On the equivalence of Lp-parabolicity and Lq-liouville property on weighted graphs

TL;DR

This paper explores the equivalence between $L^p$-parabolicity and $L^q$-Liouville properties on weighted graphs, introduces new heat kernel estimates, and establishes criteria for these properties with applications to various graph examples.

Contribution

It establishes the equivalence between $L^p$-parabolicity and $L^q$-Liouville properties on weighted graphs, introduces refined heat kernel estimates, and determines sharp volume growth criteria.

Findings

Two-sided Green function estimates on graphs

Sharp volume growth criteria for $L^q$-Liouville property

Examples illustrating the theoretical results

Abstract

We study the equivalence between the $L^p$-parabolicity, the $L^q$-Liouville property of positive super-harmonic functions, and the existence of nonharmonic positive solutions to the following elliptic differential system \begin{equation*} \left\{ \begin{array}{lr} -\Delta u\geq 0, \Delta(|\Delta u|^{p-2}\Delta u)\geq 0, \end{array} \right. \end{equation*} on weighted graphs, where $1\leq p< \infty$, and $(p, q)$ are H\"{o}lder conjugate exponent pair. Furthermore, by refining a new technique on estimate of heat kernel, we can establish two-sided estimates of Green function on graph, and find the sharp volume growth criteria for the $L^q$-Liouville property on a large class of graphs. As an application, many non-trivial interesting examples are presented.