Discrete Approximations of Metric Measure Spaces of Controlled Geometry

TL;DR

This paper establishes a precise criterion for when doubling metric spaces support a (1,p)-Poincare inequality, using discretizations and graph-based inequalities to characterize the geometric conditions involved.

Contribution

It introduces a necessary and sufficient condition linking discretizations of metric spaces with Poincare inequalities, advancing understanding of geometric analysis on metric measure spaces.

Findings

Characterizes when doubling metric spaces support (1,p)-Poincare inequalities.

Uses discretizations and graph inequalities to establish the condition.

Provides a new framework connecting geometric properties with analytic inequalities.

Abstract

We find a necessary and sufficient condition for a doubling metric space to carry a (1,p)-Poincare inequality. The condition involves discretizations of the metric space and Poincare inequalities on graphs.