Regularity Properties of Monotone Measure-preserving Maps

Yash Jhaveri, Alessio Figalli

Abstract: In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex domain and, on the low-regularity end, between domains carrying certain invariant measures.