Persistence of Rademacher-type and Sobolev-to-Lipschitz properties

TL;DR

This paper investigates the stability and transfer of Rademacher- and Sobolev-to-Lipschitz properties in Dirichlet spaces, providing conditions for their persistence and implications for metric measure space analysis.

Contribution

It introduces new conditions for the persistence and transfer of these properties in various space transformations and characterizes Dirichlet forms via Cheeger energy.

Findings

Conditions for property persistence under localization and globalization

Tensorization results for intrinsic distances and Varadhan asymptotics

Characterization of Dirichlet forms via Cheeger energy

Abstract

We consider the Rademacher- and Sobolev-to-Lipschitz-type properties for arbitrary quasi-regular strongly local Dirichlet spaces. We discuss the persistence of these properties under localization, globalization, transfer to weighted spaces, tensorization, and direct integration. As byproducts we obtain: necessary and sufficient conditions to identify a quasi-regular strongly local Dirichlet form on an extended metric topological $\sigma$-finite possibly non-Radon measure space with the Cheeger energy of the space; the tensorization of intrinsic distances; the tensorization of the Varadhan short-time asymptotics.