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Traces of Sobolev functions on fractal type sets and characterization of extension domains. (English) Zbl 0877.46025

The authors describe traces of Sobolev functions \(u\in W^{1,p}(\mathbb{R}^n)\), \(1<p\leq\infty\), on certain subsets of \(\mathbb{R}^n\) in terms of Sobolev spaces on metric spaces [see P. Hajłasz, Potential Anal. 5, No. 4, 403-415 (1996; Zbl 0859.46022)]. Their results apply to smooth submanifolds, fractal subsets, as well to open subsets of \(\mathbb{R}^n\). In particular if \(\Omega\subset\mathbb{R}^n\) is a John domain, then the authors characterize those \(W^{1,p}(\Omega)\) functions which can be extended to \(W^{1,p}(\mathbb{R}^n)\).
In the case of traces on fractal subsets their results are related to those of A. Jonsson and H. Wallin, “Function spaces on subsets of \(\mathbb{R}^n\)” (1984).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Citations:

Zbl 0859.46022
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