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

TL;DR

This paper establishes a gradient inequality for the subelliptic heat kernel on H-type groups, extending previous results from the Heisenberg group and utilizing heat kernel estimates and a novel proof approach.

Contribution

It introduces a gradient inequality for subelliptic heat kernels on H-type groups, generalizing prior work on the Heisenberg group with a new proof method.

Findings

Proved gradient inequality for subelliptic heat kernel on H-type groups

Extended Li's result from the Heisenberg group to H-type groups

Utilized pointwise heat kernel estimates and a novel proof approach

Abstract

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type: $$|\nabla P_t f| \le K P_t(|\nabla f|)$$ where $P_t$ is the heat semigroup corresponding to the sublaplacian on $G$, $\nabla$ is the subelliptic gradient, and $K$ is a constant. This extends a result of H.-Q. Li for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approach used by Bakry, Baudoin, Bonnefont, and Chafa\"i.