Harmonic functions on locally compact groups of polynomial growth

TL;DR

This paper generalizes Kleiner's theorem on harmonic functions of polynomial growth to locally compact groups with measures of non-compact support, revealing structural properties of these harmonic functions.

Contribution

It extends the finiteness result for harmonic functions of polynomial growth to a broader class of locally compact groups with non-compact support measures.

Findings

Finite dimensionality of harmonic functions of polynomial growth on these groups

Structural insights into the space of polynomially growing harmonic functions

Generalization of Kleiner's theorem to new group settings

Abstract

We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with non-compact support. This has implications to the structure of the space of polynomially growing harmonic functions.