Composition operator and Sobolev-Lorentz spaces \(WL^{n,q}\). (English) Zbl 1293.30050
Summary: Let \(\Omega,\Omega'\subset\mathbb R^n\) be domains and let \(f\colon\Omega\to\Omega'\) be a homeomorphism. We show that if the composition operator \(T_f\colon u\mapsto u\circ f\) maps the Sobolev-Lorentz space \(WL^{n,q}(\Omega')\) to \(WL^{n,q}(\Omega)\) for some \(q\neq n\) then \(f\) must be a locally bilipschitz mapping.
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.) |
