On Sobolev regularity of mass transport and transportation inequalities

Abstract: We study Sobolev a priori estimates for the optimal transportation $T = \nabla \Phi$ between probability measures $\mu=e^{-V} \ dx$ and $\nu=e^{-W} \ dx$ on $\R^d$. Assuming uniform convexity of the potential $W$ we show that $\int | D^2 \Phi|^2_{HS} \ d\mu$, 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 $|D^2 \Phi|$ 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.