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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quasi-bounded and singular functions

Authors: Maynard Arsove and Heinz Leutwiler
Journal: Trans. Amer. Math. Soc. 189 (1974), 275-302
MSC: Primary 31C05
MathSciNet review: 0379872
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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$.

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Keywords: Quasi-bounded functions, singular functions, superharmonic functions, harmonic functions, potentials, projection operators
Article copyright: © Copyright 1974 American Mathematical Society