Some convolution algebras and their multipliers

Abstract

Let G be a locally compact Abelian group (nondiscrete and non compact) with dual group \widehat{G}. For 1 \leq P < \infty, A_p (G) denotes the vector space of all complex-valued functions in L1 (G) whose Fourier transforms \hat{f} belong to Lp\widehat(G). Research on the spaces A_p (G) was initiated by Warner [20] and Larsen, Liu and Wang [14]. Later several generalizations of these spaces to the weighled case was given by Gürkanlı [6], Feichtinger and Gürkanlı [4] and Fischer, Gürkanlı and Liu [5]. One of these generalization is the space Ap_{w,\omega}(G), [4]. Also the multipliers of A_p (G) were discussed in some papers such as [14], [1], [13], [3], [9] and proved that the space of multipliers of A_p (G) is the space of all bounded complex-valued regular Borel measures on G. In the present paper we discussed the multipliers of the Banach algebra Ap_{w,\omega}(G) and proved that under certain conditions for given any multiplier T of Ap_{w,\omega}(G) there exists a unique pseudo measure \sigma such that Tf \sigma * f for all f \in Ap_{w,\omega}(G).