Lattice homomorphisms between Sobolev spaces

Abstract

Abstract. Given bounded domains Ω 1 and Ω 2 in R N and an isometry T from W 1,p (Ω 1 ) to W 1,p (Ω 2 ), we give sufficient conditions ensuring that T corresponds to a rigid motion of the space, i.e., T u = ±(u • ξ) for an isometry ξ, and that the domains are congruent. More general versions of the involved results are obtained along the way.

Riesz* homomorphisms on pre-Riesz spaces consisting of continuous functions

“…(8) is satisfied which yields the following corollary. Theorem 7.7 is very similar to a result by Biegert, which states that any Riesz homomorphisms on W 1, p 0 ( ) is a weighted composition operator, Theorem 4.4 in [4]. In his proof Biegert does not use the order structure of the space W 1, p ( ) nor the Sobolev Embedding Theorem.…”

“…Moreover, a linear operator T : E → F is a Riesz homomorphism if and only if it satisfies (4). Using this characterization and Theorem 3.2 we prove that any Riesz* homomorphism on a pointwise order dense subspace of C(X ) is automatically a Riesz homomorphism.…”

Isometries between Sobolev spaces

“…The proof of Theorem 3.5 is based on the following representation theorem for lattice isomorphisms on Sobolev spaces. It can be obtained from 9, Theorem 3.5] with arguments that are similar to those in 9, Section 4.1], and we leave the details to the reader.…”

“…Unfortunately, we do not know whether such a theorem is true. Actually, the statement that Tu = g · ( u ○ξ) for all u ∈ W 1, p (Ω 1 ) whenever T is a lattice homomorphism from W 1, p (Ω 1 ) to L q (Ω 2 ) is precisely the content of Theorem 4.13 in 9, but the proof of that theorem is faulty in that it uses the false statement that every lattice homomorphism from L ∞ (Ω 1 ) to L q (Ω 2 ) is of the form Tu = g · ( u ○ξ).…”

“…Proof. There exist ζ and h such that Tu = h · ( u ○ζ) p ‐quasi everywhere for the p ‐quasi continuous representatives whenever \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$u \in \mathscr {W}^{1,p}(\Omega _1)$\end{document} 9, Theorem 4.5], so in particular for u in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^{1,p}_0(\Omega _1)$\end{document}; we remark that in the article 9 functions in \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathscr {W}^{1,p}(\Omega _1)$\end{document} are defined on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\overline{\Omega }_1$\end{document}, and we can set u = 0 on ∂Ω 1 for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$u \in W^{1,p}_0(\Omega _1)$\end{document}.…”