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

Romanov, A. S.

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.

Reviewer: V. Grebenev (Novosibirsk)

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

Keywords:

capacity; singular measure; Sobolev space; integral relation; Hausdorff linear measure zero; regular Borel measure