  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.
