C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 100–114 (2017) D O I: 10.1501/C om mua1_ 0000000805 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
ON VECTOR-VALUED CLASSICAL AND VARIABLE EXPONENT AMALGAM SPACES
ISMAIL AYDIN
Abstract. Let 1 p; q; s 1 and 1 r(:) 1, where r(:) is a variable exponent. In this paper, we introduce …rstly vector-valued variable expo-nent amalgam spaces Lr(:)(R; E) ; `s . Secondly, we investigate some basic properties of Lr(:)(R; E) ; `s spaces. Finally, we recall vector-valued
classi-cal amalgam spaces (Lp(G; A) ; `q) ;and inquire the space of multipliers from
(Lp1(G; A) ; `q1)to Lp02(G; A ) ; `q20 :
1. Introduction
The amalgam of Lp and lq on the real line is the space (Lp; lq) (R) (or shortly
(Lp; lq) ) consisting of functions which are locally in Lp and have lq behavior at
in-…nity. Several special cases of amalgam spaces, such as L1; l2 , L2; l1 , L1; l1
and L1; l1 were studied by N. Wiener [30]. Comprehensive information about
amalgam spaces can be found in some papers, such as [16], [29], [15], [10] and [11]. Recently, there have been many interesting and important papers appeared in vari-able exponent amalgam spaces Lr(:); `s , such as Ayd¬n and Gürkanl¬ [3], Ayd¬n
[5], Gürkanli and Ayd¬n [14], Kokilashvili, Meskhi and Zaighum [17], Meskhi and Zaighum[23], Gürkanli [13], Kulak and Gürkanli [20]. Vector-valued classical amal-gam spaces (Lp(R; E) ; `q) on the real line were de…ned by Lakshmi and Ray [21] in
2009. They described and discussed some fundamental properties of these spaces, such as embeddings and separability. In their following paper [22], they investi-gated convolution product and obtained a similar result to Young’s convolution theorem on (Lp(R; E) ; `q). They also showed classical result on Fourier trans-form of convolution product for (Lp(R; E) ; `q). Vector-valued variable exponent Bochner-Lebesgue spaces Lr(:)(R; E) de…ned by Cheng and Xu [7] in 2013. They
proved dual space, the re‡exivity, uniformly convexity and uniformly smoothness of
Received by the editors: April 12, 2016; Accepted: December 06, 2016. 2010 Mathematics Subject Classi…cation. 43A15, 46E30, 43A22. Key words and phrases. Variable exponent, amalgam spaces, multipliers.
c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis t ic s .
Lr(:)(R; E). Furthermore, they gave some properties of the Banach valued
Bochner-Sobolev spaces with variable exponent. In this paper, we give some information about Lr(:)(R; E) ; `s , and obtain the generalization of some results in Sa¼g¬r [27]
and similar consequences in Avc¬and Gürkanli [1] and Öztop and Gürkanli [24]. Fi-nally, our original aim is to prove that the space of multipliers from (Lp1(G; A) ; `q1)
to Lp02(G; A ) ; `q02 is isometrically isomorphic to Aq1;q2
p1;p2(G; A) :
2. DEFINITION AND PRELIMINARY RESULTS
In this section, we give several de…nitions and theorems for vector-valued variable exponent Lebesgue spaces Lr(:)(R; E) :
De…nition 1. For a measurable function r : R ! [1; 1) (called a variable exponent on R), we put r = essinf x2R r(x), r+= esssup x2R r(x).
The variable exponent Lebesgue spaces Lr(:)(R) consist of all measurable functions f such that %r(:)( f ) < 1 for some > 0, equipped with the Luxemburg norm
kfkr(:)= inf > 0 : %r(:)( f ) 1 , where %r(:)(f ) = Z R jf(x)jr(x)dx.
If r+ < 1, then f 2 Lr(:)(R) i¤ %r(:)(f ) < 1. The space Lr(:)(R); k:kr(:) is a
Banach space. If r(x) = r is a constant function, then the norm k:kr(:) coincides
with the usual Lebesgue norm k:kr [18], [2], [4]. In this paper we assume that
r+< 1.
De…nition 2. We denote by Lr(:)loc (R) the space of ( equivalence classes of ) func-tions on R such that f restricted to any compact subset K of R belongs to Lr(:)(R):
Let 1 r(:); s < 1 and Jk = [k; k + 1), k 2 Z: The variable exponent amalgam
spaces Lr(:); `s are the normed spaces
Lr(:); `s = n f 2 Lr(:)loc (R) : kfk(Lr(:);`s) < 1 o ; where kfk(Lr(:);`s) = P k2Z f Jk sr(:) 1=s :
It is well known that Lr(:); `s is a Banach space and does not depend on the
particular choice of Jk, that is, Jk can be equal to [k; k + 1), [k; k + 1] or (k; k + 1):
[3], [5] and [14] to obtain some basic properties for Lr(:); `s spaces. It is well
known that Lr(:)(R) is not translation invariant. So, the convolution operator and
multipliers are useless in this space. By using Theorem 3.3 in [13] we also obtain Lr(:); `s is not translation invariant.
Let (E; k:kE) be a Banach space and E its dual space and ( ; ; ) be a measure
space.
De…nition 3. A function f : ! E is Bochner (or strongly) -measurable if there exists a sequence ffng of simple functions fn : ! E such that fn(x)
E
! f(x) as n ! 1 for almost all x 2 [9].
De…nition 4. A -measurable function f : ! E is called Bochner integrable if there exists a sequence of simple functions ffng such that
lim
n!1
Z
kfn f kEd = 0
for almost all x 2 [9].
Theorem 1. A -measurable function f : ! E is Bochner integrable if and only if
Z
kfkEd < 1 [9].
De…nition 5. A function F : ! E is called a vector measure, if for all sequences (An) of pairwise disjoint members of such that
1 S n=1 An 2 and F 1 S n=1 An = 1 P n=1
F (An) ; where the series converges in the norm topology of E:
Let F : ! E be a vector measure. The variation of F is the function kF k : ! [0; 1] de…ned by
kF k (A) = sup P1
B2 kF (B)kE
;
where the supremum is taken over all …nite disjoint partitions of A: If kF k ( ) < 1, then F is called a measure of bounded variation [7],[9].
De…nition 6. A Banach space E has the Radon-Nikodym property (RNP) with respect to ( ; ; ) if for each vector measure F : ! E of bounded variation, which is absolutely continuous with respect to , there exists a function g 2 L1( ; E)
such that
F (A) =R
A
gd for all A 2 [7],[9].
De…nition 7. The variable exponent Bochner- Lebesgue space Lr(:)(R; E) stands
for all (equivalence classes of ) E-valued Bochner integrable functions f on R such that
Lr(:)(R; E) =nf : kfkr(:);E < 1o; where
kfkr(:);E = inf > 0 : %r(:);E(
f ) 1 and %r(:);E(f ) = Z R kf(x)kr(:)E dx:
The following properties proved by Cheng and Xu [7]; (i) f 2 Lr(:)(R; E) , kf(x)kr(:)
E 2 L1(R) , kf(x)kE 2 Lr(:)(R)
(ii) Lr(:)(R; E) is a Banach space with respect to k:k r(:);E:
(iii) Lr(:)(R; E) is a generalization of the Lr(R; E) spaces. (iv) If E = R or C, then Lr(:)(R; E) = Lr(:)(R) :
(v) If E is re‡exive and 1 < r r+< 1, then Lr(:)(R; E) is re‡exive. Theorem 2. If E has the Radon-Nikodym Property (RNP), then the mapping g 7! 'g; r(:)1 + 1 q(:)= 1, Lq(:)(R; E ) ! Lr(:)(R; E) which is de…ned by < 'g; f >= Z R < g; f > dx
for any f 2 Lr(:)(R; E) is a linear isomorphism and
kgkq(:);E 'g (Lr(:)(R;E)) 2 kgkq(:);E :
Hence, the dual space Lr(:)(R; E) is isometrically isomorphic to Lq(:)(R; E ) ;
where E has RNP. In addition, for f 2 Lr(:)(R; E) and g 2 Lq(:)(R; E ) (g
de…nes a continuous linear functional), the dual pair < f (:); g(:) >2 L1(R) and
Hölder inequality implies Z R j< f(:); g(:) >j dx Z R kfkEkgkE dx C kfkr(:);Ekgkq(:);E for some C > 0 [7].
3. VECTOR-VALUED VARIABLE EXPONENT AMALGAM SPACES
In this section, we de…ne vector-valued variable exponent amalgam spaces Lr(:)(R; E) ; `s : We also discuss some basic and signi…cant properties of
Lr(:)(R; E) ; `s :
De…nition 8. Let 1 r(:) < 1, 1 s 1 and Jk = [k; k + 1), k 2 Z: The
vector-valued variable exponent amalgam spaces Lr(:)(R; E) ; `s are the normed space Lr(:)(R; E) ; `s = n f 2 Lr(:)loc (R; E) : kfk(Lr(:)(R;E);`s) < 1 o ; where kfk(Lr(:)(R;E);`s) = P k2Z f Jk s r(:);E 1=s ; 1 s < 1 and kfk(Lr(:)(R;E);`1) = sup k f Jk r(:);E ,s = 1:
It can be proved that Lr(:)(R; E) ; `s is a Banach space with respect to the
norm k:k(Lr(:)(R;E);`s) [21]: Moreover, Lr(:)(R; E) ; `s has some inclusions and
embeddings similar to [3].
The proof of the following Theorem is proved by using techniques in Theorem 2.6 in [11], [p. 32, 29] and [p.359,19].
Theorem 3. Let E has RNP and 1 < r r+< 1 and 1 < s < 1. Then the
dual space of Lr(:)(R; E) ; `s is isometrically isomorphic to Lq(:)(R; E ) ; `t for 1 r(:)+ 1 q(:) = 1 and 1 s+ 1 t = 1.
Proof. Let fAkgk2Zbe a family of Banach spaces. We de…ne
`s(Ak) = fx = (xk) : xk2 Ak; kxk < 1g where kxk = P k2Zkx kksAk 1 s
: It can be seen that `s(A
k) is a Banach space under
the norm k:k : It is also well known that the dual of `s(A
k) is `t(Ak). Moreover,
Lr(:)(R; E) ; `s is particular case of `s(A
k). Indeed, if we take Ak= Lr(:)(Jk; E)
and Jk = [k; k + 1), then the map f 7! (fk), fk = f Jk is an isometric isomorphism
from Lr(:)(R; E) ; `s to `s Lr(:)(J
k; E) . Hence, we have Lr(:)(R; E) ; `s =
Lq(:)(R; E ) ; `t by Theorem 2.
Corollary 1. Let 1 < r r+ < 1 and 1 < s < 1: If E is re‡exive, then Lr(:)(R; E) ; `s is re‡exive.
Theorem 4. (Generalized Hölder Inequality) Let E has RNP and m+< 1, 1 s 1. If 1 r(:)+ 1 q(:) = 1 m(:) and 1 s+ 1 t = 1
n, then there exists a C > 0 such that
k< f(:); g(:) >k(Lm(:)(R);`n) C kfk(Lr(:)(R;E);`s) kgk(Lq(:)(R;E );`t)
and < f (:); g(:) >2 Lm(:)(R) ; `n for f 2 Lr(:)(R; E) ; `s , g 2 Lq(:)(R; E ) ; `t :
Proof. Let ef (x) = kf(x)kE and eg(x) = kg(x)kE be given for any x 2 R: If f 2
Lr(:)(R; E) ; `s and g 2 Lq(:)(R; E ) ; `t ; then we have ef 2 Lr(:); `s , eg 2 Lq(:); `t and fe
(Lr(:);`s) = kfk(Lr(:)(R;E);`s), kegk(Lq(:);`t) = kgk(Lq(:)(R;E );`t).
Therefore, by using Hölder inequality for Lm(:) [18], we can write the following inequality
k< f(:); g(:) >km(:);Jk kkf(:)kEkg(:)kE km(:);Jk
C fe
r(:);Jkkegkq(:);Jk
= C f Jk r(:);E g Jk q(:);E :
By Corollary 2.4 in [3] and Jensen’s inequality for `sspaces, we obtain
k< f(:); g(:) >k(Lm(:)(R);`n) C kfk(Lr(:)(R;E);`s) kgk(Lq(:)(R;E );`t) :
This completes the proof.
De…nition 9. We de…ne c0(Z) l1 to be the linear space of (ak)k2Z such that
limkak = 0, that is, given " > 0 there exists a compact subset K of R such that
jakj < " for all k =2 K:
The vector-valued type variable exponent amalgam spaces Lr(:)(R; E) ; c0 are
the normed spaces
Lr(:)(R; E) ; c0 = f 2 Lr(:)(R; E) ; `1 : n f Jk r(:);Eo k2Z2 c0 ; where kfk(Lr(:)(R;E);`1) = sup k f Jk r(:);E for f 2 Lr(:)(R; E) ; c0 [29].
Proposition 1. Let E has RNP and m+< 1, 1 s 1: If f 2 Lr(:)(R; E) ; c 0
and g 2 Lq(:)(R; E ) ; c
0 ; then there exists a C > 0 such that
k< f(:); g(:) >k(Lm(:)(R);`1) C kfk(Lr(:)(R;E);`1) kgk(Lq(:)(R;E );`1) and < f (:); g(:) >2 Lm(:)(R) ; c 0 for r(:)1 + 1 q(:) = 1 m(:):
Proof. If f 2 Lr(:)(R; E) ; c
0 and g 2 Lq(:)(R; E ) ; c0 , then by Theorem 4 we
can write < f (:); g(:) >2 Lm(:)(R) ; `1 and
k< f(:); g(:) >km(:);Jk C f Jk r(:);E g Jk q(:);E ;
where C does not depend on for any k 2 Z. This implies that lim
k k< f(:); g(:) >km(:);Jk limk C f Jk r(:);Elimk g Jk q(:);E = 0
and < f (:); g(:) >2 Lm(:)(R) ; c0 . If we use the de…nition of the norm k:k(Lr(:)(R;E);`1),
then we get k< f(:); g(:) >k(Lr(:);`1) = sup k f Jk r(:) C sup k f Jk r(:);E g Jk q(:);E C sup k f Jk r(:);Esup k g Jk r(:);E = C kfk(Lr(:)(R;E);`1) kgk(Lq(:)(R;E );`1)
De…nition 10. Lr(:)c (R; E) denotes the functions f in Lr(:)(R; E) such that suppf
R is compact,that is, Lr(:)c (R; E) = n f 2 Lr(:)(R; E) : suppf compact o : Let K R be given. The cardinality of the set
S(K) = fJk : Jk\ K 6= ?g
is denoted by jS(K)j, where fJkgk2Z is a collection of intervals:
Proposition 2. If g belongs to Lr(:)c (R; E), then
(i) kgk(Lr(:)(R;E);`s) jS(K)j 1
skgk
r(:);E for 1 s < 1;
(ii) kgk(Lr(:)(R;E);`1) jS(K)j kgkr(:);E for s = 1,
(iii) Lr(:)c (R; E) Lr(:)(R; E) ; `s for 1 s 1;
where K is the compact support of g. Proof. (i) Since K is compact, then K
jS(K)jS i=1 Jki and kgk(Lr(:)(R;E);`s) = P k2Z g Jk sr(:);E 1=s = P Jki2S(K) g Jk sr(:);E !1=s jS(K)j1skgk r(:);E
for 1 s < 1, where the number of Jki is …nite.
(ii)Let s = 1.Then
kgk(Lr(:)(R;E);`s) = sup k2Z g Jk r(:);E = sup i=1;2;::jS(K)j g J ki r(:);E jS(K)j kgkr(:);E:
Theorem 5. (i) Lr(:)c (R; E) is subspace of Lr(:)(R; E) ; c0 for 1 s < 1:
(ii) C0(R; E) is subspace of Lr(:)(R; E) ; c0 :
Proof. (i) Firstly, we show that Lr(:)c (R; E) Lr(:)(R; E) ; c0 . Let f 2 Lr(:)c (R; E)
be given. Since f has compact support, then f Jk r(:);E is zero for all, but …nitely many Jk: By de…nition of c0; we get
n
f Jk r(:);E
o
k2Z 2 c0. Hence,
Lr(:)c (R; E) Lr(:)(R; E) ; c0 :
(ii)If f 2 C0(R; E) and given 0 < " < 1; then there exists a compact set K R
such that kf(x)kE < " for all x =2 K. Since K is compact, then K [ni=1Jki
(n is …nite) and f (:) Jk
r(:);E < " for all k 6= ki, i = 1; 2; ::n: Indeed, by using
%r(:);E(f ) ! 0 , kf(:)kr(:);E ! 0 (r+< 1) ; and jJ
kj = 1 (measure of Jk) it is written that %r(:);E(f ) = Z Jk kf(x)kr(:)E dx "r jJkj ! 0: Therefore, we obtain f 2 Lr(:)(R; E) ; c
0 due to de…nition of norm of
Lr(:)(R; E) ; c 0 .
4. VECTOR-VALUED CLASSICAL AMALGAM SPACES In this section, we consider that G is a locally compact Abelian group, and A is a commutative Banach algebra with Haar measure . By the Structure Theorem, G = Ra G1, where a is a nonnegative integer and G1is a locally compact abelian
group which contains an open compact subgroup H. Let I = [0; 1)a H and J = Za T; where T is a transversal of H in G
1, i.e. G1 =
S
t2T
(t + H) is a coset decomposition of G1. For 2 J we de…ne I = + I, and therefore G is
equal to the disjoint union of relatively compact sets I . We normalize so that (I) = (I ) = 1 for all [11], [29].
De…nition 11. Let 1 p; q < 1: The vector-valued classical amalgam spaces (Lp(G; A) ; `q) are the normed space
(Lp(G; A) ; `q) =nf 2 Lploc(G; A) : kfk(Lp(G;A);`q)< 1 o ; where kfk(Lp(G;A);`q)= P 2J f I qp;A 1=q ; 1 p; q < 1: Now we give Young’s inequality for vector-valued amalgam spaces.
Theorem 6. ([22])Let 1 p1; q1; p2; q2 < 1: If f 2 (Lp1(G; A) ; `q1) and g 2
(Lp2(G; A) ; `q2), then f g 2 (Lr1(G; A) ; `r2), where 1 p1 + 1 p2 1, 1 q1 + 1 q2 1, 1 r1 = 1 p1 + 1 p2 1 and 1 r2 = 1 q1 + 1
q2 1: Moreover, there exists a C > 0 such that
kf gk(Lr1(G;A);`r2) C kfk(Lp1(G;A);`q1)kgk(Lp2(G;A);`q2):
De…nition 12. Let A be a Banach algebra. A Banach space B is said to be a Banach A module if there exists a bilinear operation : A B ! B such that
(i) (f g) h = f (g h) for all f; g 2 A, h 2 B:
(ii) For some constant C 1; kf hkB C kfkAkhkB for all f 2 A; h 2 B: By Theorem 6, we have the following inequality
kf gk(Lp(G;A);`q) C kfk(Lp(G;A);`q)kgk(L1(G;A);`1)= C kfk(Lp(G;A);`q)kgkL1(G;A)
for all f 2 (Lp(G; A) ; `q) and g 2 L1(G; A), where C 1, i.e. the amalgam space (Lp(G; A) ; `q) is a Banach L1(G; A) module with respect to convolution. More-over, it is easy to see that the amalgam space Lp(G; A) ; `1 is a Banach algebra under convolution p 1, if we de…ne the norm jkfkj(Lp(G;A);`1)= C kfk(Lp(G;A);`1)
for Lp(G; A) ; `1 . Recall that Lp(G; A) ; `1 L1(G; A).
De…nition 13. Let V and W be two Banach modules over a Banach algebra A. Then a multiplier from V into W is a bounded linear operator T from V into W , which commutes with module multiplication, i.e. T (av) = aT (v) for a 2 A and v 2 V . We denote by HomA(V; W ) the space of all multipliers from V into 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 [6], [28]. If K is the closed linear subspace of V W , which is spanned by all elements of the form av w v aw, a 2 A, v 2 V; w 2 W , then the A module tensor product V AW is de…ned to be
the quotient Banach space (V W ) =K: Every element t of (V W ) =K can be written t = P1 i=1 vi wi; vi2 V; wi2 W; where P1 i=1kvik kwik < 1. Moreover, (V
W ) =K is a normed space according to ktk = inf P1
i=1kv
where the in…mum is taken over all possible representations for t [26]. It is known that
HomA(V; W ) = (V AW ) ;
where W is dual of W (Corollary 2.13, [25]). The linear functional T on HomA(V; W ), which corresponds to t 2 V AW gets the value
< t; T >= P1
i=1
< wi; T vi > :
Moreover, it is well known that the ultraweak -operator topology on HomA(V; W )
corresponds to the weak -topology on (V AW ) [26].
5. THE MULTIPLIERS OF THE SPACE Aq1;q2
p1;p2(G; A)
By Theorem 6, a linear operator b : (Lp1(G; A) ; `q1) (Lp2(G; A) ; `q2) !
(Lr1(G; A) ; `r2) can be de…ned by
b (f; g) = ef g; f 2 (Lp1(G; A) ; `q1) ; g 2 (Lp2(G; A) ; `q2) ;
where ef (x) = f ( x) and fe
(Lp1(G;A);`q1)= kfk(Lp1(G;A);`q1): It is easy to see that
kbk C: Furthermore, there exists a bounded linear operator B from (Lp1(G; A) ; `q1)
(Lp2(G; A) ; `q2) into (Lr1(G; A) ; `r2) such that B (f g) = b (f; g), where
f 2 (Lp1(G; A) ; `q1) ; g 2 (Lp2(G; A) ; `q2) and kBk C by Theorem 6 in [6].
De…nition 14. Aq1;q2
p1;p2(G; A) denotes the range of B with the quotient norm.
Hence, we write Aq1;q2 p1;p2(G; A) = h = 1 P i=1 e fi gi : 1 P i=1kf ik(Lp1(G;A);`q1)kgik(Lp2(G;A);`q2)< 1 for fi2 (Lp1(G; A) ; `q1) ; gi2 (Lp2(G; A) ; `q2) and kjhjk = inf P1 i=1kf ik(Lp1(G;A);`q1)kgik(Lp2(G;A);`q2): h = 1 P i=1 e fi gi : It is clear that Aq1;q2 p1;p2(G; A) (L r1(G; A) ; `r2) and khk (Lr1(G;A);`r2) C kjhjk :
Moreover, by using the technique given in Theorem 2.4 by [12], Aq1;q2
p1;p2(G; A) can
be showed a Banach space with respect to kj:jk.
Let K be the closed linear subspace of (Lp1(G; A) ; `q1) (Lp2(G; A) ; `q2),
which is spanned by all elements of the form (' f ) g f ('e g), where f 2 (Lp1(G; A) ; `q1) ; g 2 (Lp2(G; A) ; `q2) and ' 2 Lp1(G; A). Then the L1(G; A)–
module tensor product (Lp1(G; A) ; `q1)
L1(G;A)(Lp2(G; A) ; `q2) is de…ned to be
the quotient Banach space ((Lp1(G; A) ; `q1) (Lp2(G; A) ; `q2)) =K: We de…ne the
norm khk = inf P1 i=1kf ik(Lp1(G;A);`q1)kgik(Lp2(G;A);`q2): h = 1 P i=1 fi gi :
for h 2 (Lp1(G; A) ; `q1)
L1(G;A)(Lp2(G; A) ; `q2) Also, it is well known that this
space is a Banach space [26].
In the following Lemma, we use numbers p and q given in Lemma 3.2 in [1]. Lemma 1. Let 1 p1; q1; p2; q2< 1, p11 +p12 1, q11 +q12 1, r11 = p11 +p12 1 and r1 2 = 1 q1 + 1 q2 1; p = p1p02 p1p02+p1 p02 and q = q1q02 q1q02+q1 q02, where 1 p2 + 1 p0 2 = 1 and q1 2 + 1 q0 2 = 1: If we de…ne T'f = f ' for f 2 (L p1(G; A) ; `q1) and ' 2
CC(G; A), then we have T' 2 HomL1(G;A) (Lp1(G; A) ; `q1) ; Lp 0
2(G; A ) ; `q02
and the inequality
kT'k C k'k(Lp(G;A);`q)
for some C > 0.
Proof. Let f 2 (Lp1(G; A) ; `q1) and ' 2 C
C(G; A) (Lp(G; A) ; `q) be given. Due to p = p1p02 p1p02+p1 p02 and q = q1q20 q1q02+q1 q02, 1 p1 + 1 p = 1 + 1 p0 2 > 1 and 1 r1 = 1 p1 + 1 p 1, 1 q1 + 1 q = 1 + 1 q0 2 > 1 and 1 r2 = 1 q1 + 1
q 1, we have r1= p02 and r2= q20: Therefore,
by Theorem 6 we obtain f ' 2 (Lr1(G; A) ; `r2) = Lp02(G; A) ; `q20 and
kT'f k Lp02(G;A);`q02 C kfk(Lp1(G;A);`q1)k'k(Lp(G;A);`q);
i.e. T'f is continuous. Also, we can write the inequality
kT'k C k'k(Lp(G;A);`q)
for some C > 0.
De…nition 15. A locally compact Abelian group G is said to satisfy the property P(q1;q2)
(p1;p2) if every element of HomL1(G;A) (L
p1(G; A) ; `q1) ; Lp02(G; A ) ; `q02 can
be approximated in the ultraweak -operator topology by operators T'; ' 2 CC(G; A) :
Theorem 7. Let G be a locally compact Abelian group. If p1
1+ 1 p2 1, 1 q1+ 1 q2 1, 1 r1 = 1 p1+ 1 p2 1 and 1 r2 = 1 q1+ 1
q2 1, then the following statements are equivalent:
(i) G satis…es the property P(q1;q2)
(p1;p2):
(ii) The kernel of B is K such that (Lp1(G; A) ; `q1)
L1(G;A)(Lp2(G; A) ; `q2) = Aqp1;q2
1;p2(G; A) :
Proof. Since B ((' f ) g f ('e g)) = (' f ) g fe (e' g) = 0, then K KerB: Assume that G satis…es the property P(q1;q2)
(p1;p2): To display KerB K
it is enough to prove that K? (KerB)? due to the fact that K is the closed
linear subspace of (Lp1(G; A) ; `q1) (Lp2(G; A) ; `q2) : It is well known that
[8]. Also, we write
K? = (Lp1(G; A) ; `q1)
L1(G;A)(Lp2(G; A) ; `q2) :
Moreover, by using Corollary 2.13 in [25], we obtain K?= HomL1(G;A) (Lp1(G; A) ; `q1) ; Lp
0
2(G; A ) ; `q02 :
Thus, there is a multiplier T 2 HomL1(G;A) (Lp1(G; A) ; `q1) ; Lp 0
2(G; A ) ; `q02
which corresponds to F 2 K? such that
< t; F >= P1 i=1 < gi; T fi>; where t 2 KerB, t = P1 i=1 fi gi, 1 P i=1kf
ik(Lp1(G;A);`q1)kgik(Lp2(G;A);`q2) < 1 and
fi2 (Lp1(G; A) ; `q1) ; gi 2 (Lp2(G; A) ; `q2). Also, due to t 2 KerB we get
B P1 i=1 fi gi = 1 P i=1 e fi gi= 0: (5.1)
Now, we show that
< t; F >= P1
i=1
< gi; T fi>= 0. (5.2)
Furthermore, since G satis…es the property P(q1;q2)
(p1;p2), then there exists a net
'j : j 2 I CC(G; A) such that the operators T'j de…ned in Lemma 1
con-verges to T in the ultraweak -operator topology, that is, lim j2I 1 P i=1 < gi; T'jfi>= limj 2I 1 P i=1 < gi; fi 'j >= 1 P i=1 < gi; T fi> :
So, to obtain (5.2), it is enough to show that
1
P
i=1
< gi; fi 'j >= 0
for all j 2 I. It can be seen easily
1 P i=1 < gi; fi 'j>= 1 P i=1 < efi gi; 'j> : (5.3)
If we use the equalities (5.1), (5.3) and Hölder inequality for amalgam spaces, then we have 1 P i=1 < gi; fi 'j> 1 P i=1 e fi gi (Lr1(G;A);`r2) 'j Lrp1(G;A);`rp2 = 0; where P1 i=1 e fi gi2 (Lr1(G; A) ; `r2) and 'j2 CC(G; A) Lr p 1(G; A) ; `rp2 :
By de…nition of Aq1;q2
p1;p2(G; A) and the First Isomorphism Theorem, we get
(Lp1(G; A) ; `q1) (Lp2(G; A) ; `q2) =K = Aq1;q2 p1;p2(G; A) and (Lp1(G; A) ; `q1) L1(G;A)(Lp2(G; A) ; `q2) = (Lp1(G; A) ; `q1) (Lp2(G; A) ; `q2) =K: This proves (Lp1(G; A) ; `q1) L1(G;A)(Lp2(G; A) ; `q2) = Aqp1;q2 1;p2(G; A) :
Suppose conversely that KerB = K: If we show that the operators of the form T'
for ' 2 CC(G; A) are dense in HomL1(G;A) (Lp1(G; A) ; `q1) ; Lp 0
2(G; A ) ; `q02
in the ultraweak -operator topology, then we …nish the proof. Hence, it is su¢ cient to prove that the corresponding functionals are dense in
(Lp1(G; A) ; `q1)
L1(G;A)(Lp2(G; A) ; `q2)
in the weak topology by Theorem 1.4 in [26]. Let M be set of the linear functionals corresponding to the operators T'. Let t 2 KerB and F 2 M: Since
(Lp1(G; A) ; `q1)
L1(G;A)(Lp2(G; A) ; `q2) = K?;
then F 2 M K? and < t; F >= 0: Hence we …nd KerB M?. Conversely, let t 2 M?: Since M? (Lp1(G; A) ; `q1)
L1(G;A)(Lp2(G; A) ; `q2), then there exist
fi2 (Lp1(G; A) ; `q1) ; gi2 (Lp2(G; A) ; `q2) such that t = P1 i=1 fi gi; 1 P i=1kf ik(Lp1(G;A);`q1)kgik(Lp2(G;A);`q2)< 1
and < t; F >= 0 for all F 2 M. Also, there is an operator T'2 HomL1(G;A) (Lp1(G; A) ; `q1) ; Lp
0
2(G; A ) ; `q20
corresponding to F such that < t; F >=< t; T'>= 1 P i=1 < gi; T'fi>= 1 P i=1 < efi gi; 'j > =< P1 i=1 e fi gi; 'j>= 0
by using (5.2). Therefore, we obtain B(t) =P1
i=1
e
fi gi= 0
and M? KerB: Consequently, KerB = M?: This completes the theorem.
Corollary 2. Let G be a locally compact Abelian group. If p1
1+ 1 p2 1, 1 q1+ 1 q2 1, 1 r1 = 1 p1 + 1 p2 1 and 1 r2 = 1 q1 + 1
q2 1 and G satis…es the property P
(q1;q2) (p1;p2), then HomL1(G;A) (Lp1(G; A) ; `q1) ; Lp 0 2(G; A ) ; `q20 = Aq1;q2 p1;p2(G; A) :
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Current address, Ismail AYDIN: Sinop University, Faculty of Sciences and Letters Department of Mathematics, Sinop, Turkey.
