# A quasiconformal composition problem for the $Q$-spaces

### Pekka Koskela

University of Jyväskylä, Finland### Jie Xiao

Memorial University of Newfoundland, St. John’s, Canada### Yi Ru-Ya Zhang

Beijing University of Aeronautics and Astronautics, China and University of Jyväskylä, Finland### Yuan Zhou

Beijing University of Aeronautics and Astronautics, China

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## Abstract

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 $0<\alpha<1$.

## Cite this article

Pekka Koskela, Jie Xiao, Yi Ru-Ya Zhang, Yuan Zhou, A quasiconformal composition problem for the $Q$-spaces. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 1159–1187

DOI 10.4171/JEMS/690