Calculus of Variations and Geometric Measure Theory

B. Han - E. Milman

Sharp Poincare inequality under Measure Contraction Property

created by han1 on 15 May 2019
modified on 29 Sep 2024

[BibTeX]

Published Paper

Inserted: 15 may 2019
Last Updated: 29 sep 2024

Journal: Ann. Sc. Norm. Super. Pisa Cl. Sci.
Year: 2021

Abstract:

We prove a sharp Poincare inequality for subsets $\Omega$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property 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 Poincare 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 Poincare inequality. In particular, our results apply to geodesic balls in ideal sub-Riemannian manifolds, such as the Heisenberg group.


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