On global H\"older estimates for optimal transportation

We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation $T$ between uniformly log-concave probability measures. Let $T:{\mathbb{R}}^d\to {\mathbb{R}}^d$ 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 $A_q\mathrm{\mid }x{\mathrm{\mid }}^{1+q}$, $q\ge 1$. 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 $K\subset {\mathbb{R}}^d$. We get estimates for the Lipschitz constant of $T$ in terms of $d$, $diam(K)$ and $D V, D^2 V$.