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 : → R to a class of second order hypoelliptic ultraparabolic partial differential equations in the formwhere is any open subset of R N+1 , and the vector fields X 1 , . . . , X m and X 0 − ∂ t are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x 0 , t 0 ) ∈ we give a geometric sufficient condition on the compact sets K ⊆ for which the Harnack inequalityholds for all non-negative solutions u to the equation L u = 0. We also compare our result with an abstract Harnack inequality from potential theory.

A boundary estimate for non-negative solutions to Kolmogorov operators in non-divergence form

“…Indeed, according to [7,Proposition 4.5], the cylinder Q − R (x 0 , t 0 ) has to be contained in Int A (x 0 ,t 0 ) .…”

“…We follow the same argument used in [7,Theorem 3.2]. We summarize the proof for the reader's convenience.…”

Kolmogorov–Fokker–Planck equations: Comparison principles near Lipschitz type boundaries

“…In this section we briefly discuss, without giving the complete proofs, to the extent one can generalize Theorems 1.1, 1.2 and 1.3 to the context of a subset of the more general operators of Kolmogorov type considered in [CNP1], [CNP2] and [CNP3]. In [CNP1], [CNP2] and [CNP3] we considered Kolmogorov operators of the form …”

“…In this case (1.19) A (0,0,0) (Ω) = (x, y, t) ∈ Ω : |y| ≤ −tR) , and one can prove, see [CNP1], that there exists a non-negative solution u to Ku = 0 in Ω such that u ≡ 0 in A (0,0,0) (Ω) and such that u > 0 in Ω \ A (0,0,0) (Ω).…”

“…In [CNP1], [CNP2] and [CNP3], we developed a number of important preliminary estimates concerning the boundary behavior of non-negative solutions to equations of Kolmogorov-Fokker-Planck type in Lipschitz type domains. These papers were the results of our ambition to understand to the extent, and in what sense, scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures, previously established for uniformly parabolic equations with bounded measurable coefficients in Lipschitz type domains, see [FS], [FSY], [SY], [FGS], [N], [Sa], can be established for non-negative solutions to the equation Ku = 0 and for more general equations of Kolmogorov-Fokker-Planck type.…”