Superharmonic functions associated with hypoelliptic non-Hörmander operators

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.

Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings

“…In that paper, however, the Poisson-Jensen formula is established only for Ω = Ω r (x). (2) Taking into account the results presented at the end of Section 3, we deduce that all the representation theorems established in [2] when Ω ∈ B L are particular cases of the ones proved in this section.…”

“…It is worth mentioning that Theorem 1.3 is exploited in [13] in order to prove the validity of a Weak Maximum Principle on unbounded domains for L. Moreover, Theorems 1.1-1.2 extend and generalize some results contained in [2].…”

Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

“…Under these assumptions, a satisfactory Potential Theory for L can be constructed (see, e.g., [3,4]). In this theory, the "harmonic" functions are the L-harmonic functions, that is, the (smooth) solutions to…”

“…A simple yet remarkable consequence of (1.2) is the fact that a function u in C 2 (Ω, R) is L-subharmonic in Ω if and only if Lu ≥ 0 on Ω (see, e.g., [3]).…”

“…It is proved in [3] that, if Ω ⊆ R N is an open set and u ∈ L(Ω) (not necessarily bounded above), then u ∈ L 1 loc (Ω) and Lu ≥ 0 in the sense of distribution on Ω. Hence, the L-Riesz measure µ of u is defined by µ := Lu.…”