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Transactions of the American Mathematical Society

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

 

 

Sets of determination for harmonic functions


Author: Stephen J. Gardiner
Journal: Trans. Amer. Math. Soc. 338 (1993), 233-243
MSC: Primary 31B05
DOI: https://doi.org/10.1090/S0002-9947-1993-1100694-2
MathSciNet review: 1100694
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Abstract: Let $ h$ denote a positive harmonic function on the open unit ball $ B$ of Euclidean space $ {{\mathbf{R}}^n}\;(n \geq 2)$. This paper characterizes those subsets $ E$ of $ B$ for which $ {\sup _E}H/h = {\sup _B}H/h$ or $ {\inf _E}H/h = {\inf _B}H/h$ for all harmonic functions $ H$ belonging to a specified class. In this regard we consider the classes of positive harmonic functions, differences of positive harmonic functions, and harmonic functions with a one-sided quasi-boundedness condition. We also consider the closely related question of representing functions on the sphere $ \partial B$ as sums of Poisson kernels corresponding to points in $ E$.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1100694-2
Article copyright: © Copyright 1993 American Mathematical Society