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In this paper we consider a class of hypoelliptic second-order partialdifferential operators $\mathcal{L}$ in divergence form on $\mathbb{R}^N$,arising from CR geometry and Lie group theory, and we prove the Strong and WeakMaximum Principles and the Harnack Inequality for $\mathcal{L}$. The involvedoperators are not assumed to belong to the H\"ormander hypoellipticity class,nor to satisfy subelliptic estimates, nor Muckenhoupt-type estimates on thedegeneracy of the second order part; indeed our results hold true in theinfinitely-degenerate case and for operators which are not necessarily sums ofsquares. We use a Control Theory result on hypoellipticity in order to recovera meaningful geometric information on connectivity and maxima propagation, yetin the absence of any H\"ormander condition. For operators $\mathcal{L}$ with$C^\omega$ coefficients, this control-theoretic result will also imply a UniqueContinuation property for the $\mathcal{L}$-harmonic functions. The (Strong)Harnack Inequality is obtained via the Weak Harnack Inequality by means of aPotential Theory argument, and by a crucial use of the Strong Maximum Principleand the solvability of the Dirichlet problem for $\mathcal{L}$ on a basis ofthe Euclidean topology.

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