A quasiconformal composition problem for the -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

Abstract

Given a quasiconformal mapping with , we show that (un-)boundedness of the composition operator on the spaces depends on the index and the degeneracy set of the Jacobian . We establish sharp results in terms of the index and the local/global self-similar Minkowski dimension of the degeneracy set of . 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 of an arbitrary quasisymmetric mapping is shown to preserve for any . Moreover, is shown to be invariant under inversions for all .

Cite this article

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

DOI 10.4171/JEMS/690