Remarks on the results by Koskela concerning the radial uniqueness for Sobolev functions

Author:
Yoshihiro Mizuta

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1043-1047

MSC (1991):
Primary 31B25, 31B15, 46E35

DOI:
https://doi.org/10.1090/S0002-9939-98-04296-8

MathSciNet review:
1443397

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we aim to complete the results by Koskela concerning the radial uniqueness for Sobolev functions.

Let be a positive nonincreasing function on the interval , and let denote the unit ball of . Consider a -precise function on such that

where . We give conditions on which assure that whenever has vanishing fine boundary limits on a set of positive -capacity.

We are also concerned with the sharpness.

**1.**P. Koskela, A radial uniqueness theorem for Sobolev functions, Bull. London Math. Soc. 27 (1995), 460-466. MR**96e:31010****2.**N. G. Meyers, A theory of capacities for potentials in Lebesgue classes, Math. Scand. 26 (1970), 255-292. MR**43:3474****3.**N. G. Meyers, Taylor expansion of Bessel potentials, Indiana Univ. Math. J. 23 (1974), 1043-1049. MR**50:980****4.**Y. Mizuta, Existence of various boundary limits of Beppo Levi functions of higher order, Hiroshima Math. J. 9 (1979), 717-745. MR**81d:31013****5.**Y. Mizuta, Boundary behavior of -precise functions on a half space of , Hiroshima Math. J. 18 (1988), 73-94. MR**89d:31014****6.**Y. Mizuta, Continuity properties of potentials and Beppo-Levi-Deny functions, Hiroshima Math. J. 23 (1993), 79-153. MR**94d:31005****7.**Y. Mizuta, Potential theory in Euclidean spaces, Gakktosyo, Tokyo, 1996. CMP**97:06****8.**M. Ohtsuka, Extremal length and precise functions in -space, Lecture Notes, Hiroshima University, 1973.**9.**Yu. G. Reshetnyak, The concept of capacity in the theory of functions with generalized derivatives, Siberian Math. J. 10 (1969), 818-842. MR**43:2234****10.**W. P. Ziemer, Extremal length as a capacity, Michigan Math. J. 17 (1969), 117-128. MR**42:3299****11.**W. P. Ziemer, Weakly differentiable functions, Springer-Verlag, New York, 1989. MR**91e:46046**

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Additional Information

**Yoshihiro Mizuta**

Affiliation:
The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739, Japan

Email:
mizuta@mis.hiroshima-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-98-04296-8

Keywords:
$p$-precise functions,
Sobolev functions,
capacity,
fine boundary limits

Received by editor(s):
September 18, 1996

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 1998
American Mathematical Society