Composition operator and Sobolev–Lorentz spaces $WL^{n,q}$
Volume 221 / 2014
Studia Mathematica 221 (2014), 197-208
MSC: Primary 30C65; Secondary 46E35, 46E30.
DOI: 10.4064/sm221-3-1
Abstract
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.
