On Limiting Relations for Capacities

Abstract

The paper is devoted to the study of limiting behaviour of Besov capacities \(\text{cap}(E;B_{p,q}^\alpha),\ (0<\alpha <1)\) of sets in \(\mathbb{R}^n\) as \(\alpha\to 1\) or \(\alpha\to 0.\) Namely, let \(E\subset \mathbb{R}^n\) and \[ J_{p,q}(\alpha, E)=[\alpha(1-\alpha)q]^{p/q}\,\text{cap}(E;B_{p,q}^\alpha). \] It is proved that if \(1\le p<n,\,\,1\le q<\infty,\) and the set \(E\) is open, then \(J_{p,q}(\alpha, E)\) tends to the Sobolev capacity \(\text{cap}(E;W_p^1)\) as \(\alpha\to 1\). This statement fails to hold for compact sets. Further, it is proved that if the set \(E\) is compact and \(1\le p,q<\infty\), then \(J_{p,q}(\alpha, E)\) tends to \(2n^p|E|\) as \(\alpha\to 0\) (\(|E|\) is the measure of \(E\)). For open sets it is not true.