A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators

Abstract

In this paper we are concerned with Harnack inequalities for non-negative solutions u:Ω→ℝ to a class of second order hypoelliptic ultraparabolic partial differential equations in the form $$ \mathcal{L} u:=\sum\limits_{j=1}^m X_j^2u+X_0u-\partial_tu=0 $$ where Ω is any open subset of ℝN + 1, and the vector fields X1, ..., Xm and \(X_0 - \partial_t\) are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x0,t0) ∈ Ω we give a geometric sufficient condition on the compact sets \(K\subseteq {\Omega}\) for which the Harnack inequality $$ \sup\limits_{K}u\le C_K\, u(x_0,t_0) $$ holds for all non-negative solutions u to the equation \(\mathcal{L} u=0\). We also compare our result with an abstract Harnack inequality from potential theory.