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Weighted graphs in the sense of John and a global Poincaré inequality

Fernando López-García, John Rodriguez

TL;DR

This work extends John-domain ideas to weighted graphs by introducing a discrete John-domain condition: there exists a rooted spanning tree with shadows $S_t$ satisfying $\mu(S_t) \le c\mu(t)$. Under finite measure, the authors establish a global $\ell^p(V,\mu)$-Poincaré inequality for $1\le p<\infty$, derived via a local-to-global strategy that hinges on a Hardy-type averaging operator $T$ along the tree and a decomposition of functions into edge-local pieces $f_t$ on segments $\{t,t_p\}$. Key contributions include sharp continuum-like control of the Poincaré constant $C_P$ in terms of the geometric constant $c$ and the tree degree $M$, along with explicit bounds for $p>1$ and the endpoint $p=1$. The results connect to spectral estimates for the graph Laplacian and provide a discrete framework paralleling Whitney cube and John-domain theories in the Euclidean setting.

Abstract

In this paper, we establish a condition on weighted graphs with finite measure that guarantees the validity of a global Poincaré inequality. This condition can be viewed as a discrete analogue of the criterion introduced by J. Boman in 1982 for Whitney cubes, which in turn characterizes the condition originally proposed by F. John in his seminal 1961 work.

Weighted graphs in the sense of John and a global Poincaré inequality

TL;DR

This work extends John-domain ideas to weighted graphs by introducing a discrete John-domain condition: there exists a rooted spanning tree with shadows satisfying . Under finite measure, the authors establish a global -Poincaré inequality for , derived via a local-to-global strategy that hinges on a Hardy-type averaging operator along the tree and a decomposition of functions into edge-local pieces on segments . Key contributions include sharp continuum-like control of the Poincaré constant in terms of the geometric constant and the tree degree , along with explicit bounds for and the endpoint . The results connect to spectral estimates for the graph Laplacian and provide a discrete framework paralleling Whitney cube and John-domain theories in the Euclidean setting.

Abstract

In this paper, we establish a condition on weighted graphs with finite measure that guarantees the validity of a global Poincaré inequality. This condition can be viewed as a discrete analogue of the criterion introduced by J. Boman in 1982 for Whitney cubes, which in turn characterizes the condition originally proposed by F. John in his seminal 1961 work.

Paper Structure

This paper contains 4 sections, 9 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Some basic definitions and an averaging operator on Graphs with Rooted Spanning Trees
  3. A Decomposition of Functions Technique
  4. A global weighted Poincaré Inequality on Graphs with Rooted Spanning Trees

Key Result

Theorem 1

Let $(V,\mu)$ be a weighted graph that satisfies the condition in Definition def: John domain, whose spanning tree has degree uniformly bounded. Also, let $p\in [1,\infty)$. Then, there exists a constant $C_P>0$ such that the inequality holds for any $f\in \ell^p(V,\mu)$ that sums zero with respect to $\mu$ on $V$.

Theorems & Definitions (25)

  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 15 more