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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.

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 $W{L}^{n,q}\left({\Omega }^{\text{'}}\right)$ to $W{L}^{n,q}\left(\Omega \right)$ for some q ≠ n then f must be a locally bilipschitz mapping.