Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 31B15, 31B25

Retrieve articles in all journals with MSC (1991): 31B15, 31B25

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

A. A. Borichev
Affiliation: Department of Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden

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