Sharp Poincaré inequalities under Measure Contraction Property

Abstract

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

24 pages; addressed comments by referee, to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. Changed conv(A) notation to geo(A)