Global Hölder estimates for optimal transportation
Tóm tắt
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.
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