Global Hölder Estimates for Optimal Transportation

We generalize the well-known result due to Caffarelli concerning Lipschitz estimates for the optimal transportation $T$ of logarithmically concave probability measures. Suppose that $T\colon\mathbb R^d\to\mathbb R^d$ is the optimal transportation mapping $\mu=e^{-V}\,dx$ to $\nu=e^{-W}\,dx$. Suppose that the second difference-differential $V$ is estimated from above by a power function and that the modulus of convexity $W$ is estimated from below by the function $A_q|x|^{1+q}$, $q\ge1$. We prove that, under these assumptions, the mapping $T$ is globally Hölder with the Hölder constant independent of the dimension. In addition, we study the optimal mapping $T$ of a measure $\mu$ to Lebesgue measure on a convex bounded set $K\subset\mathbb R^d$. We obtain estimates of the Lipschitz constant of the mapping $T$ in terms of $d$, $\operatorname{diam}(K)$, and $DV$, $D^2V$.