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

Abstract

Let Omega aS, a"e (n) be an open set and X(Omega) be any rearrangement invariant function space close to L (q) (Omega), i.e. X has the q-scaling property.

We prove that each homeomorphism f which induces the composition operator u a dagger broken vertical bar u a"' 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.

Keywords

• Sobolev space
• Rearrangement invariant space
• Quasiconformal mappings
• Composition operator