Improved regularity estimates for Lagrangian flows on $\text{RCD}(K,N)$ spaces

TL;DR

This paper advances the understanding of Lagrangian flow regularity on non-smooth spaces with Ricci curvature bounds, providing improved local and time-behavior estimates crucial for analyzing RCD spaces.

Contribution

It introduces sharper regularity estimates for Lagrangian flows on RCD spaces, enhancing previous results and enabling the construction of parallel transport in this setting.

Findings

Improved regularity estimates with better time behavior.

Local nature of the regularity estimates.

Applications to the study and analysis of RCD spaces.

Abstract

This paper gives a contribution to the study of regularity of Lagrangian flows on non-smooth spaces with lower Ricci curvature bounds. The main novelties with respect to the existing literature are the better behaviour with respect to time and the local nature of the regularity estimates. These are obtained by sharpening previous results of the first and third authors, in combination with some tools recently developed by the second author (adapting to the synthetic framework ideas introduced in [Colding-Naber 12]. The estimates are suitable for applications to the fine study of $\text{RCD}$ spaces and play a central role in the construction of a parallel transport in this setting.