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Multipliers of Banach valued weighted function spaces. (English) Zbl 0968.43004

Let \(G\) be an \(LCA\) group and \(X\) a Banach space with dual space \(X^*\) having the wide Radon Nikodým property. The author considers the weighted \(p\)-Lebesgue space \[ L^p_\psi(G,X) =\left[\int_G \bigl\|f(t)\bigr \|^p_X \psi(t)^p dt\right]^{1/p}, \quad 1\leq p<\infty, \] where \(\psi\) is a locally bounded weight function satisfying \(\psi(s+t) \leq\psi (s)\psi(t)\), \(s,t \in G\). The author studies the multiplier spaces concerning the weighted space \(L^p_\psi (G,X)\), \(1\leq p<\infty\). It is shown that the multiplier space \[ \operatorname{Hom}_{L^1_\psi (G,A)} \bigl\{L^p_\psi (G,X),L^{q'}_{\psi-1} (G,X^*)\bigr\} \] is isometrically isomorphic to the dual of a tensor product space \[ \Bigl[ L^p_\psi(G,X) \otimes_{L^1_\psi (G,A)}L^q_\psi (G,X)\Bigr]^*,\;{1\over p}+{1 \over q}\geq 1,\;{1\over q}+{1 \over q'}=1, \] where \(A\) is a commutative Banach algebra with identity of norm 1 and \(X\) is an \(A\)-module Banach space. The other characterization \[ \operatorname{Hom}_{L^1_\psi (G,A)}\bigl[L^p_\psi (G,X),\;L^{q'}_{\psi-1} (G,X^*)\bigr] \cong\bigl[ A_\psi^{p,q} (G,X)\bigr]^* \] is also proved here.
Reviewer: H.-C.Lai (Taiwan)

MSC:

43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
46E40 Spaces of vector- and operator-valued functions
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