Estimates for capacities and traces of potentials

Abstract

It is shown that “isoperimetric” inequalities, relating measures and capacities, hold for all sets in ℝn if they are valid for all balls. As a corollary, the necessary and sufficient conditions for the continuity of some imbeddings of M. Riesz and Bessel potential spaces are obtained. The introduction contains a survey of known results on inequalities of Sobolev type.

Keywords

• Sobolev spaces
• Bessel potential spaces
• imbedding theorems
• capacities
• isoperimetric inequalities.