[Paper Review] Lq Harmonic Functions on Graphs

Bobo Hua, Jürgen Jost|arXiv (Cornell University)|Jan 15, 2013

Nonlinear Partial Differential Equations7 references4 citations

TL;DR

This paper establishes an $L^q$ Liouville theorem for harmonic and nonnegative subharmonic functions on weighted graphs, proving that nonconstant $L^q$ functions ($1 \leq q < \infty$) cannot exist on any infinite, connected, locally finite weighted graph, even without curvature or degree bounds. The key contribution is a new $L^q$ Caccioppoli-type inequality that enables a quantitative growth estimate and resolves a gap in prior work for $1 < q < 2$, while also showing counterexamples for $0 < q < 1$. The result extends to polyharmonic functions via iterative application of the main theorem.

ABSTRACT

We prove an analogue of Yau's Caccioppoli-type inequality for nonnegative subharmonic functions on graphs. We then obtain a Liouville theorem for harmonic or non-negative subharmonic functions of class Lq, 1&lt;=q 1. Also, we provide counterexamples for Liouville theorems for 0 &lt; q &lt; 1.

Motivation & Objective

• To establish a Liouville-type theorem for $L^q$ harmonic and nonnegative subharmonic functions on general weighted graphs without curvature or degree bounds.
• To resolve the open case $1 < q < 2$ in the graph setting, previously unresolved by Rigoli-Salvatori-Vignati.
• To provide a quantitative version of the growth condition for nonnegative subharmonic functions in $L^q$ ($q > 1$).
• To show that the $L^q$ Liouville theorem fails for $0 < q < 1$ by constructing explicit counterexamples on graphs of finite and infinite volume.
• To extend the $L^q$ Liouville result to higher-order polyharmonic operators on graphs of infinite volume.

Proposed method

• Derive a new $L^q$ Caccioppoli-type inequality for nonnegative subharmonic functions on graphs, analogous to Yau’s result in Riemannian geometry.
• Use the Caccioppoli inequality to prove a quantitative growth estimate: $\liminf_{R\to\infty} \frac{1}{R^2} \sum_{B_R(p)} f^q(x) \mu_x = \infty$ for nonconstant nonnegative subharmonic $f$ in $L^q$, $1 < q < \infty$.
• Prove the $L^1$ Liouville theorem by showing that any nonnegative $L^1$ subharmonic function must be harmonic, and then use truncation and harmonicity to deduce constancy.
• Construct explicit counterexamples for $0 < q < 1$ using graphs with finite and infinite volume, where nonconstant harmonic functions exist in $L^q$.
• Apply the $L^q$ Liouville theorem to higher-order operators by showing that if $\Delta^m f = 0$ and $f \in L^q$, then $f \equiv 0$ for $1 \leq q < \infty$ on graphs of infinite volume.

Experimental results

Research questions

• RQ1 Does an $L^q$ Liouville theorem hold for nonnegative subharmonic functions on general weighted graphs without curvature or degree bounds, for $1 \leq q < \infty$?
• RQ2 Can the gap in the $1 < q < 2$ case of the graph $L^q$ Liouville theorem be closed using a Caccioppoli-type inequality?
• RQ3 What happens for $0 < q < 1$? Are there nonconstant $L^q$ harmonic functions on graphs?
• RQ4 Can the $L^q$ Liouville theorem be extended to solutions of higher-order elliptic operators on graphs?
• RQ5 Is the $L^1$ Liouville theorem valid on graphs without additional curvature assumptions?

Key findings

• The paper proves that any nonnegative subharmonic function in $L^q$ for $1 < q < \infty$ on any infinite, connected, locally finite weighted graph must satisfy $\liminf_{R\to\infty} \frac{1}{R^2} \sum_{B_R(p)} f^q(x) \mu_x = \infty$, establishing a quantitative growth condition.
• For $q = 1$, the paper proves that any nonnegative $L^1$ subharmonic function on a graph must be constant, without requiring curvature or degree bounds.
• The authors construct a counterexample of a nonconstant harmonic function in $L^q$ for $0 < q < 1$ on a finite-volume graph, showing the $L^q$ Liouville theorem fails in this regime.
• Another counterexample is given on an infinite-volume graph, demonstrating that nonconstant $L^q$ harmonic functions exist for $0 < q < 1$, again invalidating the Liouville theorem.
• The $L^q$ Liouville theorem is extended to polyharmonic functions: if $f \in L^q$ and $\Delta^m f = 0$ for $m \geq 2$ on a graph of infinite volume, then $f \equiv 0$ for $1 \leq q < \infty$.