Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and $RCD(K,\infty)$ spaces

TL;DR

This paper proves new Lusin-type approximation results for Sobolev functions by Lipschitz functions in Gaussian and RCD(K,∞) spaces, using heat semigroup estimates, leading to stability results for Lagrangian flows.

Contribution

It introduces novel approximation techniques for Sobolev functions in non-doubling metric measure spaces, expanding the understanding of regularity and stability in these contexts.

Findings

Lusin approximation results for Sobolev functions in Gaussian and RCD spaces

Quantitative stability of regular Lagrangian flows in Gaussian spaces

Application of heat semigroup estimates to non-doubling spaces

Abstract

We establish new approximation results, in the sense of Lusin, of Sobolev functions by Lipschitz ones, in some classes of non-doubling metric measure structures. Our proof technique relies upon estimates for heat semigroups and applies to Gaussian and $RCD(K, \infty)$ spaces. As a consequence, we obtain quantitative stability for regular Lagrangian flows in Gaussian settings.