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

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.

17 pages, to appear in Math. Ann

References in corpus (2)

• Rademacher-type Theorems and Sobolev-to-Lipschitz Properties for Strongly Local Dirichlet Spaces
• Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry

Cited by in corpus (2)

• Harmonic functions and gravity localization
• Integral Varadhan formula for non-linear heat flow