Abstract

We prove a boundary Harnack inequalities for non-negative solutions to a class of second order degenerate differential operators of Kolmogorov type. Due to the degeneracy of the differential operator, we require geometric conditions on the boundary of the domain and on the compact set where the Harnack inequality applies. Specifically, we rely on the existence of interior and exterior cones to the boundary of the domain, and we show that an "attainability condition" on the compact set is sufficient for the validity of the boundary Harnack inequality. These conditions are satisfied by a wide class of Lipschitz domains. Our main result is a scale-invariant boundary estimate, usually referred to as Carleson type estimate, that generalizes previous results valid for second order uniformly parabolic equations.