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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions
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by H. Aikawa and A. A. Borichev PDF
Trans. Amer. Math. Soc. 348 (1996), 1013-1030 Request permission

Abstract:

We show that if a set $E$ is dispersely decomposed into subsets, then the capacity of $E$ is comparable to the summation of the capacities of the subsets. From this fact it is derived that the Lebesgue measure of a certain expanded set is estimated by the capacity of $E$. These properties hold for classical capacities, $L^{p}$-capacities and energy capacities of general kernels. The estimation is applied to the boundary behavior of harmonic functions. We introduce a boundary thin set and show a fine limit type boundary behavior of harmonic functions. We show that a thin set does not meet essentially Nagel-Stein and Nagel-Rudin-Shapiro type approaching regions at almost all bounary points.
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Additional Information
  • H. Aikawa
  • Affiliation: Department of Mathematics, Faculty of Science, Kumamoto University, Kumamoto 860, Japan
  • Address at time of publication: Department of Mathematics and Computer Science, Shimane University, Matsue 690, Japan
  • Email: haikawa@shimane-u.ac.jp
  • A. A. Borichev
  • Affiliation: Department of Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden
  • Email: borichev@math.uu.se
  • Received by editor(s): April 25, 1994
  • Additional Notes: This work was started when the first author visited the Department of Mathematics, University of Uppsala. He acknowledges support from the Royal Swedish Academy of Sciences and the Japan Society of Promotion of Science.

  • Dedicated: Dedicated to Professor F.-Y. Maeda on the occasion of his sixtieth birthday
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 1013-1030
  • MSC (1991): Primary 31B15, 31B25
  • DOI: https://doi.org/10.1090/S0002-9947-96-01554-1
  • MathSciNet review: 1340166