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Singular measures and \((1,p)\)-capacity on weighted Sobolev classes. (Russian, English) Zbl 1035.28004

Sib. Mat. Zh. 44, No. 2, 433-437 (2003); translation in Sib. Math. J. 44, No. 2, 346-349 (2003).
The purpose of the article is to study the conditions which guarantee that a contribution of the so-called singular part of a measure into \((1,p)\)-capacity of an arbitrary capacitor is zero.
The main result reads as follows: Let \(E\subset D \subset \mathbb R^n\) be a set of Hausdorff linear measure zero. Then every finite regular Borel measure with support in \(E\) is \(p\)-trivial.

MSC:

28A12 Contents, measures, outer measures, capacities
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E27 Spaces of measures
28A78 Hausdorff and packing measures
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