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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 ${\bf C}_f$ on the spaces $Q_α(\mathbb R^n)$ depends on the index $α$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $α$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] 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 $Q_α (\mathbb R^n)$ for any $(α,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}$. Moreover, $Q_α(\mathbb R^n)$ is shown to be invariant under inversions for all $0<α<1$.