Convolution Properties of Orlicz Spaces on hypergroups

Abstract

In this paper, for a locally compact commutative hypergroup $K$ and for a pair $(\Phi_1, \Phi_2)$ of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of $K,$ for the convolution $f\ast g$ to exist a.e., where $f$ and $g$ are arbitrary elements of Orlicz spaces $L^{\Phi_1}(K)$ and $L^{\Phi_2}(K)$, respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from $L^1_w(K)$ into $L^\Phi_w(K)$ for a weight $w$ on a locally compact hypergroup $K$.