An Analysis on Harmonic Function For Measuring Hypergroups: Applications and Properties

Abstract

We initiate a study of harmonic functions on hypergroups.In particular, we introduce the concept of a nilpotent hypergroup and show suchhypergroup admits an invariant measure as well as a Liouville theorem forbounded harmonic functions. Further, positive harmonic functions on nilpotenthypergroups are shown to be integrals of exponential functions. For arbitraryhypergroups, we derive a Harnack inequality for positive harmonic functions andprove a Liouville theorem for compact hypergroups. We discuss an application toharmonic spherical functions.