Convolution in Orlicz Spaces on Hypergroups

Let G be a locally compact hypergroup, and let K be a compact sub-hypergroup of G. \((G,K)\) is a Gelfand pair if \(M_{c}(G//K)\) , the algebra of measures with compact support on the double coset \(G//K\) , is commutative for the convolution. In this chapter, we assume that \((G,K)\) is a Gelfand pair, and we consider a pair \((\Phi _{1},\Phi _{2})\) of Young functions satisfying the sequence condition. We prove a necessary condition in terms of aperiodic elements of the center of G, for the existence always everywhere of convolution \(f\ast g\) , where f and g are arbitrary elements of Orlicz spaces \(L^{\Phi _{1}}(G)\) and \(L^{\Phi _{2}}(G).\)