On limiting relations for capacities

TL;DR

This paper investigates the limiting behavior of Besov capacities for sets in ^n as the smoothness parameter approaches 0 or 1, revealing different limits for open and compact sets.

Contribution

It establishes the asymptotic limits of scaled Besov capacities as  or 1, highlighting distinctions between open and compact sets.

Findings

For open sets, scaled capacities tend to Sobolev capacities as 1.

For compact sets, scaled capacities tend to a multiple of the measure as 0.

The convergence behavior differs between open and compact sets.

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.