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Composition of \(q\)-quasiconformal mappings and functions in Orlicz-Sobolev spaces. (English) Zbl 1291.30133

Summary: Let \(\Omega \subset \mathbb{R}^{n}, q\geq n\) and \({\alpha}\geq 0\) or \(1<q\leq n\) and \(\alpha\leq 0\). We prove that the composition of \(q\)-quasiconfomal mapping \(f\) and function \[ u\in WL^{q}\log^{\alpha}L_{\text{loc}}(f(\Omega)) \] satisfies \(u \circ f\in WL^{q}\log^{\alpha}L_{\text{loc}}(\Omega)\). Moreover, each homeomorphism \(f\) which introduces continuous composition operator from \(WL^{q}\log^{\alpha}L\) to \(WL^{q}\log^{\alpha}L\) is necessarily a \(q\)-quasiconformal mapping. As a new tool, we prove a Lebesgue density type theorem for Orlicz spaces.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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Full Text: Euclid

References:

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