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arXiv · 2009.04118

Uniform Poincar\'e inequalities on measured metric spaces

Gautam Neelakantan Memana·Soma Maity
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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.

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Gautam Neelakantan Memana, Soma Maity. 2020-09-09. Uniform Poincar\'e inequalities on measured metric spaces. https://doi.org/10.1007/s00229-022-01436-5

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