Superharmonic functions associated with hypoelliptic non-Hörmander operators

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.

A Decomposition Theorem for Positive Superharmonic Functions

Let X be a harmonic space in the sense of C. Constantinescu and A. Cornea. We show that, for any subset E of X, a positive superharmonic function u on X has a representation u = p + h, where p is the greatest specific minorant of u satisfying . This result is a generalization of a theorem of M. Brelot. We also state some characterizations of extremal superharmonic functions.

A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces

Harmonic functions with certain boundedness properties on a given open Riemann surface R attain their maxima and minima on the harmonic boundary ΔB of R. The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].For the corresponding problem in higher dimensions we shall first show that the complement of ΔR with respect to the Royden boundary ΓR of a Riemannian N-space R is harmonically negligible: given any non-empty compact subset E of ΓR – ΔR there exists an Evans superharmonic function v, i.e., a positive continuous function on R* = R ∪ ΓR, superharmonic on R, with v = 0 on ΔR, v ≡ ∞ on E, and with a finite Dirichlet integral over R.

Positively Infinite Singularities of a Superharmonic Function

Let E be a compact set of logarithmic capacity zero in the complex plane. Then the following is well-known as Evans-Selberg’s theorem [1] [8]: there is a measure with support contained in E such that its logarithmic potential is positively infinite at each point of E. But such a potential does not exist for E of logarithmic positive capacity. Now suppose that E is contained in the circumference of the unit disc |z| < 1 and is of linear measure zero.

Superharmonic Functions in a Domain of a Riemann Surface

Let R be a Riemann surface. Let G be a domain in R with relative boundary ∂G of positive capacity. Let U(z) be a positive superharmonic function in G such that the Dirichlet integral D(min(M,U(z))) < ∞ for every M. Let D be a compact domain in G.