Multipliers and Tensor Products of Weighted Lp-Spaces

Abstract

Let G be a locally compact unimodular group with Haar measure rmdx and ω be the Beurling's weight function on G (Reiter, [10]). In this paper the authors define a space Ap,q<inf>ω</inf> (G) and prove that Ap,q<inf>ω</inf> (G) is a translation invariant Banach space. Furthermore the authors discuss inclusion properties and show that if G is a locally compact abelian group then Ap,q<inf>ω</inf> (G) admits an approximate identity bounded in L1<inf>ω</inf>. (G). It is also proved that the space Lp<inf>ω</inf> (G) ⊗<inf>L1ω</inf> L̄q<inf>ω</inf> (G) is isometrically isomorphic to the space Ap,q<inf>ω</inf> (G) and the space of multipliers from Lp<inf>ω</inf> (G) to Lq′<inf>ω-1</inf> (G) is isometrically isomorphic to the dual of the space Ap,q<inf>ω</inf> (G) iff G satisfies a property Pq<inf>p</inf>. At the end of this work it is showed that if G is a locally compact abelian group then the space of all multipliers from L1<inf>ω</inf> (G) to Ap,q<inf>ω</inf> (G) is the space Ap,q<inf>ω</inf> (G).