Global Hölder estimates for optimal transportation

Author: Kolesnikov A.  

Publisher: MAIK Nauka/Interperiodica

ISSN: 0001-4346

Source: Academy of Sciences of the USSR Mathematical Notes, Vol.88, Iss.5-6, 2010-12, pp. : 678-695

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Abstract

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 Hölder with the Hölder 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.