On Global Hölder Estimates for Optimal Transportation
نویسنده
چکیده
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 → R be an optimal transportation pushing forward μ = e dx to ν = e 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 Aq|x|, q ≥ 1. Under these assumptions we show that T is globally Hölder with a dimension-free coefficient. In addition, we study optimal transportation T between μ and the uniform measure on a bounded convex set K ⊂ R. We get estimates for the Lipschitz constant of T in terms of d, diam(K) and DV,DV . We recover in this way some functional and concentration inequalities for log-concave measures.
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