Poincar\'e inequalities on graphs

TL;DR

This paper establishes local and global Poincaré inequalities on graphs with specific measure conditions, advancing understanding of functional inequalities in discrete geometric settings.

Contribution

It introduces new local and global $L^p$-Poincaré inequalities on graphs with locally doubling measures and analyzes their optimality.

Findings

Proves local $L^p$-Poincaré inequalities on quasiconvex sets in graphs.

Establishes global $L^p$-Poincaré inequalities on connected sets for flow measures on trees.

Discusses the optimality of the derived inequalities.

Abstract

We prove local $L^p$-Poincar\'e inequalities, $ p\in[1,\infty]$, on quasiconvex sets in infinite graphs endowed with a family of locally doubling measures, and global $L^p$-Poincar\'e inequalities on connected sets for flow measures on trees. We also discuss the optimality of our results.