On limiting relations for capacities

Abstract

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