Quasi-bounded and singular functions

A general formulation is given for the concepts of quasi-bounded and singular functions, thereby extending to a much broader class of functions the concepts initially formulated by Parreau in the harmonic case. Let Ω be a bounded Euclidean region. With the underlying space taken as the class M M of all nonnegative functions u on Ω admitting superharmonic majorants, an operator S is introduced by setting Su equal to the regularization of the infimum over λ ≥ 0 0 of the regularized reduced functions for (u − λ) + (u -) ^ +. Quasi-bounded and singular functions are then defined as those u for which S u = 0 Su = 0 and S u = u Su = u, respectively. A development based on properties of the operator S leads to a unified theory of quasi-bounded and singular functions, correlating earlier work of Parreau (1951), Brelot (1967), Yamashita (1968), Heins (1969), and others. It is shown, for example, that a nonnegative function u on Ω is quasi-bounded if and only if there exists a nonnegative, increasing, convex function φ on 0, ∞ 0, such that φ (x) / x → + ∞ (x) /x