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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quasiadditivity and measure property of
capacity and the tangential boundary
behavior of harmonic functions


Authors: H. Aikawa and A. A. Borichev
Journal: Trans. Amer. Math. Soc. 348 (1996), 1013-1030
MSC (1991): Primary 31B15, 31B25
MathSciNet review: 1340166
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01554-1
PII: S 0002-9947(96)01554-1
Keywords: Quasiadditivity of capacity, boundary behavior of harmonic functions, thin set, fine limit, approach region
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
Article copyright: © Copyright 1996 American Mathematical Society