Hypoelliptic heat kernel inequalities on H-type groups

TL;DR

This paper derives precise bounds for the heat kernel on H-type groups, applies these to gradient estimates of heat solutions, and explores geometric properties of these groups.

Contribution

It provides new explicit heat kernel bounds and gradient estimates for the hypoelliptic sublaplacian on H-type groups, advancing understanding of their geometric analysis.

Findings

Explicit pointwise heat kernel bounds established

Gradient estimates for heat equation solutions derived

Geometric analysis of H-type groups discussed

Abstract

We study inequalities related to the heat kernel for the hypoelliptic sublaplacian on an H-type Lie group. Specifically, we obtain precise pointwise upper and lower bounds on the heat kernel function itself. We then apply these bounds to derive an estimate on the gradient of solutions of the heat equation, which is known to have various significant consequences including logarithmic Sobolev inequalities. We also present a computation of the heat kernel, and a discussion of the geometry of H-type groups including their geodesics and Carnot-Carath\'eodory distance functions.