Graphs of bounded degree and the $p$-harmonic boundary

TL;DR

This paper introduces the concept of the $p$-harmonic boundary for bounded degree graphs, characterizes when only constant $p$-harmonic functions exist, and relates these boundaries to $ ext{l}^p$-cohomology of groups.

Contribution

It defines the $p$-harmonic boundary for graphs, explores its properties, and links it to the $ ext{l}^p$-cohomology of finitely generated groups, providing new insights into harmonic analysis on graphs.

Findings

Characterization of graphs with only constant $p$-harmonic functions.

Extension of boundary functions to $p$-harmonic functions on graphs.

Relation between $p$-harmonic boundary and $ ext{l}^p$-cohomology of groups.

Abstract

Let $p$ be a real number greater than one and let $G$ be a connected graph of bounded degree. In this paper we introduce the $p$-harmonic boundary of $G$. We use this boundary to characterize the graphs $G$ for which the constant functions are the only $p$-harmonic functions on $G$. It is shown that any continuous function on the $p$-harmonic boundary of $G$ can be extended to a function that is $p$-harmonic on $G$. Some properties of this boundary that are preserved under rough-isometries are also given. Now let $\Gamma$ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced $\ell^p$-cohomology of $\Gamma$ in terms of the cardinality of its $p$-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $\Gamma$, the $p$-harmonic boundary of $\Gamma$ with the first reduced $\ell^p$-cohomology of $\Gamma$.