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 | References | Similar Articles | Additional Information

Abstract: We show that if a set is dispersely decomposed into subsets, then the capacity of 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 . These properties hold for classical capacities, -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:
https://doi.org/10.1090/S0002-9947-96-01554-1

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