The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic divergence-form operators

Abstract

In this paper we consider a class of hypoelliptic second-order partial differential operators $\mathcal{L}$ in divergence form on ${\mathbb{R}}^N$, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality for $\mathcal{L}$. 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 $\mathcal{L}$ with $C^{\omega }$ coefficients, this control-theoretic result will also imply a Unique Continuation property for the $\mathcal{L}$-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 $\mathcal{L}$ on a basis of the Euclidean topology.