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We consider a class of degenerate Ornstein-Uhlenbeck operators in R N , of the kindTo get this estimate we use in a crucial way the left invariance of L with respect to a Lie group structure in R N +1 and some results on singular integrals on nonhomogeneous spaces recently proved in Bramanti (Revista Matematica Iberoamericana, 2009, in press).

We show an invariant Harnack inequality for a class of hypoelliptic ultraparabolic operators with underlying homogeneous Lie group structures. As a byproduct we prove a Liouville type theorem for the related “stationary” operators. We also introduce a notion of link of homogeneous Lie Groups that allows us to show that our results apply to wide classes of operators

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