Convolution properties of Orlicz spaces on hypergroups

Abstract

In this paper, for a locally compact commutative hypergroup K K and for a pair ( Φ 1 , Φ 2 ) (\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 , K, for the convolution f ∗ g f\ast g to exist a.e., where f f and g g are arbitrary elements of Orlicz spaces L Φ 1 ( K ) L^{\Phi _1}(K) and L Φ 2 ( K ) 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 w 1 ( K ) L^1_w(K) into L w Φ ( K ) L^\Phi _w(K) for a weight w w on a locally compact hypergroup K K .