International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 1, Pages 41-63
doi:10.1155/S0161171284000053
Estimates for capacities and traces of potentials
1Institute of Mathematics and Mechanics, Leningrad University, Leningrad, Russia
2Department of Mathematics, Leningrad Polytechnical Institute, Leningrad, Russia
Received 4 June 1981
Copyright © 1984 V. G. Maz'ja and S. P. Preobrazenskii. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.