Geodesic
A quasiconformal composition problem for the $Q$-spaces
Journal of the European Mathematical Society, Tome 19 (2017) no. 4, pp. 1159-1187.

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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 01.
DOI : 10.4171/jems/690
Classification : 42-XX, 30-XX, 46-XX, 47-XX
Keywords: Quasiconformal mappings, compositions, Q-spaces 
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     title = {A quasiconformal composition problem for the $Q$-spaces},
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Pekka Koskela; Jie Xiao; Yi Ru-Ya Zhang; Yuan Zhou. A quasiconformal composition problem for the $Q$-spaces. Journal of the European Mathematical Society, Tome 19 (2017) no. 4, pp. 1159-1187. doi : 10.4171/jems/690. http://geodesic.mathdoc.fr/articles/10.4171/jems/690/

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