Dimension-independent functional inequalities by tensorization and projection arguments

TL;DR

This paper establishes dimension-independent functional inequalities for Markov semigroups on metric spaces by leveraging tensorization and projection techniques, with applications to hypoelliptic diffusions and Lie groups.

Contribution

It introduces a novel approach to achieve dimension-independent gradient and functional inequalities using tensorization and projection arguments.

Findings

Constants in gradient estimates can be made dimension-independent.

Applicable to hypoelliptic diffusions on sub-Riemannian manifolds.

Provides dimension-independent reverse Poincaré and logarithmic Sobolev inequalities.

Abstract

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