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On Weiner criterion for massiveness on weighted graphs

Lu Hao

Abstract

This paper investigates $p$-harmonic functions on infinite, connected, and locally finite weighted graphs. We focus on the concept of $p$-massiveness, establishing its equivalent characterization with the non-uniqueness of bounded solutions to the Dirichlet boundary value problem. Furthermore, for graphs satisfying the volume doubling condition and the weak $(1,p)$-Poincaré inequality, we establish a Wiener-type criterion at infinity to determine the $p$-massiveness of an infinite set.

On Weiner criterion for massiveness on weighted graphs

Abstract

This paper investigates -harmonic functions on infinite, connected, and locally finite weighted graphs. We focus on the concept of -massiveness, establishing its equivalent characterization with the non-uniqueness of bounded solutions to the Dirichlet boundary value problem. Furthermore, for graphs satisfying the volume doubling condition and the weak -Poincaré inequality, we establish a Wiener-type criterion at infinity to determine the -massiveness of an infinite set.

Paper Structure

This paper contains 7 sections, 25 theorems, 203 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lemmas
  4. Theorems on Uniqueness and Liouville Property
  5. criterion for $p$-massive
  6. criterion for $D_p$-massive
  7. Example

Key Result

Theorem 1.2

Let $(V, \mu)$ be an infinite, connected, and locally finite graph. Then the following statements are equivalent:

Figures (1)

  • Figure 1: Region $A_n$ and $B_n$.

Theorems & Definitions (65)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7: Wiener's criterion at infinity
  • Remark 1.8
  • Theorem 1.9
  • Proposition 2.1
  • ...and 55 more