On global H\"older estimates for optimal transportation

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 $D V, D^2 V$.