Dimension‐Independent Functional Inequalities by Tensorization and Projection Arguments

ABSTRACT We study stability under tensorization and projection‐type operations of gradient estimates and other functional inequalities for Markov semigroups on metric spaces. Using transportation inequalities obtained by Baudoin and Eldredge in 2021, we prove that constants in the gradient estimates can be chosen to be dimension‐independent. Our results are applicable to hypoelliptic diffusions on sub‐Riemannian manifolds and some hypocoercive diffusions. As a byproduct, we obtain dimension‐independent reverse Poincaré inequality, reverse logarithmic Sobolev inequality, and gradient bounds on Lie groups with transverse symmetries and for non‐isotropic Heisenberg groups.