A note on gradient estimates for the heat semigroup on nonisotropic Heisenberg groups

Abstract

In this note we obtain gradient estimates for the heat semigroup on nonisotropic Heisenberg groups. More precisely, our aim is to get the H.-Q. Li inequality on nonisotropic Heisenberg groups, which is a generalization of the original result on the classical Heisenberg group of dimension 3 and a counterpart of the inequality on the H-type groups. Our proof is based on a Cheeger type inequality, which is an approach proposed by Bakry et al.