Generalized potentials on commutative hypergroups

TL;DR

This paper introduces generalized potential operators on commutative hypergroups and proves their boundedness from Lebesgue spaces to Orlicz spaces, extending classical results to a broader algebraic setting.

Contribution

It extends the Hardy-Littlewood-Sobolev theorem to generalized potentials on commutative hypergroups, providing new boundedness results.

Findings

Boundedness of generalized potential operators established

Extension of classical theorems to hypergroup setting

Operators map Lebesgue spaces into Orlicz spaces

Abstract

By the Hardy-Littlewood-Sobolev theorem the classical Riesz potential is bounded on Lebesgue spaces. E. Nakai and H. Sumitomo [16] extended that theorem to the Orlicz spaces. We introduce generalized potential operators on commutative hypergroups and under some assumptions on the kernel we showed the boundedness of these operators from Lebesgue space into certain Orlicz space. Our result is an analogue of Theorem 1.3 in [16].