Uniform Gaussian estimates for the fundamental solutions for heat operators on Carnot groups

Abstract

In this paper, we are concerned with existence, qualitative properties, and uniform Gaussian estimates of the global fundamental solutions of a family of heat operators on Carnot groups. As a byproduct, we obtain existence and uniqueness theorems of Thychonov type for the Cauchy problem. Our effort here is also to provide simple and direct proofs relying on few basic tools such as invariant Harnack inequalities and maximum principles. In our study, we thoroughly exploit some structural properties of Carnot groups pointed out in the previous paper [4].