Convolution Properties of Orlicz Spaces on hypergroups

TL;DR

This paper investigates the convolution properties of Orlicz spaces on hypergroups, establishing necessary conditions for convolution existence, characterizing compactness of convolution operators, and applying results to locally compact abelian groups.

Contribution

It provides new necessary conditions for convolution in Orlicz spaces on hypergroups and characterizes compact convolution operators, extending classical harmonic analysis results.

Findings

Necessary conditions for a.e. convolution existence on hypergroups.

Equivalent conditions for compactness of certain groups.

Characterization of compact convolution operators.

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 $K$.