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Sobolev spaces with only trivial isometries, II

Authors:	Geoff Diestel

Institution:	(1) Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

Abstract:	In this article, we obtain a canonical form for surjective linear isometries $T : W^k_p(U) \rightarrow W^k_p(U)$ provided U is an open, bounded, connected, domain with Lipschitz boundary, $1\leq p < \infty, p \neq 2$ and $TC(\overline{U})] = C(\overline{U})$ . We will show there exists \|c\| = 1 and mapping τ that is a composition of a translation and a sign-changing permutation of coordinates such that Tf = cf(τ). As a corollary, if $k > \frac{n}{p}$ , all surjective isometries $T : W^k_p(U) \rightarrow W^k_p(U)$ have this trivial form by the Sobolev Imbedding Theorem.

Keywords:	Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 42B20 42B25 Secondary 46B70 47B38
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