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 : Omega --> R to a class of second order hypoelliptic ultraparabolic partial differential equations in the form Lu: = Sigma(m)(j=1) X(j)(2)u + X0u - partial derivative(i)u = 0 where Omega is any open subset of RN+1, and the vector fields X-1,..., X-m and X-0 - partial derivative(i) are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x(0), t(0)) is an element of Omega we give a geometric sufficient condition on the compact sets K subset of Omega for which the Harnack inequality sup u <= C-K u(x(0), t(0)) K holds for all non-negative solutions u to the equation Lu = 0. We also compare our result with an abstract Harnack inequality from potential theory.