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arXiv · 1905.05465

Sharp Poincar\'e inequalities under Measure Contraction Property

Bang-Xian Han·Emanuel Milman
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Abstract

We prove a sharp Poincar\'e inequality for subsets $\Omega$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by identifying the corresponding one-dimensional model densities and a localization argument, ensuring that the Poincar\'e constant we obtain is best possible as a function of $K$, $N$ and $D$. Another new feature of our work is that we do not need to assume that $\Omega$ is geodesically convex, by employing the geodesic hull of $\Omega$ on the energy side of the Poincar\'e inequality. In particular, our results apply to geodesic balls in ideal sub-Riemannian manifolds, such as the Heisenberg group.

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Bang-Xian Han, Emanuel Milman. 2019-05-14. Sharp Poincar\'e inequalities under Measure Contraction Property. https://arxiv.org/abs/1905.05465

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