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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Polydiscs and nontangential limits


Author: Kohur GowriSankaran
Journal: Proc. Amer. Math. Soc. 115 (1992), 977-984
MSC: Primary 31B25; Secondary 32A40
DOI: https://doi.org/10.1090/S0002-9939-1992-1113640-7
MathSciNet review: 1113640
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Abstract: A well-known result states that for all bounded $ n$-harmonic functions on the polydisc $ {\mathbb{D}^n}$ the nontangential limits exist for (Lebesgue) almost every element of the $ n$-torus. In this paper it is shown that a similar result is not in general valid for bounded quotients of two positive $ n$-harmonic functions. Necessary and sufficient conditions on a $ n$-harmonic function $ u > 0$ are given to ensure the existence "almost everywhere" of the nontangential limits of the quotients $ w/u$ in the case (i) for all $ n$-harmonic functions $ w$ such that $ w/u$ is bounded and in the case (ii) for all $ n$-harmonic functions $ w$ that are $ u$-quasi-bounded.'


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DOI: https://doi.org/10.1090/S0002-9939-1992-1113640-7
Keywords: Nontangential limit, $ n$-harmonic functions, polydisc, quasi-bounded functions, fine limit
Article copyright: © Copyright 1992 American Mathematical Society