# Sobolev spaces with only trivial isometries, II

Sobolev spaces with only trivial isometries, II 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 $$T[C(\overline{U})] = C(\overline{U})$$ . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Sobolev spaces with only trivial isometries, II

, Volume 13 (4) – Feb 20, 2009
10 pages

/lp/springer_journal/sobolev-spaces-with-only-trivial-isometries-ii-ACSE2lyAIX
Publisher
Springer Journals
Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2242-7
Publisher site
See Article on Publisher Site

### 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 $$T[C(\overline{U})] = C(\overline{U})$$ . 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.

### Journal

PositivitySpringer Journals

Published: Feb 20, 2009

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