**
J. Rákosník (ed.), **

Function spaces, differential operators and nonlinear analysis.

Proceedings of the conference held in Paseky na Jizerou, September 3-9, 1995.

Mathematical Institute, Czech Academy of Sciences, and Prometheus Publishing House, Praha 1996

p. 17 - 26

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On compact embeddings of Sobolev spaces and extension operators which preserve some smoothness

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Viktor I. Burenkov, W. Desmond Evans

Department of Mathematics College of Cardiff, University of Wales, 23 Senghennydd Rd, Cardiff CF2 4YH, United Kingdom burenkov@cardiff.ac.uk Department of Mathematics College of Cardiff, University of Wales, 23 Senghennydd Rd, Cardiff CF2 4YH, United Kingdom w.d.evans@cardiff.ac.uk

**Abstract:** It is well-known that there are bounded domains $\Omega \subset {\Bbb R^{n}}$ whose boundaries $\partial \Omega $ are not smooth enough for there to exist a bounded linear extension of the Sobolev space $ W_{p}^{1}(\Omega)$ into $W_{p}^{1}({\Bbb R^{n}})$ but the embedding $W_{p}^{1}(\Omega) \subset L_{p}(\Omega)$ is nevertheless compact. For a Lip $\gamma$ $(0<\gamma<1)$ boundary there ]exists an extension of $W_{p}^{1}(\Omega)$ into $W_{p}^{\gamma}({\Bbb R^{n}})$, but not into $W_{p}^{1}({\Bbb R^{n}})$ in general, and the smoothness retained by this extension is enough to ensure that the embedding $W_{p}^{1}(\Omega) \subset L_{p}(\Omega)$ is compact. It is natural to ask if this is typical for bounded domains which are such that $W_{p}^{1}(\Omega) \subset L_{p}(\Omega)$ is compact, that is, there exists a bounded extension into a space of functions in ${\Bbb R^{n}}$ which enjoy adequate smoothness. This is the question which will be discussed in the lecture. A central feature on the analysis is a Hardy-type inequality for differences.

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