Precise estimates for the subelliptic heat kernel on H-type groups

TL;DR

This paper derives precise upper and lower bounds for the subelliptic heat kernel on H-type groups, including explicit formulas for the distance and geodesics, advancing understanding of heat distribution in these geometric structures.

Contribution

It provides the first sharp bounds for the heat kernel on H-type groups, including explicit formulas for the distance and geodesics, with bounds on the gradient of the heat kernel.

Findings

Established bounds for the heat kernel involving polynomial corrections

Derived explicit formulas for the Carnot-Carathéodory distance and geodesics

Obtained bounds on the norm of the subelliptic gradient of the heat kernel

Abstract

We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type. Specifically, we show that there exist positive constants $C_1$, $C_2$ and a polynomial correction function $Q_t$ on $G$ such that $$C_1 Q_t e^{-\frac{d^2}{4t}} \le p_t \le C_2 Q_t e^{-\frac{d^2}{4t}}$$ where $p_t$ is the heat kernel, and $d$ the Carnot-Carath\'eodory distance on $G$. We also obtain similar bounds on the norm of its subelliptic gradient $|\nabla p_t|$. Along the way, we record explicit formulas for the distance function $d$ and the subriemannian geodesics of H-type groups.