On limiting relations for capacities

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