A Bakry-Émery approach to Lipschitz transportation on manifolds

Abstract

On weighted Riemannian manifolds we prove the existence of globally Lipschitz transport maps between the weight (probability) measure and log-Lipschitz perturbations of it, via Kim and Milman's diffusion transport map, assuming that the curvature-dimension condition $\mathrm{CD}({\rho }_1,\infty )$ holds, as well as a second order version of it, namely ${\Gamma }_3\ge {\rho }_2{\Gamma }_2$. We get new results as corollaries to this result, as the preservation of Poincaré's inequality for the exponential measure on $(0,+\infty )$ when perturbed by a log-Lipschitz potential and a new growth estimate for the Monge map pushing forward the gamma distribution on $(0,+\infty )$ (then getting as a particular case the exponential one), via Laguerre's generator.

21 pages. Minor corrections. Subsection 5.3. improved, in particular Proposition 5.2. To appear in Potential Analysis