On Sobolev regularity of mass transport and transportation inequalities. (English. Russian original) Zbl 1279.60032

Kolesnikov, A. V.

Theory Probab. Appl. 57, No. 2, 243-264 (2013); translation from Teor. Veroyatn. Primen. 57, No. 2, 296-321 (2012).

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.

Reviewer: Ludger Rüschendorf (Freiburg i. Br.)

MSC:

60E15	Inequalities; stochastic orderings
49Q20	Variational problems in a geometric measure-theoretic setting
35B65	Smoothness and regularity of solutions to PDEs
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

Monge-Kantorovich problem; Monge-Ampère equation; Sobolev a priori estimates; Gaussian measures; log-concave measures; transportation inequalities; log-Sobolev inequality; Lipschitz mappings