Hausdorff measures, capacities, and Sobolev spaces with weights.(English)Zbl 0723.46024

In the first part of the dissertation an $$\alpha$$-dimensional weighted spherical Hausdorff type measure is constructed and a weighted Hausdorff dimension of a set in $${\mathbb{R}}^ n$$ is introduced. Upper bound for ordinary Hausdorff dimension in terms of the weighted Hausdorff dimension is expressed and the relationship between weighted Hausdorff measures and content densities is investigated. Next upper bounds for content densities in terms of capacity densities is considered. It is shown that if the capacity density of a set is zero at a given point, then the point also has zero content density. After that a relationship between capacity density, linearly increasing gauges and Hausdorff dimensions with respect to a given set is demonstrated. Finally upper bounds for capacity densities in terms of content densities are obtained. In the second part weighted Sobolev spaces as weighted Bessel potentials have been characterized.

MSC:

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