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 $L^1 (G)$ whose Fourier transforms $hat{f}$ belong to $L^pwidehat(G)$. 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 $A^p_{w,omega}(G)$, [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 $A_p (G)$ isthespaceofallbounded complex-valued regular Borel measures on G. In the present paper we discussed the multipliers of the Banach algebra $A^p_{w,omega}(G)$ and proved that under certain conditions for given any multiplier T of $A^p_{w,omega}(G)$ there exists a unique pseudo measure $sigma$ such that Tf = $sigma$ * f for all f $in$ $A^p_{w,omega}(G)$.