Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups

TL;DR

This paper provides precise bounds and asymptotic estimates for the heat kernel on both isotropic and nonisotropic Heisenberg groups, extending existing results and completing previous work on Grushin operators.

Contribution

It offers new bounds and asymptotic behaviors for the heat kernel on various Heisenberg groups and completes prior results on Grushin operators.

Findings

Precise bounds for heat kernel on isotropic Heisenberg groups.

Asymptotic estimates at infinity for nonisotropic Heisenberg groups.

Complete short-time behavior of the heat kernel on these groups.

Abstract

The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals-Gaveau-Greiner. Third, we complete the results obtained in \cite{Li12} about the heat kernel of Grushin operators.