A Bakry-Émery approach to Lipschitz transportation on manifolds

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 $\mathrm{C}\mathrm{D}({\rho }_1,\mathrm{\infty })$ holds, as well as a second order version of it, namely ${\mathrm{\Gamma }}_3\ge {\rho }_2{\mathrm{\Gamma }}_2$. We get new results as corollaries to this result, as the preservation of Poincaré’s inequality for the exponential measure on $(0,+\mathrm{\infty })$ when perturbed by a log-Lipschitz potential and a new growth estimate for the Monge map pushing forward the gamma distribution on $(0,+\mathrm{\infty })$ (then getting as a particular case the exponential one), via Laguerre’s generator.