International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 1, Pages 41-63

Estimates for capacities and traces of potentials

V. G. Maz'ja1 and S. P. Preobrazenskii2

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.


It is shown that “isoperimetric” inequalities, relating measures and capacities, hold for all sets in n 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.