On gradient bounds for the heat kernel on the Heisenberg group

TL;DR

This paper provides simple proofs of gradient bounds for the heat kernel on the Heisenberg group and explores their implications for functional inequalities, enhancing understanding of hypoelliptic diffusion processes.

Contribution

It offers simplified proofs of existing gradient bounds and investigates their applications to isoperimetric inequalities in the sub-Riemannian setting.

Findings

Simplified proofs of gradient bounds for the heat kernel.

Derivation of Cheeger and Bobkov type isoperimetric inequalities.

Enhanced understanding of hypoelliptic diffusion on the Heisenberg group.

Abstract

It is known that the couple formed by the two dimensional Brownian motion and its L\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.