A quasiconformal composition problem for the $Q$-spaces

Given a quasiconformal mapping f:{\mathbb R}^n\to{\mathbb R}^n with n\ge2 , we show that (un-)boundedness of the composition operator {\mathbf C}_f on the spaces Q_{\alpha}({\mathbb R}^n) depends on the index \alpha and the degeneracy set of the Jacobian J_f . We establish sharp results in terms of the index \alpha and the local/global self-similar Minkowski dimension of the degeneracy set of J_f . This gives a solution to [3, Problem 8.4] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel–Lizorkin and Besov spaces. Consequently, Tukia–Väisälä's quasiconformal extension f:{\mathbb R}^n\to{\mathbb R}^n of an arbitrary quasisymmetric mapping g:{\mathbb R}^{n-p}\to {\mathbb R}^{n-p} is shown to preserve Q_{\alpha} ({\mathbb R}^n) for any (\alpha,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\} . Moreover, Q_{\alpha}({\mathbb R}^n) is shown to be invariant under inversions for all 0<\alpha<1 .

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