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

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 where Ω is any open subset of ℝ N + 1 , and the vector fields X 1 , ..., X m and are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed ( x 0 , t 0 ) ∈ Ω we give a geometric sufficient condition on the compact sets for which the Harnack inequality holds for all non-negative solutions u to the equation . We also compare our result with an abstract Harnack inequality from potential theory.