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Harmonic functions on hypergroups |
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| Authors: | Massoud Amini Cho-Ho Chu |
| Institution: | a School of Mathematics, Institute for Research in Fundamental Sciences, P.O. Box 19395-5746, Tehran, Iran b Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran c School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK |
| Abstract: | We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions. |
| Keywords: | Harmonic function Spherical function Nilpotent hypergroup Spherical hypergroup Liouville theorem Harnack inequality |
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