A quasiconformal composition problem for the Q-spaces

Given a quasiconformal mapping $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ with $n\ge 2$, 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 [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-V"ais"al"a’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 $(\alpha,p)\in (0,1)\times[2,n)\cup(0,1/2)\times{1}$\mathrm{.}Moreover,$Q_{\alpha}(\mathbb R^n)$ is shown to be invariant under inversions for all $0<\alpha <1$.