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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.
Positivity – Springer Journals
Published: Feb 20, 2009
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