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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Yoshihiro Mizuta
Journal: Proc. Amer. Math. Soc. 126 (1998), 1043-1047.
MSC (1991): Primary 31B25, 31B15, 46E35
MathSciNet review: 1443397
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Abstract: In this note we aim to complete the results by Koskela concerning the radial uniqueness for Sobolev functions.

Let $\varphi$ be a positive nonincreasing function on the interval $(0,\infty)$ , and let $\mathbf{B}$ denote the unit ball of $R^n$ . Consider a $p$ -precise function $u$ on $\mathbf{B}$ such that

$\begin{displaymath}\int _{U(\varepsilon)} |\nabla u(x)|^p dx \leqq \varepsilon^p \varphi(\varepsilon) \hspace{2em} \text{for any $\varepsilon > 0$,} \end{displaymath}$

where $U(\varepsilon) = \{x \in \mathbf{B} : |u(x)|< \varepsilon\}$ . We give conditions on $\varphi$ which assure that $u = 0$ whenever $u$ has vanishing fine boundary limits on a set of positive $p$ -capacity.

We are also concerned with the sharpness.

References:

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 $p$ -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, Gakk $\overline{{o}}$ tosyo, Tokyo, 1996. CMP 97:06
8.: M. Ohtsuka, Extremal length and precise functions in $3$ -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: 10.1090/S0002-9939-98-04296-8
PII: S 0002-9939(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
Copyright of article: Copyright 1998, American Mathematical Society

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