The strong maximum principle and the Harnack inequality for a class of hypoelliptic non-H"ormander operators

We consider a class of hypoelliptic second-order operators L in divergence form, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality. The operators are not assumed in the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates or Muckenhoupt-type degeneracy conditions; 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 to recover a meaningful geometric information on connectivity and maxima propagation, in the absence of any maximal rank condition. For operators L with C^ω coefficients, this control- theoretic result also implies a Unique Continuation property for the L–harmonic functions. The Harnack theorem is obtained via a weak Harnack inequality by means of a Potential Theory argument and the solvability of the Dirichlet problem for L.