On global H\"older estimates for optimal transportation

TL;DR

This paper extends Caffarelli's Lipschitz estimates for optimal transportation between log-concave measures to a broader class, establishing global Hölder continuity under new convexity assumptions and providing Lipschitz bounds related to domain geometry.

Contribution

It generalizes Lipschitz estimates to cases with weaker convexity conditions, introducing global Hölder regularity results for optimal transport maps.

Findings

Optimal transport map is globally Hölder continuous under new assumptions.

Derived dimension-free Hölder estimates for transportation between log-concave measures.

Provided Lipschitz bounds depending on domain diameter and potential derivatives.

Abstract

We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation $T$ between uniformly log-concave probability measures. Let $T : \R^d \to \R^d$ be an optimal transportation pushing forward $\mu = e^{-V}dx$ to $\nu = e^{-W}dx$. Assume that 1) the second differential quotient of $V$ can be estimated from above by a power function, 2) modulus of convexity of $W$ can be estimated from below by $A_q |x|^{1+q}$, $q \ge 1$. Under these assumptions we show that $T$ is globally H\"older with a dimension-free coefficient. In addition, we study optimal transportation $T$ between $\mu$ and the uniform measure on a bounded convex set $K \subset \R^d$. We get estimates for the Lipschitz constant of $T$ in terms of $d$, ${diam(K)}$ and $D V, D^2 V$.