A quasiconformal composition problem for the Q-spaces
Koskela, P., Xiao, J., Zhang, Y. R.-Y., & Zhou, Y. (2017). A quasiconformal composition problem for the Q-spaces. Journal of the European Mathematical Society, 19(4), 1159-1187. https://doi.org/10.4171/JEMS/690
Julkaistu sarjassa
Journal of the European Mathematical SocietyPäivämäärä
2017Tekijänoikeudet
© 2019 EMS Publishing House
Given a quasiconformal mapping f:Rn→Rn with n≥2, we show that (un-)boundedness of the composition operator Cf on the spaces Qα(Rn) depends on the index α and the degeneracy set of the Jacobian Jf. We establish sharp results in terms of the index α and the local/global self-similar Minkowski dimension of the degeneracy set of Jf. 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:Rn→Rn of an arbitrary quasisymmetric mapping g:Rn−p→Rn−p is shown to preserve Qα(Rn) for any (α,p)∈(0,1)×[2,n)∪(0,1/2)×{1}. Moreover, Qα(Rn) is shown to be invariant under inversions for all 0<α<1.
Julkaisija
EMS Publishing HouseISSN Hae Julkaisufoorumista
1435-9855Asiasanat
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https://converis.jyu.fi/converis/portal/detail/Publication/26960719
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