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Currently displaying 1 – 2 of 2

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Composition operators on W 1 X are necessarily induced by quasiconformal mappings

Luděk Kleprlík — 2014

Open Mathematics

Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to L q(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W 1 X to W 1 X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.

Composition operator and Sobolev-Lorentz spaces $W L^{n, q}$

Stanislav Hencl; Luděk Kleprlík; Jan Malý — 2014

Studia Mathematica

Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator $T_{f} : u \mapsto u \circ f$ maps the Sobolev-Lorentz space $W L^{n, q} (Ω^{'})$ to $W L^{n, q} (Ω)$ for some q ≠ n then f must be a locally bilipschitz mapping.

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