Article Electronically published on February 25, 2017
Com mun.Fac.Sci.Univ.Ank.Series A1 Volum e 66, Numb er 2, Pages 100–114 (2017) DOI: 10.1501/Com mua1_ 0000000805 ISSN 1303–5991 http://com munications.science.ankara.edu.tr/index.php?series=A1
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 exponent amalgam spaces Lr(:) (R; E) ; `s . Secondly, we investigate some basic properties of Lr(:) (R; E) ; `s spaces. Finally, we recall vector-valued classical amalgam spaces (Lp (G; A) ; `q ) ; and inquire the space of multipliers from (Lp1 (G; A) ; `q1 ) to
0
0
Lp2 (G; A ) ; `q2 :
1. Introduction The amalgam of Lp and lq on the real line is the space (Lp ; lq ) (R) (or shortly (L ; 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 variable 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 amalgam 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 investigated convolution product and obtained a similar result to Young’s convolution theorem on (Lp (R; E) ; `q ). They also showed classical result on Fourier transform 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 p
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 rsité d ’A n ka ra . S é rie s A 1 . M a th e m a tic s a n d S ta tistic s.
100
ON VECTOR-VALUED CLASSICAL AND VARIABLE EXPONENT
101
Lr(:) (R; E). Furthermore, they gave some properties of the Banach valued BochnerSobolev spaces with variable exponent. In this paper, we give some information about Lr(:) (R; E) ; `s , and obtain the generalization of some results in Sa¼ g¬r [27] and similar consequences in Avc¬and Gürkanli [1] and Öztop and Gürkanli [24]. Finally, our original aim is to prove that the space of multipliers from (Lp1 (G; A) ; `q1 ) 0
0
to Lp2 (G; A ) ; `q2
2 is isometrically isomorphic to Apq11;q ;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 = essinfr(x), r+ = esssupr(x). x2R
x2R r(:)
The variable exponent Lebesgue spaces L (R) consist of all measurable functions f such that %r(:) ( f ) < 1 for some > 0, equipped with the Luxemburg norm kf kr(:) = inf where %r(:) (f ) =
f > 0 : %r(:) ( ) Z
jf (x)j
r(x)
1 ,
dx.
R
If r
+
r(:)
< 1, then f 2 L
(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. r(:)
De…nition 2. We denote by Lloc (R) the space of ( equivalence classes of ) functions 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 n o r(:) Lr(:) ; `s = f 2 Lloc (R) : kf k(Lr(:) ;`s ) < 1 ; where
kf k(Lr(:) ;`s ) =
P
k2Z
f
Jk
s r(:)
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): Thus, we have same spaces Lr(:) ; `s [15]. Furthermore, it can be seen in references
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[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 E there exists a sequence ffn g of simple functions fn : ! E such that fn (x) ! 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 ffn g such that Z lim kfn f kE d = 0 n!1
for almost all x 2
[9].
Theorem 1. A -measurable function f : Z if kf kE d < 1 [9].
! E is Bochner integrable if and only
De…nition 5. A function F :
! E is called a vector measure, if for all sequences 1 1 S S (An ) of pairwise disjoint members of such that An 2 and F An = 1 P
n=1
n=1
F (An ) ; where the series converges in the norm topology of E:
n=1
Let F : ! E be a vector measure. The variation of F is the function kF k : ! [0; 1] de…ned by 1 P kF (B)kE ; kF k (A) = sup B2
where the supremum is taken over all …nite disjoint partitions 1, then F is called a measure of bounded variation [7],[9].
of A: If kF k ( )
0 : %r(:);E ( )
kf kr(:);E = inf and %r(:);E (f ) =
Z
1
r(:)
kf (x)kE dx:
R
The following properties proved by Cheng and Xu [7]; r(:) (i) f 2 Lr(:) (R; E) , kf (x)kE 2 L1 (R) , kf (x)kE 2 Lr(:) (R) (ii) Lr(:) (R; E) is a Banach space with respect to k:kr(:);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 1 1 g 7! 'g ; r(:) + q(:) = 1, Lq(:) (R; E ) ! Lr(:) (R; E) which is de…ned by < 'g ; f >=
Z
< g; f > dx
R
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
kf kE kgkE dx
R
C kf kr(:);E kgkq(:);E for some C > 0 [7].
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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 n o r(:) Lr(:) (R; E) ; `s = f 2 Lloc (R; E) : kf k(Lr(:) (R;E);`s ) < 1 ; where
kf k(Lr(:) (R;E);`s ) = and
P
f
k2Z
Jk
kf k(Lr(:) (R;E);`1 ) = sup f k
s r(:);E
1=s
Jk r(:);E
;1
s 0 such that k< f (:); g(:) >k(Lm(:) (R);`n )
C kf k(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 fe(x) = kf (x)kE and ge(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 fe 2 Lr(:) ; `s , ge 2 = kf k(Lr(:) (R;E);`s ) , ke g k(Lq(:) ;`t ) = kgk(Lq(:) (R;E );`t ) . Lq(:) ; `t and fe r(:) (L ;`s ) Therefore, by using Hölder inequality for Lm(:) [18], we can write the following inequality k< f (:); g(:) >km(:);Jk
kkf (:)kE kg(:)kE km(:);Jk
=
C fe C f
r(:);Jk
ke g kq(:);Jk
Jk r(:);E
g
Jk q(:);E
:
By Corollary 2.4 in [3] and Jensen’s inequality for `s spaces, we obtain k< f (:); g(:) >k(Lm(:) (R);`n )
C kf k(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 limk ak = 0, that is, given " > 0 there exists a compact subset K of R such that jak j < " for all k 2 = K: The vector-valued type variable exponent amalgam spaces Lr(:) (R; E) ; c0 are the normed spaces n o Lr(:) (R; E) ; c0 = f 2 Lr(:) (R; E) ; `1 : f Jk r(:);E 2 c0 ; k2Z
where
kf k(Lr(:) (R;E);`1 ) = sup f k
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) ; c0 and g 2 Lq(:) (R; E ) ; c0 ; then there exists a C > 0 such that k< f (:); g(:) >k(Lm(:) (R);`1 )
C kf k(Lr(:) (R;E);`1 ) kgk(Lq(:) (R;E
and < f (:); g(:) >2 Lm(:) (R) ; c0 for
1 r(:)
+
1 q(:)
=
1 m(:) :
);`1 )
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ISM AIL AYDIN
Proof. If f 2 Lr(:) (R; E) ; c0 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
g
Jk r(:);E
Jk q(:);E
;
where C does not depend on for any k 2 Z. This implies that lim k< f (:); g(:) >km(:);Jk
lim C f
k
k
Jk r(:);E
lim g
Jk q(:);E
k
=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 f
Jk r(:)
k
C sup f k
C sup f k
Jk r(:);E Jk r(:);E
g
Jk q(:);E
sup g k
Jk r(:);E
= C kf k(Lr(:) (R;E);`1 ) kgk(Lq(:) (R;E r(:)
);`1 )
De…nition 10. Lc (R; E) denotes the functions f in Lr(:) (R; E) such that suppf R is compact,that is, n o r(:) Lr(:) (R; E) = f 2 L (R; E) : suppf compact : c Let K
R be given. The cardinality of the set
S(K) = fJk : Jk \ K 6= ?g
is denoted by jS(K)j, where fJk gk2Z is a collection of intervals: r(:)
Proposition 2. If g belongs to Lc (R; E), then 1 (i) kgk(Lr(:) (R;E);`s ) jS(K)j s kgkr(:);E for 1 s < 1; (ii) kgk(Lr(:) (R;E);`1 ) jS(K)j kgkr(:);E for s = 1, r(:) Lr(:) (R; E) ; `s for 1 s 1; (iii) Lc (R; E) where K is the compact support of g.
Proof. (i) Since K is compact, then K
jS(K)j S
Jki and
i=1
kgk(Lr(:) (R;E);`s )
=
P
g
k2Z
=
P
Jk
Jki 2S(K) 1
s r(:);E
g
Jk
jS(K)j s kgkr(:);E
1=s
s r(:);E
!1=s
ON VECTOR-VALUED CLASSICAL AND VARIABLE EXPONENT
107
for 1 s < 1, where the number of Jki is …nite. (ii) Let s = 1.Then kgk(Lr(:) (R;E);`s )
=
sup g k2Z
=
Jk r(:);E
sup
g
i=1;2;::jS(K)j
Jki
r(:);E
jS(K)j kgkr(:);E : r(:)
Theorem 5. (i) Lc (R; E) is subspace of Lr(:) (R; E) ; c0 for 1 (ii) C0 (R; E) is subspace of Lr(:) (R; E) ; c0 :
s < 1:
r(:)
r(:)
Proof. (i) Firstly, we show that Lc (R; E) Lr(:) (R; E) ; c0 . Let f 2 Lc (R; E) be given. Since f has compact support, then f Jk r(:);E is zero for all, but n o …nitely many Jk : By de…nition of c0 ; we get f Jk r(:);E 2 c0 . Hence, k2Z
r(:)
(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=1 Jki (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 jJk j = 1 (measure of Jk ) it is written that Z r(:) %r(:);E (f ) = kf (x)kE dx Lc
Jk
"r jJk j ! 0: Therefore, we obtain f 2 Lr(:) (R; E) ; c0 due to de…nition of norm of Lr(:) (R; E) ; c0 . 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 G1 is a locally compact abelian a group which contains an open compact subgroup H. Let I = [0; 1) H and S J = Za T; where T is a transversal of H in G1 , i.e. G1 = (t + H) is a t2T
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].
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De…nition 11. Let 1 p; q < 1: The vector-valued classical amalgam spaces (Lp (G; A) ; `q ) are the normed space n o (Lp (G; A) ; `q ) = f 2 Lploc (G; A) : kf k(Lp (G;A);`q ) < 1 ; where
kf k(Lp (G;A);`q ) =
P
f
2J
I
q p;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 p2 q2 (L (G; A) ; ` ), then f g 2 (Lr1 (G; A) ; `r2 ), where p11 + p12 1, q11 + q12 1, 1 1 1 1 1 1 = + 1 and = + 1: Moreover, there exists a C > 0 such that r1 p1 p2 r2 q1 q2 kf
gk(Lr1 (G;A);`r2 )
C kf k(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 kf kA khkB for all f 2 A; h 2 B: By Theorem 6, we have the following inequality kf
gk(Lp (G;A);`q )
C kf k(Lp (G;A);`q ) kgk(L1 (G;A);`1 ) = C kf k(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. Moreover, 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 jkf kj(Lp (G;A);`1 ) = C kf k(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 A W is de…ned to be the quotient Banach space (V W ) =K: Every element t of (V W ) =K can be written 1 P t= vi wi ; vi 2 V; wi 2 W; i=1
where
1 P
i=1
kvi k kwi k < 1. Moreover, (V ktk = inf
1 P
i=1
W ) =K is a normed space according to
kvi k kwi k < 1 ;
ON VECTOR-VALUED CLASSICAL AND VARIABLE EXPONENT
109
where the in…mum is taken over all possible representations for t [26]. It is known that HomA (V; W ) = (V A W ) ; where W is dual of W (Corollary 2.13, [25]). The linear functional T on HomA (V; W ), which corresponds to t 2 V A W gets the value < t; T >=
1 P
< wi ; T vi > :
i=1
Moreover, it is well known that the ultraweak -operator topology on HomA (V; W ) corresponds to the weak -topology on (V A W ) [26]. 2 5. THE MULTIPLIERS OF THE SPACE Apq11;q ;p2 (G; A)
By Theorem 6, a linear operator b : (Lp1 (G; A) ; `q1 ) (L (G; A) ; `r2 ) can be de…ned by r1
(Lp2 (G; A) ; `q2 ) !
b (f; g) = fe g; f 2 (Lp1 (G; A) ; `q1 ) ; g 2 (Lp2 (G; A) ; `q2 ) ;
where fe(x) = f ( x) and fe
(Lp1 (G;A);`q1 )
= kf k(Lp1 (G;A);`q1 ) : It is easy to see that
C: Furthermore, there exists a bounded linear operator B from (Lp1 (G; A) ; `q1 ) (L (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].
kbk
p2
2 De…nition 14. Aqp11;q ;p2 (G; A) denotes the range of B with the quotient norm. Hence, we write 2 Aqp11;q ;p2 (G; A) =
p1
for fi 2 (L
h=
1 1 P P fei gi : kfi k(Lp1 (G;A);`q1 ) kgi k(Lp2 (G;A);`q2 ) < 1
i=1 q1
(G; A) ; ` ) ; gi 2 (L
kjhjk = inf
1 P
i=1
2 Aqp11;q ;p2
i=1
p2
(G; A) ; `q2 ) and
kfi k(Lp1 (G;A);`q1 ) kgi k(Lp2 (G;A);`q2 ) : h = r1
r2
1 P fei gi :
i=1
(G; A) (L (G; A) ; ` ) and khk(Lr1 (G;A);`r2 ) C kjhjk : It is clear that 2 Moreover, by using the technique given in Theorem 2.4 by [12], Aqp11;q (G; A) can ;p2 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 1 1 P P khk = inf kfi k(Lp1 (G;A);`q1 ) kgi k(Lp2 (G;A);`q2 ) : h = fi gi : i=1
i=1
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ISM AIL AYDIN
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 and and
1 r2 1 q2
= +
1 q1 1 q20
1 q2
+
1 1 p1 + p2 p1 p02 p1 p02 +p1 p02 and q
p1 ; q1 ; p2 ; q2 < 1, 1; p =
= 1: If we de…ne T' f = f
1 1 q1 + q2 q1 q20 q1 q20 +q1 q20 , p1
1, =
' for f 2 (L
1,
1 r1
1 1 p1 + p2 1 1 p2 + p02 = q1
=
where
1 1
(G; A) ; ` ) and ' 2 0
0
CC (G; A), then we have T' 2 HomL1 (G;A) (Lp1 (G; A) ; `q1 ) ; Lp2 (G; A ) ; `q2 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 CC (G; A) (Lp (G; A) ; `q ) be given. Due q1 q20 p1 p02 1 1 1 1 = p11 + p1 1, to p = p1 p0 +p 0 and q = q q 0 +q 0 , p + p = 1 + p0 > 1 and r 1 p 1 1 q 1 1 1 q1
+
1 q
2
=1+
2
1 q20
> 1 and
1 r2
by Theorem 6 we obtain f kT' f k
0
2
1 q1
=
2
+
2
1 q
1, we have r1 = p02 and r2 = q20 : Therefore, 0
0
' 2 (Lr1 (G; A) ; `r2 ) = Lp2 (G; A) ; `q2
and
C kf k(Lp1 (G;A);`q1 ) k'k(Lp (G;A);`q ) ;
0
Lp2 (G;A);`q2
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 0 0 (q ;q ) P(p11;p22) if every element of HomL1 (G;A) (Lp1 (G; A) ; `q1 ) ; Lp2 (G; A ) ; `q2 can be approximated in the ultraweak -operator topology by operators T' ; ' 2 CC (G; A) : 1, q11 + q12 1, Theorem 7. Let G be a locally compact Abelian group. If p11 + p12 1 1 1 1 1 1 = + 1 and = + 1, then the following statements are equivalent: r1 p1 p2 r2 q1 q2 (q ;q )
(i) G satis…es the property P(p11;p22) : (ii) The kernel of B is K such that (Lp1 (G; A) ; `q1 )
L1 (G;A)
2 (Lp2 (G; A) ; `q2 ) = Apq11;q ;p2 (G; A) :
Proof. Since B ((' f ) g f (e ' g)) = (' f ) g fe (e ' g) = 0, then (q1 ;q2 ) K KerB: Assume that G satis…es the property P(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 K ? = (((Lp1 (G; A) ; `q1 )
(Lp2 (G; A) ; `q2 )) =K)
ON VECTOR-VALUED CLASSICAL AND VARIABLE EXPONENT
111
[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 0
0
K ? = HomL1 (G;A) (Lp1 (G; A) ; `q1 ) ; Lp2 (G; A ) ; `q2
: 0
0
Thus, there is a multiplier T 2 HomL1 (G;A) (Lp1 (G; A) ; `q1 ) ; Lp2 (G; A ) ; `q2 which corresponds to F 2 K ? such that 1 P < t; F >= < gi ; T fi >; i=1
where t 2 KerB, t = p1
fi 2 (L
q1
1 P
fi
gi ,
i=1
(G; A) ; ` ) ; gi 2 (L B
p2
1 P
1 P
i=1
kfi k(Lp1 (G;A);`q1 ) kgi k(Lp2 (G;A);`q2 ) < 1 and
(G; A) ; `q2 ). Also, due to t 2 KerB we get fi
gi
i=1
Now, we show that
=
1 P fei gi = 0:
(5.1)
i=1
< t; F >=
1 P
< gi ; T fi >= 0.
(5.2)
i=1 (q ;q )
Furthermore, since G satis…es the property P(p11;p22) , then there exists a net 'j : j 2 I CC (G; A) such that the operators T'j de…ned in Lemma 1 converges to T in the ultraweak -operator topology, that is, 1 1 1 P P P lim < gi ; T'j fi >= lim < gi ; fi 'j >= < gi ; T fi > : j2I i=1
j2I i=1
i=1
So, to obtain (5.2), it is enough to show that 1 P < gi ; fi 'j >= 0 i=1
for all j 2 I. It can be seen easily 1 1 P P < gi ; fi 'j >= < fei gi ; 'j > : i=1
(5.3)
i=1
If we use the equalities (5.1), (5.3) and Hölder inequality for amalgam spaces, then we have 1 1 P P < gi ; fi 'j > fei gi 'j = 0; p p r r i=1
where
i=1
(Lr1 (G;A);`r2 )
1 P fei gi 2 (Lr1 (G; A) ; `r2 ) and 'j 2 CC (G; A)
L
1 (G;A);` 2
p
p
Lr1 (G; A) ; `r2 : There-
i=1
?
fore, < t; F >= 0 for all t 2 KerB. That means F 2 (KerB) : Hence KerB = K:
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ISM AIL AYDIN
2 By de…nition of Aqp11;q ;p2 (G; A) and the First Isomorphism Theorem, we get
(Lp1 (G; A) ; `q1 )
2 (Lp2 (G; A) ; `q2 ) =K = Aqp11;q ;p2 (G; A)
and (Lp1 (G; A) ; `q1 )
p2 L1 (G;A) (L
(G; A) ; `q2 ) = (Lp1 (G; A) ; `q1 )
(Lp2 (G; A) ; `q2 ) =K:
This proves (Lp1 (G; A) ; `q1 )
L1 (G;A)
2 (Lp2 (G; A) ; `q2 ) = Apq11;q ;p2 (G; A) :
Suppose conversely that KerB = K: If we show that the operators of the form T' 0
0
for ' 2 CC (G; A) are dense in HomL1 (G;A) (Lp1 (G; A) ; `q1 ) ; Lp2 (G; A ) ; `q2 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)
= K ?;
(Lp2 (G; A) ; `q2 )
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 fi 2 (Lp1 (G; A) ; `q1 ) ; gi 2 (Lp2 (G; A) ; `q2 ) such that 1 1 P P t= fi gi ; kfi k(Lp1 (G;A);`q1 ) kgi k(Lp2 (G;A);`q2 ) < 1 i=1
i=1
and < t; F >= 0 for all F 2 M . Also, there is an operator 0
0
T' 2 HomL1 (G;A) (Lp1 (G; A) ; `q1 ) ; Lp2 (G; A ) ; `q2
corresponding to F such that < t; F >=< t; T' >=
1 P
< gi ; T' fi >=
i=1 1 P
=
1 P fei gi = 0
i=1
and M ?
KerB: Consequently, KerB = M ? : This completes the theorem.
Corollary 2. Let G be a locally compact Abelian group. If 1 r1
=
1 p1
+
1 p2
1 and
1 r2
=
1 q1
+
1 q2
1 p1
+ p12
1 and G satis…es the property 0
0
HomL1 (G;A) (Lp1 (G; A) ; `q1 ) ; Lp2 (G; A ) ; `q2
1 1 1, q1 + q2 (q1 ;q2 ) P(p1 ;p2 ) , then
1,
2 = Aqp11;q ;p2 (G; A)
:
ON VECTOR-VALUED CLASSICAL AND VARIABLE EXPONENT
113
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[26] Rie¤el, M. A. Multipliers and tensor products of Lp spaces of locally compact geroups, Stud Math, 1969, 33, 71-82. [27] Sa¼ g ¬r, B. Multipliers and tensor products of vector-valued Lp (G; A) spaces, Taiwan J Math, 7(3), 2003, 493-501. [28] Schatten, R. A Theory of Cross-Spaces, Annal Math Stud, 26, 1950. [29] Squire, M. L. T. Amalgams of Lp and `q , Ph.D. Thesis, McMaster University, 1984. [30] Wiener, N. On the representation of functions by trigonometric integrals, Math. Z., 24, 1926, 575-616. Current address, Ismail AYDIN: Sinop University, Faculty of Sciences and Letters Department of Mathematics, Sinop, Turkey. E-mail address :
[email protected] [email protected]
