show ljb(1 5 3r. Then there exists a bounded linear operator B from L ( p 1 , q1)(G)gj7 L ( p 2 , qZ)(G)into L(T,s ) ( G ) such that

B ( f @ g )= b ( f , g ) ,

f E L(pi,ql)(G),gE L(pz,qz)(G)

and llBll 5 3r by Theorem 6 in Bonsall-Duncan[2]. Definition 2.2 The range of B with the quotient norm is denoted by A;;$;(G). Thus w

h E L(r,s ) ( G ): h = A P 2 742 P l , q l (GI

=

-

C fi*gi, i=l

c II

fi E

gi E J%%qz)(G),

fill(p~,~l)ll~ill(pz,q2)

i=l

< 00

and

IIIhIII = inf

C II

I

L(pi,qi)(G),

W

1

: f i E L ( ~ l , q l ) ( Ggi ) ,E L ( p 2 , ~ 2 ) ( G ).

f~II(pl,ql)II~iII(p2,qz)

{ i=l w

Evidently A;::%;(G)c L ( r , s ) ( G )and JJhJJ(,,) 5 3rJIJh J ) I . Also by the technique of the proof used in Theorem 2.4 (Gaudry [6]), one can prove that (A:;;:; ( G ) ,111.111) is a Banach space. Let S be the set of simple functions defined on G. It is known that S c L ( p , q)(G)and S is dense in L(p,q)(G)with respect to the norm IIII(p,q) (Hunt [7]). Define

i CSiw

B ( S @ S )= t = Y

W

*ti : %,ti E

s,C i=l

i=l

<

I

lISi(l(pl,ql)lltill(pz,qz) 00

As S is dense in L(p,q)(G),it is not difficult to prove that B ( S B 7S ) is dense in APp::9q;(G). Proposition 2.3 a) The space A;:;:; ( G ) is invariant under translation. b) For every h E B ( S €31~ S ) the map s L,h is continuous from G into A;;:9q;(G). --f

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Proof a) Let h E AF:;:f(G) be given. Then we have

M

Thus we have Lsh E A;:::: (G). b) Let h E B ( S @ ,S ) and E

> 0 be given. Then we have the expansion

h=

En * gi i=l

c oci

M ..

7

Ilfill(p1,41)I19ill(pz,qz) <001

i=l

where fi, gi E S. For the proof of this part it is suffices to show that s + L,h is continuous at the unit e E G. As

c 00

IIfiII(p1,q1)I19~Il(pz,qz) < 00,

i=l

there exists a natural number rt. such that

i=n+l

Let k =

5

J”i

i=l

* gi and C =

5~ ~ g ~ ~ ~ pAlso z , q because 2).

i=l

r x

i=n+l

we have

c lIfiII(pl,ql)Il~ill(pz,q,) <3. M

Ill h - k Ill 5

E

i=n+l

We also write

Hence For the last term of (2.2), we write n

n

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It is known that the operator L , is an isometry on L(p1, q1)(G) and s + L, f i are continuous from G into L(p~,ql)(G)for all f i E L(pl,ql)(G) and i E {1,2;..,n} by Lemma 3.2 in Chen - and Lai [3]. As L,fi = (R,fi), we write llLsfi - fiII(P1rql) = IIR,fi - f i ~ ~ ( p l , q l ) , and the maps s + R, fi are continuous from G into L(p1, q1)(G) for all i E { 1 , 2 , . . . , n } . Then there exist neighbourhoods V , of the unit e E G such that IlRsfi - fill(pl,ql) <

&

(2.4)

3c

-)

for all s E V , and i E { 1 , 2 , . . . , n } . We take V = n:==,V,. Thus, using (2.1), (2.3) and (2.4), we see from (2.2) that

for all s E V . This completes the proof. As ( 1 ) L,h 111 5 111 h 1 1 1 and ' B ( S@,-I S ) is dense in A;;::: (G), it follows that the function s + L,h is continuous from G into A;;;;;(G) for every h E AF;;:;(G). Proposition 2.4 There exist approximate identities {a,} and { b p } of L1(G) such that ((6, * balll 5 1 and lim,,p 111 h - 6, * bp * h 111 = 0 for every h E AF;;:;(G). Proof It follows from Lemma 3.3 of Chen and Lai [3] that there exist approximate identities {a,} and { b p } of L1(G) such that Ilaalll = llbplll = 1 and

for all f E L(pi,qi)(G), g E L(p2,q2)(G). Thus we have for every a E I , p E J . Let h E A;;;;; (G) be given. Then 00

00

i=1

i= 1

Hence

and so 6,

* bp * h E A;:;:; (G). Also we may write

*

b p E L'-(G) and 116,

* bp/ll 5 1

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Finally from (2.5) we have lim 111 h - 6, * bp

* h 111 = 0.

Proposition 2.5 There is an approximate identity {hu}of L1(G)with compact support such that llhulll = 1 and

“u” Ilf * hu fll(p,q) = 0, -

for every f E L(p, q)(G). Proof One can easily prove this proposition by the technic used in the proof of Theorem 5.3 in Wang [15] and by Lemma 3.3 in Chen and Lai [3]. Corollary 2.6 There exists approximate identities {a,} and { b p } of L1(G)with compact support such that 116, * bpc(ll 5 1 and lim I I / h a,4

* bp * h 111 = 0,

for every h f A;:;Z:(G). Proof We can easily prove this corollary by using Propositions 2.4 and 2.5. By Corollary 2.6, it is easy to see that A;;$;(G) is a Banach module over L1(G).

3 Multipliers from L ( p l , ql)(G) to L(p’,,q i ) ( G ) Let K be the closed linear subspace of L(p1,qI)(G)BTL(p2,q2)(G)spanned by all elements of the form (P * f ) @ g - f 8 (‘P * 9) where f E L ( m ,qi)(G), g E L(p2,q2)(G)and ‘p E L1(G). Then the L1(G)-module tensor product L(p1,ql)(G) @LI(G) L(p2,42)(G)is defined to be the

qz)(G))/K. quotient Banach space (L(p1,ql)(G)8-,L(Pz, Theorem 3.1 Let G be a locally compact abelian group with p (G) < m. If + > 1, 1 -1 = and s 2 1 is any number such that PI pa 2 and either p l < 00 or p 2 < 00, q 2 5 S, then the space L ( p 1 , ql)(G) @ L I ( G )L(p2, q2)(G) is isometrically isomorphic to the space 2,92 G

+

4,41(

& &

5,

+&

1.

Proof For the proof of this theorem, it suffices to show that the kernel of B is exactly K . As B((P * f ) w - f w *g))=(cp * f j * g - P * ( @

*g)=o,

for all (‘p * f ) @ g - f 8 ($3 * g) E K ,the kernel of B contains K . Conversely, suppose that t is an element of the kernel of B. Then

i=l

i=l

(4,) be an approximate identity of L’(G) satisfying the conditions in Proposition 2.5. Suppose that 0 < p2 < 00 (one has an analogous proof for the case in which p l < 0;) instead). We define t , = C (fi * &) 8 gi. where the summation converges absolutely in L(r, s ) ( G ) . Let

00

It is clear that t , E L ( p 1 , qI)(G)B7L(p2,q z ) ( G ) . As p2

fi

* 4,

i=l

converges to

< m, one can easily prove that t , converges to t in L(r,s)(G). As

in L(r,s ) ( G )and B ( t )= 0, then given

E

rn

fi

for each i, and

C f; * gi is convergent

i=l > 0 there exists no E N such that

Avci & Giirkanli: MULTIPLIERS AND TENSOR PRODUCTS OF L ( p , q )

No.1

whenever n > no. Also we can choose n1

for all n

113

> no such that

> n1. In the equality

i=l

i=l

i=l

the second term on the right side is in K . We denote this term by k. We also write

< 00, by genaral assumption, we write p > 1 or p = 1. Thus we have &--15 0. Using 1 - 1 = ;, we obtain & 2 ;, then p2 5 r. Also because p(G) < co,42 5 s, we write

As p z 1

+

1

L(r,s ) ( G )C L(p2, q2)(G)from 1.8 in Hunt [7]. Using (3.2) we can obtain

Then (3.3) and (3.5) can be written as

That means t , E K . As K is closed, t E K . Thus KerB c K . This completes the proof. 1 > 1, 2PI- P2 2- - 1 = and s 2 1 is any number such that Lemma 3.2 Let P l P2

+

l + L > L 41

42

A.

PlP' < 41-4; Given any cp E C c ( G ) and = PlP;+PT-P;, * cp. Then Tq E HomLl(G) ( L ( p l ,ql)(G),L(p',, d ) ( G ) )

'3' p

by Tvf = f

+

f E L(pl,ql)(G)define Tv

I I T ~I I 3I ~ 2cpii(p,q). 1

1 I

Proof Let f E L(p1,ql)(G)and cp E C c ( G )c L(p, q)(G)be given. By the inequalities = 1 + 4 > 1 and + - 1 = we write r = p:. Also because q 5 we have

+f

P2

$ $

2 41-42 and + 2 1. If let s = qh, we obtain 41 cp * f E L ( r ,s ) ( G )= qP:, 4 X G ) and 4

&,

i,

&+f

2

Thus T9 is continuous. Also from this inequality one may write

Again it is easily shown that

$.

By Theorem 2.1 we have

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Definition 3.3 A locally compact abelian group G is said to satisfy the property PF:;:: if every element of HomLi(G) ( L ( p 1 ,q1)(G),L(p!,,, q&)(G))can be approximated in the ultraweak* operator topology by operators T,, cp E Cc(G). Theorem 3.4 Let G be a locally compact abelian group. If > 1 and 1Pl + 1P2- 1 = and let s 2 1 be such that 2 then the following statements are equivalent. i) G satisfies the property Pi;;:;.

&+&

& + 5 i,

ii) The kernel of B is K so that L ( p l , q l ) ( G ) @ ~ 1L(pz,qz)(G) (~) AE;;:;(G) (that is, isometrically isomorphic). Proof As in Theorem 3.1, it is easily seen that K is contained in the kernel of B. Suppose that G satisfies the property Pp”1”;;;. To show that the kernel of B is contained in K , it suffices, by the Hahn-Banach Theorem, to show that any bounded linear functional on L(pi, qi)(G) @ L ~ ( G L(p2, ) q2)(G) which annihilates K also annihilates the kernel of B. It is known that (Conway [4])

Therefore, we may write

On the other hand, it is known that

Now let F E K I . F’rom (3.12) there corresponds t o F a n operator T E HornLl(G)(L(pl,ql)(G), L(pk, q&)(G))such that

c m

( t , F )=

(%,Tfi),

(3.13)

i=l

for all t E L(p1, q1)(G)

L(p2, qz)(G) with the expansion 00

(3.14)

(3.15)

i=l the summation converging in the norm of L(r,s)(G). Also we write ca

(3.16) i=l

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We want to show that ( t ,F ) = 0 or equivalently (3.17) i=l

As G satisfies the property PF;;,$t, there exists a net ((9,) : (Y E I } c C c ( G )such that the family of operators Tpadefined in Lemma 3.2 converges to T in the ultraweak*-operator topology In particular (3.18) Thus to prove (3.17), it suffices to show that 00

(3.19) i=l

for each a. It is easy to see M

M

(3.20) i=l

i=l

As Cc(G)c L(T’,s’)(G),using 13-15)](3.20), we may write (3.21) by Theorem 3.5 (Saeki and Thome [12]). Therefore ( t ,F ) = 0 for all t E KerB. That means F E (KerB)I and thus KerB c K . Hence K = KerB. This proves L(P1, Ql)(G) 8 L(PZ1 qz)(G)

-4:;

(G).

Ll(G)

Suppose conversely that the kernel of B is K . We have to show that the operators of the form Tp for cp E Cc(G) are dense in HomLl(G) ( L ( p 1 ,q l ) ( G ) ,L(p’,,qL)(G))in the ultraweak*operator topology. It is sufficient, according to Theorem 1.4 in Rieffel [lo], to show that the corresponding functionals are dense in (L(p1,ql)(G)@LI(G) L ( p z , qz)(G))*in the weak*topology. Let M be the set of the linear functionals corresponding to the operators Tp. If we could prove that M I = K = KerB, then this completes the proof by Corollary 6.14 in Conway [4]. As (L(P1, s1)(G)

LFG)L(Pz1 qz)(Gb)*

=

(L(Pl1d ( G )63w Y

= (KerB)’-,

2 , qz)(G)IW*”=

KL (3.22)

) qz)(G))*2 (KerB)’-, then ( t ,F ) = 0 for all t E KerB and and M C ( L ( p i ,qi)(G)@ L ~ ( GL(pzl F E M . Thus t E M I . That means KerB c M I . Conversely, let t E M I . As M’- c L(p1,q1)(G) @ L I ( G ) L(pz1qz)(G),there exist fi E G l , q l ) ( G ) , si E L(pZ,qz)(G)such that 00

00

(3.23)

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(3.24) a=1

i=l

It is clear that there corresponds to every cp E Cc(G) an F E M . Also because Cc(G)is dense in L(T’,s’), using the Hahn-Banach Theorem, we can easily prove that m

(3.25) i=l

Therefore M I C KerB. Finally M I = KerB = K . This proves the assertion. Corollary 3.6 Let G be a locally compact abelian group. If > 1, - 1= and s 2 1 is any number such that 2 and G satisfies the property Pf;;:;, then

& +&

5+

H O ~ L ~ ( G (L(Pi, ) 4i)(G),L(PL,d ) ( G ) )

+&

(AE::;f(G))*.

Proof One can easily prove this corollary by using

and Theorem 3.4.

References 1 Blozinski A P. On a convolution Theorem for L ( p , q ) spaces. Trans of the Amer Math Society, 1972, 255-265 2 Bonsall F F, Duncan J. Complete Normed Algebras. Belin, Heidelberg, New York: Springer-Verlag, 1973 3 Chen Y K , Lai H C. Multipliers of Lorentz spaces. Hokkaido Math J, 1975, 4: 247-260 4 Conway J B. A Course in F‘unctional Analysis. New York: Springer-Verlag, 1985 5 Doran R S, Wichmann J. Approximate Identities and Factorization in Banach Modules. In: Lecture Notes in Mathematics, 768. Springer Verlag, 1979 6 Gaudry G I. Quasimeasures and operators commuting with convolution. Pacific Journal of Mathematics, 1965, 13(3): 461-476 7 Hunt R A. On L ( p , q ) spaces. Extrait de L’Enseignement Mathematique, 1966, 12(2): 249-277 8 O’neil R. Convolution operators and L ( p , q ) spaces. Duke Math J, 1963; 30: 129-142 9 Rieffel M A. Induced Banach representation of Banach algebras and locally compact groups. Journal of Functional Analysis, 1967, 1: 443-491 10 Rieffel M A. Multipliers and tensor products of LP spaces of locally compact groups. Studia Math, 1969, 33: 71-82 11 Oztop S, Gurkanli A T. Multipliers and tensor product of weighted Lp-spaces. Acta Mathernatica Scientia, 2001, 21B:41-49 12 Saeki S, Thome E L. Lorentz spaces as L I modules and multipliers. Hokkaido Math J, 1994, 23: 55-92 13 Schatten R. Theory of cross-spaces. Annals of Mathematics Studies, 1950, 26 14 Yap L Y H. Some remarks on convolution operators and L(p, 9) spaces. Duke Math J, 1969, 36: 647-658 15 Wang H C. Homogeneous Banach Algebras. New York, Basel: Marcel1 Dekker INC, 1977