Fundamental solution and Harnack inequality for subelliptic evolution operators on Carnot groups

Abstract

In this paper we study regularity properties of a class of subelliptic evolution operators. We first prove the existence of the fundamental solution by means of Levi's parametrix method, establishing also several key properties. We then employ these results, together with some mean value formulas, to prove a maximum principle and an invariant Harnack inequality for the classical solutions to the equations under study. Our analysis critically relies on the Carnot Caratheodory geometry naturally induced by the vector fields defining these operators.