Superharmonic functions associated with hypoelliptic non-H"ormander operators

In this paper, we consider a class of degenerate-elliptic linear operators L 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 L-superharmonic function and we characterize the L-superharmonicity of a function u in terms of the sign of the distribution Lu; we also establish some Riesz- type decomposition theorems and we prove a Poisson–Jensen formula. The operators involved are C∞-hypoelliptic but they do not satisfy the H ̈ormander Rank Condition nor subelliptic estimates or Muckenhoupt-type degeneracy conditions.