A quasiconformal composition problem for the Q-spaces

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.