On Sobolev regularity of mass transport and transportation inequalities

We study Sobolev a priori estimates for the optimal transportation $T = \nabla \Phi$betweenprobabilitymeasures$\mu=e^{-V} \ dx$and$\nu=e^{-W}
dx$on$\R^d$\mathrm{.}Assuminguniformconvexityofthepotential$W$weshowthat$\int | D^2 \Phi|^2_{HS} \ d\mu$,where$|\cdot|_{HS}$ is the Hilbert-Schmidt norm, is controlled by the Fisher information of $\mu$. In addition, we prove similar estimate for the $L^p(\mu )$-norms of $\mathrm{\parallel }{D}^2\mathrm{\Phi }\mathrm{\parallel }$ and obtain some $L^p$-generalizations of the well-known Caffarelli contraction theorem. We establish a connection of our results with the Talagrand transportation inequality. We also prove a corresponding dimension-free version for the relative Fisher information with respect to a Gaussian measure.