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: ℝ d → ℝ d is the optimal transportation mapping µ = e −V dx to ν = 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 ≥ 1. We prove that, under these assumptions, the mapping T is globally Holder with the Holder constant independent of the dimension. In addition, we study the optimal mapping T of a measure µ to Lebesgue measure on a convex bounded set K ⊂ ℝ d . We obtain estimates of the Lipschitz constant of the mapping T in terms of d, diam(K), and DV, D 2 V.