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

TL;DR

The paper proves an equivalence on weighted graphs among $L^p$-parabolicity, $L^q$-Liouville property, and the nonexistence of nonharmonic nonnegative solutions to the elliptic system $-\Delta u \ge 0$, $\Delta(|\Delta u|^{p-2}\Delta u) \ge 0$ with $1/p+1/q=1$. It introduces $L^p$-capacity on graphs and shows $C_p(K,U)=\hat{C}_p(K,U)=\bar{C}_p(K,U)$, linking capacity to Green operator norms. Using refined heat-kernel techniques, it derives two-sided Green function bounds and sharp volume-growth criteria, extending manifold results to the graph setting. The main result yields practical criteria for $L^p$-parabolicity and $L^q$-Liouville properties via volume growth and heat-kernel bounds, with substantial examples including Cayley graphs and the discrete Heisenberg group to demonstrate applicability and sharpness.

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} -Δu\geq 0, Δ(|Δu|^{p-2}Δu)\geq 0, \end{array} \right. \end{equation*} on weighted graphs, where $1\leq p< \infty$, and $(p, q)$ are Hölder 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.

Theorems & Definitions (61)

• Definition 1.1
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• Theorem 1.4
• Remark 1.5
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• Theorem 1.10