Uniform Poincaré inequalities on measured metric spaces

Abstract

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

19 pages, revised version