On limiting relations for capacities. (English) Zbl 1276.46027

Kolyada, V. I.

Real Anal. Exch. 38(2012-2013), No. 1, 211-240 (2013).

Let \(\mathrm{cap}(E; B^\alpha_{p,q})\) be the Besov capacity of a set \(E\) in \(\mathbb{R}^n\). The limiting behavior of this entity is studied as \(\alpha \to 1\) or \(\alpha \to 0\). In particular, it is proved that if \(1\leq p<n\), \(1\leq q<\infty\), and \(E\) is open, then \[ J_{p,q}(\alpha, E)=[\alpha (1- \alpha)q]^{p/q} \mathrm{cap}(E; B^\alpha_{p,q}) \] tends to the Sobolev capacity \(\mathrm{cap}(E; W^1_{p})\) as \(\alpha \to 1\). The case when \(E\) is compact is also considered.

Reviewer: Boris Rubin (Baton Rouge)

MSC:

46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
31B15	Potentials and capacities, extremal length and related notions in higher dimensions

Keywords:

capacity; moduli of continuity; Sobolev spaces; Besov spaces

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