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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; gurkanli@omu. edu. 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 43A15

1 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. (1.3)

Also the average function of f is defined by

l t f * * ( t ) = -1 f*(z)dz.

t o

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

108 ACTA MATHEMATICA SCIENTIA Vo1.27 Ser.B

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 1 1 f l / ; p , q ) < 00 where

Yap proved in [14] that if 1 < p < 00, 0 < q I 00 and f is a measurable function defined on a measure space ( G, p ) , then

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 to W is a bounded linear operator T from V to W , which commutes with module multiplication, that is, T (su) = sT(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 is the dual of W (Rieffel to t E V@A W takes the

i=l (Rieffel [lo]). It is known that HomA(V, W " ) (V@A W)* where W*

[9]). Then the linear functional on HomA(V, W * ) , which corresponds 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]).

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

No. 1 Avci & Giirkanli: MULTIPLIERS AND TENSOR PRODUCTS OF L(p, q) 109

2 The Space A P z y q a ( G ) and Some Properties

Throughout this article, let G be a locally compact abelian group and 1 < p1,pz < co, 1 5

We need the following theorem for the definition of the spaces A;;::; (G). One can find the

Theorem 2.1 If T is a convolution operator h = T ( f , g) = f * g for f E L(pl,ql)(G), and s 2 1 is any

f & 2 $. Moreover

Pl A 1

q1,qz I rn orp1,pz = 1 = q 1 , q 2 , p1,pz = 00 = q 1 , q 2 .

proof of this theorem in O’Neil [8].

9 E L(p2, qz)(G) with & + number such that

> 1, then h E L(r, s)(G), where 2- + 2 - 1 = Pl PZ

I1 hll (T, , ) 5 3r II f II (PI ,ql 1 II 9 I I (pz ,q2).

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

IIJ* gll(T,s) 5 ~~llfll(p,,q,)11~11(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(pz , q2)(G). It is easy to show ljb(1 5 3r. Then there exists a bounded linear operator B from L(p1, 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),g E 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

I w - h E L(r, s)(G) : h = C fi*gi, fi E L(pi, qi)(G),

i=l W

gi E J%%qz)(G), c II fill(p~,~l)ll~ill(pz,q2) < 00 i=l

AP2 742 Pl ,q l (GI =

and

IIIhIII = inf C II f~II(pl,ql)II~iII(p2,qz) : f i E L ( ~ l , q l ) ( G ) , gi E L ( p 2 , ~ 2 ) ( G ) . { w i=l 1

Evidently A;::%;(G) c L(r,s)(G) and JJhJJ(,,) 5 3rJIJ h J ) I . Also by the technique of the proof used in Theorem 2.4 (Gaudry [6]), one can prove that (A:;;:; (G) , 1 1 1 . 1 1 1 ) 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 W

B ( S @ S ) = t = CSi *ti : %,ti E s,C lISi(l(pl,ql)lltill(pz,qz) < 00 Y i w i= l i=l

As S is dense in L(p, q)(G), it is not difficult to prove that B(SB7 S ) is dense in APp::9q;(G). Proposition 2.3 b) For every h E B(S €31~ S ) the map s --f L,h is continuous from G into A;;:9q;(G).

a) The space A;:;:; (G) is invariant under translation.

Vo1.27 Ser.B 110 ACTA MATHEMATICA SCIENTIA

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

M oci ..

h = En * gi 7 c Ilfill(p1,41)I19ill(pz,qz) < 0 0 1

i=l 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

00 c 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 * gi and C = 5 ~ ~ g ~ ~ ~ p z , q 2 ) . Also because i=l

r x

i=n+l

i=l

we have

We also write

Hence

M E

I l l h - k I l l 5 c lIfiII(pl,ql)Il~ill(pz,q,) <3. i=n+l

For the last term of (2.2), we write

n n

No. 1 Avci & Giirkanli: MULTIPLIERS AND TENSOR PRODUCTS OF L(p , q ) 111

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,f i) , 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

- -

(2.4) &

IlRsfi - fill(pl,ql) < -) 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.

s + L,h is continuous from G into A;;;;;(G) for every h E AF;;:;(G).

((6, * balll 5 1 and lim,,p 1 1 1 h - 6, * bp * h 1 1 1 = 0 for every h E AF;;:;(G).

identities {a,} and { b p } of L1(G) such that Ilaalll = llbplll = 1 and

As ( 1 ) L,h 1 1 1 5 1 1 1 h 1 1 1 and 'B(S @I-, S ) is dense in A;;::: (G), it follows that the function

Proposition 2.4 There exist approximate identities {a,} and { b p } of L1(G) such that

Proof It follows from Lemma 3.3 of Chen and Lai [3] that there exist approximate

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

* b p E L'-(G) and 116, * bp/l l 5 1

00 00

i=1 i= 1

Hence

and so 6, * bp * h E A;:;:; (G). Also we may write

112 ACTA MATHEMATICA SCIENTIA Vo1.27 Ser.B

Finally from (2.5) we have lim 1 1 1 h - 6, * bp * h 1 1 1 = 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).

5.3 in Wang [15] and by Lemma 3.3 in Chen and Lai [3].

support such that 116, * bpc(ll 5 1 and

Proof One can easily prove this proposition by the technic used in the proof of Theorem

There exists approximate identities {a,} and { b p } of L1(G) with compact Corollary 2.6

lim I I / h - a,4

* bp * h 1 1 1 = 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) BT L(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

quotient Banach space (L(p1, ql)(G) 8-, L(Pz, qz)(G)) /K. Theorem 3.1 Let G be a locally compact abelian group with p (G) < m. If & + & > 1,

- + + & 2 and either p l < 00 or p2 < 00, PI pa q 2 5 S, then the space L(p1, ql)(G) @ L I ( G ) L(p2, q2)(G) is isometrically isomorphic to the space

Proof For the proof of this theorem, it suffices to show that the kernel of B is exactly

1 - 1 = 5 , and s 2 1 is any number such that

2,92 G 4 , 4 1 ( 1.

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

where the summation converges absolutely in L(r, s)(G). Let (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.

It is clear that t , E L(p1, qI)(G) B7 L(p2, qz)(G). As fi * 4, converges to fi for each i, and p2 < m, one can easily prove that t , converges to t in L(r, s)(G). As C f; * gi is convergent

in L(r, s)(G) and B(t ) = 0, then given E > 0 there exists no E N such that

00

i=l

rn

i=l

No.1 Avci & Giirkanli: MULTIPLIERS AND TENSOR PRODUCTS OF L(p , q) 113

whenever n > no. Also we can choose n1 > no such that

for all n > 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

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

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

1 1 1 +

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. Lemma 3.2 Let + 1 > 1, 2- + 2- - 1 = and s 2 1 is any number such that

< A. Given any cp E Cc(G) and f E L(pl,ql)(G) define Tv Pl P2 PI P2

l + L > L

by Tvf = f * cp. Then Tq E HomLl(G) (L(pl , ql)(G), L(p',, d ) ( G ) )

PlP' 41 42 ' 3' p = PlP;+PT-P;, - 41-4;

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

Proof Let f E L(p1, ql)(G) and cp E Cc(G) c L(p, q)(G) be given. By the inequalities

2 41-42 and $ + $ 2 1. If let s = qh, we obtain & + f 2 $. By Theorem 2.1 we have

1 + f = 1 + 4 > 1 and + - 1 = i, we write r = p: . Also because q 5 &, we have P2

I

4 41

cp * f E L(r , s)(G) = qP:, 4 X G ) and

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

Again it is easily shown that

ACTA MATHEMATICA SCIENTIA V01.27 Ser.B 114

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).

and let s 2 1 be such that & + 5 2 i, then the following statements are equivalent. Theorem 3.4 Let G be a locally compact abelian group. If &+& > 1 and 1 + 1 - 1 =

i) G satisfies the property Pi;;:;. ii) The kernel of B is K so that L ( p l , q l ) ( G ) @ ~ 1 ( ~ ) L(pz,qz)(G)

Proof

Pl P2

AE;;:;(G) (that is, isometrically isomorphic).

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 to F an operator T E HornLl(G)(L(pl, ql)(G), L(pk, q&)(G)) such that

m

( t , F ) = c ( % , T f i ) , (3.13) i = l

for all t E L(p1, q1)(G) L(p2, qz)(G) with the expansion

00

(3.14)

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

(3.15)

ca

(3.16) i=l

No. 1 Avci & Giirkanli: MULTIPLIERS AND TENSOR PRODUCTS OF L(p , q) 115

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 Cc(G) such that the fam- ily of operators Tpa defined in Lemma 3.2 converges to T in the ultraweak*-operator topology

In particular

Thus to prove (3.17), it suffices to show that

00

i=l

for each a. It is easy to see M M

i=l i=l

As Cc(G) c L(T’, s’)(G), using 13-15)] (3.20), we may write

(3.18)

(3.19)

(3.20)

(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). L l ( 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(p1 , 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(Pl1 d ( G ) 63 w 2 , qz)(G)IW* ”= KL Y

= (KerB)’-, (3.22)

and M C ( L ( p i , qi)(G) @ L ~ ( G ) L(pzl qz)(G))* 2 (KerB)’-, then ( t , F ) = 0 for all t E KerB and 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) @LI(G) L(pz1 qz)(G), there exist fi E

G l , q l ) ( G ) , si E L(pZ,qz)(G) such that 00 00

(3.23)

ACTA MATHEMATICA SCIENTIA Vo1.27 Ser.B 116

(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.

and s 2 1 is any number such that 5 + Corollary 3.6 Let G be a locally compact abelian group. If & + & > 1, + & - 1 =

2 and G satisfies the property Pf;;:;, then

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.

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