Composition operators on W1X are necessarily induced by quasiconformal mappings

Abstract

Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to Lq(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 W1X to W1X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.