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Proceedings of the American Mathematical Society

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

 

 

An example concerning parts and Newtonian capacity


Author: James Li Ming Wang
Journal: Proc. Amer. Math. Soc. 75 (1979), 218-220
MSC: Primary 46J10
DOI: https://doi.org/10.1090/S0002-9939-1979-0532139-3
MathSciNet review: 532139
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Abstract: We prove the following theorem: Let $ \phi $ be an admissible function with $ \phi ({0^ + }) = 0$ and p a nonnegative integer. Then there is a compact set X in the plane and $ x \in X$ such that x is a nonpeak point of $ R(X)$,

$\displaystyle \sum {{2^{(p + 1)n}}\phi {{({2^{ - n}})}^{ - 1}}\mathcal{C}({A_n}(x)\backslash P(x)) < \infty ,} $

while $ \Sigma {2^{(p + 1)n}}\phi {({2^{ - n}})^{ - 1}}\gamma ({A_n}(x)\backslash X) = \infty $, where $ {A_n}(x) = \{ {2^{ - n - 1}} \leqslant \vert z - x\vert \leqslant {2^{ - n}}\} ,P(x)$ denotes the Gleason part of $ x,\gamma $ the analytic capacity and $ \mathcal{C}$ the Newtonian capacity.

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DOI: https://doi.org/10.1090/S0002-9939-1979-0532139-3
Keywords: Gleason part, Newtonian capacity, analytic capacity
Article copyright: © Copyright 1979 American Mathematical Society