On Sobolev regularity of mass transport and transportation inequalities (Q2845211)

Let \(T=\nabla \Phi\) be the optimal transportation between probability measures \(\mu = e^{-V} \, dx \) and \(\nu = e^{-W} \, dx\) where \(W\) is uniformly convex. It is established that \(\int \| D^2 \Phi \|_{HS}\), where \(\| \;\|_{HS}\) is the Hilbert-Schmidt norm, is controlled by the Fisher information \(I_{\mu} = \int |\nabla V|^2 \, d\mu\) of \(\mu\). This result can be considered as a Sobolev a priori estimate for the Monge-Ampère equation \(e^{-V} = e^{-W (\nabla \Phi)} \det D^2 \Phi\). The paper gives also a similar estimate for the \(L^p(\mu)\)-norm of \(\| D^2 \Phi \|\) and gives an \(L^p\)-generalization of the Caffarelli contraction theorem. The author finally establishes a relation to a generalized Talagrand inequality and gives a dimension-free version of the main inequality. The results in the paper are based essentially on probabilistic arguments.