Improved regularity estimates for Lagrangian flows on RCD(K,N) spaces

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 [CoN12]). The estimates are suitable for applications to the fine study of RCD spaces and play a central role in the construction of a parallel transport in this setting.