Sobolev-to-Lipschitz Property on QCD-spaces and Applications

TL;DR

This paper establishes the Sobolev-to-Lipschitz property for certain metric measure spaces satisfying a quasi curvature-dimension condition, with applications to heat semigroup properties on sub-Riemannian manifolds.

Contribution

It proves the Sobolev-to-Lipschitz property under the quasi curvature-dimension condition and explores its implications for heat semigroup behavior in sub-Riemannian spaces.

Findings

Sobolev-to-Lipschitz property holds under the quasi curvature-dimension condition

Varadhan short-time asymptotics for heat semigroup established

Heat semigroup irreducibility proved in the setting of sub-Riemannian manifolds

Abstract

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature dimension condition recently introduced in E. Milman, The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifolds, Comm. Pure Appl. Math. (to appear, arXiv:1908.01513v5). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These result apply in particular to large classes of (ideal) sub-Riemannian manifolds.