Sobolev-to-Lipschitz property on $${\mathsf {QCD}}$$-spaces and applications

Dello Schiavo, Lorenzo; Suzuki, Kohei

Abstract

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). 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 results apply in particular to large classes of (ideal) sub-Riemannian manifolds.

Citation

Dello Schiavo, L., & Suzuki, K. (2022). Sobolev-to-Lipschitz property on $${\mathsf {QCD}}$$-spaces and applications. Mathematische Annalen, 384, 1815-1832. https://doi.org/10.1007/s00208-021-02331-2

• Journal Article Type: Article
• Acceptance Date: Nov 25, 2021
• Online Publication Date: Dec 23, 2021
• Publication Date: 2022-12
• Deposit Date: Oct 20, 2024
• Journal: Mathematische Annalen
• Print ISSN: 0025-5831
• Electronic ISSN: 1432-1807
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 384
• Pages: 1815-1832
• DOI: https://doi.org/10.1007/s00208-021-02331-2
• Public URL: https://durham-repository.worktribe.com/output/2977756
• Additional Information: Received: 12 October 2021; Revised: 22 November 2021; Accepted: 25 November 2021; First Online: 23 December 2021; : ; : The authors have no conflicts of interest to declare that are relevant to the content of this article.