Quasi-bounded and singular functions

Abstract:

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 $\Omega$ be a bounded Euclidean region. With the underlying space taken as the class $\mathcal {M}$ of all nonnegative functions u on $\Omega$ admitting superharmonic majorants, an operator S is introduced by setting Su equal to the regularization of the infimum over $\lambda \geq 0$ of the regularized reduced functions for ${(u - \lambda )^ + }$. Quasi-bounded and singular functions are then defined as those u for which $Su = 0$ and $Su = u$, 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 $\Omega$ is quasi-bounded if and only if there exists a nonnegative, increasing, convex function $\varphi$ on $[0,\infty ]$ such that $\varphi (x)/x \to + \infty$ as $x \to \infty$ and $\varphi \circ u$ admits a superharmonic majorant. Extensions of the theory are made to the vector lattice generated by the positive cone of functions u in $\mathcal {M}$ satisfying $Su \leq u$.