DICHOTOMY OF GLOBAL CAPACITY DENSITY
Abstract
Let 1 < p < infinity and let d mu(x) = w(x) dx be a p-admissible weight in R-n, n >= 2. By Cap(p, mu)(E, D) we denote the variational (p, mu)-capacity of condenser (E, D). We show a dichotomy of the global density with respect to Cap(p, mu). One of our results is as follows: Let lambda > 1 and let B(x, r) stand for the open ball with center at x and radius r. Then lim(r ->infinity) (inf(x is an element of Rn) Cap(p, mu)(E boolean AND B(x, r), B(x, lambda r))/Cap(p, mu)(B(x, r), B(x, lambda r))) is equal to either 0 or 1; the first case occurs if and only if inf(x is an element of Rn) Cap(p, mu)(E boolean AND B(x, r(0)), B(x, lambda r))/Cap(p, mu)(B(x, r), B(x, lambda r)) is identically equal to 0. This provides a sharp contrast between capacity and Lebesgue measure.
Journal
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- Proceedings of the American mathematical society
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Proceedings of the American mathematical society 143 (12), 5381-5393, 2015-12
American Mathematical Society
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Keywords
Details 詳細情報について
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- CRID
- 1050282813996063104
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- NII Article ID
- 120005682128
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- HANDLE
- 2115/60452
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- ISSN
- 00029939
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- Text Lang
- en
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- Article Type
- journal article
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- Data Source
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- IRDB
- CiNii Articles