JournalsjemsVol. 19, No. 4pp. 1159–1187

A quasiconformal composition problem for the QQ-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
A quasiconformal composition problem for the $Q$-spaces cover
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Abstract

Given a quasiconformal mapping f:RnRnf:{\mathbb R}^n\to{\mathbb R}^n with n2n\ge2, we show that (un-)boundedness of the composition operator Cf{\mathbf C}_f on the spaces Qα(Rn)Q_{\alpha}({\mathbb R}^n) depends on the index α\alpha and the degeneracy set of the Jacobian JfJ_f. We establish sharp results in terms of the index α\alpha and the local/global self-similar Minkowski dimension of the degeneracy set of JfJ_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:RnRnf:{\mathbb R}^n\to{\mathbb R}^n of an arbitrary quasisymmetric mapping g:RnpRnpg:{\mathbb R}^{n-p}\to {\mathbb R}^{n-p} is shown to preserve Qα(Rn)Q_{\alpha} ({\mathbb R}^n) for any (α,p)(0,1)×[2,n)(0,1/2)×{1}(\alpha,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}. Moreover, Qα(Rn)Q_{\alpha}({\mathbb R}^n) is shown to be invariant under inversions for all 0<α<10<\alpha<1.

Cite this article

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

DOI 10.4171/JEMS/690