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Integrability of absolutely continuous transformations of measures and applications to optimal mass transportation. (English. Russian original) Zbl 1118.26015
Theory Probab. Appl. 50, No. 3, 367-385 (2006); translation from Teor. Veroyatn. Primen. 50, No. 3, 433-456 (2005).
The authors establish various extensions of Talagrand’s inequality of the following form. Let $$T(x) = x + F (x)$$ be an optimal mass transportation from meausure $$\mu$$ to measure $$\nu = g \cdot \mu$$ on $$\mathbb{R}^d$$, then $$\int \Phi_1 (| F(x)| ) \,d \mu (x) \leq \int \Phi_2 (g (x)) \,d \mu (x)$$ for certain functions $$\Phi_1, \Phi_2$$. Talagrand’s result is the case, where $$\Phi_1 (x) = x^2$$ and $$\Phi_2 (x) = x\log x$$ and $$\mu$$ is a Gaussian measure on $$\mathbb{R}^d$$. Examples included are, where $$\mu$$ is Gaussian (or satisfies the logarithmic Sobolev inequality), $$\Phi_2 (x) = x^p$$ and $$\Phi_1 (x) = \exp (\alpha x^2)$$, or $$\Phi_2 (x) = x | \log x| ^p$$ and $$\Phi_1 (x) = | x| ^r$$. Some general inequalities of this type are established for $$\mu$$ Gaussian or strictly convex measures and for optimal transformations w.r.t. cost functions $$| x-y| ^p$$, $$p \in (1,2]$$. For the case $$p=2$$ also measures satisfying the logarithmic Sobolev inequality and the Poincaré inequality are considered and, e.g., $$L^p$$-estimates of the optimal transportation are given.

##### MSC:
 26D10 Inequalities involving derivatives and differential and integral operators 60E15 Inequalities; stochastic orderings
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