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

TL;DR

This paper establishes maximum principles and Harnack inequalities for a broad class of hypoelliptic divergence-form operators, including infinitely-degenerate cases, using control theory and potential theory without requiring Hörmander conditions.

Contribution

It extends maximum principles and Harnack inequalities to hypoelliptic operators beyond traditional assumptions, including infinitely-degenerate and non-sum-of-squares operators, via a novel control-theoretic approach.

Findings

Maximum principles hold for a wide class of hypoelliptic operators.

Harnack inequalities are established without Hörmander or subelliptic assumptions.

Results include operators with infinitely-degenerate structures and non-sum-of-squares forms.

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 H\"ormander 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 H\"ormander 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.