The Liouville theorem for discrete symmetric averaging operators

TL;DR

This paper introduces a unified framework for averaging operators on lattices and proves a Liouville theorem for various classes of harmonic functions, offering a simpler proof for positive p-harmonic functions on integer lattices.

Contribution

It develops a general approach to Liouville properties for discrete averaging operators, encompassing multiple types of harmonic functions and providing an elementary proof.

Findings

Unified framework for averaging operators on lattices

Liouville theorem extended to various harmonic functions

Elementary proof for positive p-harmonic functions

Abstract

We introduce averaging operators on lattices $\mathbb{Z}^d$ and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, $p$-harmonic, $\infty$-harmonic and the so-called game $p$-harmonic functions. Our approach provides an elementary alternative proof of the Liouville Theorem for positive $p$-harmonic functions on $\mathbb{Z}^d$.