The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic non-Hörmander operators

Abstract

In this paper we consider a class of hypoelliptic second-order partial differential operators L in divergence form on R N , arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality for L. The involved operators are not assumed to belong to the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates, nor Muckenhoupt-type estimates on the degeneracy of the second order part; indeed our results hold true in the infinitely-degenerate case and for operators which are not necessarily sums of squares. We use a Control Theory result on hypoellipticity in order to recover a meaningful geometric information on connectivity and maxima propagation, yet in the absence of any Hörmander condition. For operators L with C ω coefficients, this control-theoretic result will also imply a Unique Continuation property for the L-harmonic functions. The (Strong) Harnack Inequality is obtained via the Weak Harnack Inequality by means of a Potential Theory argument, and by a crucial use of the Strong Maximum Principle and the solvability of the Dirichlet problem for L on a basis of the Euclidean topology.

Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings

“…By combining Theorem 3.11 with the Harnack inequality for L proved in [3], we easily obtain the following important result. Theorem 3.12.…”

“…Remark 2.1. We explicitly highlight that the C ∞ -hypoellipticity of L η = L − η is crucially exploited in [3] in order to establish a homogeneous non-invariant Harnack inequality for L. As we will see, this result ensures that L endows R N with a structure of harmonic space, in which the Harnack Axiom holds.…”

“…One can argue exactly as in the proof of [19, Thm. 9.1.2], by using the Strong Maximum Principle for H L -harmonic functions established in [3].…”

“…A comparison between two notions of L-Green function. We conclude the section by briefly comparing the notion of L-Green function introduced here with the one introduced by Bony [20] in the context of Hörmander operators (and later extended in [3] to general divergence-form operators).…”

“…To begin with, let B L be the topological basis of R N appearing in the statement of [3,Lem. 1.7] (remind that the elements of B L are H L -regular open sets, see Section 2), and let Ω ∈ B L .…”