No 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.

No abstract

In this paper, we consider a class of degenerate-elliptic linear operators [Formula: see text] in quasi-divergence form and we study the associated cone of superharmonic functions. In particular, following an abstract Potential-Theoretic approach, we prove the local integrability of any [Formula: see text]-superharmonic function and we characterize the [Formula: see text]-superharmonicity of a function [Formula: see text] in terms of the sign of the distribution [Formula: see text]; we also establish some Riesz-type decomposition theorems and we prove a Poisson–Jensen formula. The operators involved are [Formula: see text]-hypoelliptic but they do not satisfy the Hörmander Rank Condition nor subelliptic estimates or Muckenhoupt-type degeneracy conditions.

We characterize the sphere with radius tan 2 r = 2n + 1 in the complex projective space CP n as the unique stable hypersurface subject to certain bounds on the curvatures.