Fig$$\grave{\hbox {a}}$$–Talamanca–Herz–Orlicz Algebras and Convoluters of Orlicz Spaces

Abstract

Let G be a locally compact group and $$(\Phi , \Psi )$$ be a complementary pair of N-functions. In this paper, the Fig $$\grave{\hbox {a}}$$ –Talamanca–Herz algebras and the space of p-pseudomeasures on G are extended to Orlicz spaces. Indeed, the Banach algebra of $$\Phi $$ -pseudomeasures $${\mathop {\mathrm{PM}}}_{\Phi }(G)$$ and the Fig $$\grave{\hbox {a}}$$ –Talamanca–Herz–Orlicz algebras $${\mathop {\mathrm{A}}}_{\Phi }(G)$$ are defined. Then, it is shown that $${\mathop {\mathrm{A}}}_{\Phi }(G)^*={\mathop {\mathrm{PM}}}_{\Psi }(G)$$ . Furthermore, we characterize $$\mathop {\mathrm{Cv}}_{\Phi }(G)$$ , the space of $$\Phi $$ -convoluters, in terms of right translation invariant operators on $$M^{\Phi }(G)$$ . Then when G is amenable, we show that $$\mathop {\mathrm{Cv}}_{\Phi }(G)$$ , is equal to $$\mathop {\mathrm{PM}}_{\Phi }(G)$$ , a generalization of the classical p-version. Finally, we study $$B_{\Phi }(G)$$ , the space of pointwise multipliers of $$\mathop {\mathrm{A}}_{\Phi }(G)$$ when G is amenable.