Uniform Poincar\'e inequalities on measured metric spaces

TL;DR

This paper proves uniform Poincaré inequalities on metric spaces under certain growth and geometric conditions, extending classical results to hyperbolic and universal cover spaces with applications to measure and geometric analysis.

Contribution

It establishes a family of uniform Poincaré inequalities on metric spaces satisfying local inequalities and volume growth conditions, including hyperbolic and universal cover spaces.

Findings

Uniform Poincaré inequalities hold under local inequalities and volume growth conditions.

Doubling measures support $(\sigma,eta,\sigma)$-Poincaré inequalities.

Hyperbolic spaces exhibit exponential growth in Poincaré constants.

Abstract

Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $\mu.$ We establish a family of uniform Poincar\'e inequalities on $(X,d,\mu)$ if it satisfies a local Poincar\'e inequality ($P_{loc}$) and a condition on growth of volume. Consequently if $\mu$ is doubling and supports $(P_{loc})$ then it satisfies a $(\sigma,\beta,\sigma)$-Poincar\'e inequality. If $(X,d,\mu)$ is a $\delta$-hyperbolic space then using the volume comparison theorem in \cite{BCS} we obtain a uniform Poincar\'e inequality with exponential growth of the Poincar\'e constant. If $X$ is the universal cover of a compact $CD(K,\infty)$ space then it supports a uniform Poincar\'e inequality and the Poincar\'e constant depends on the growth of the fundamental group.