科研之友
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摘要
Given a quasiconformal mapping integral : R-n -> R-n with n >= 2, we show that (un) boundedness of the composition operator C-f on the spaces Q(alpha)(R-n) depends on the index alpha and the degeneracy set of the Jacobian J(f). We establish sharp results in terms of the index alpha and the local/global self-similar Minkowski dimension of the degeneracy set of J(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-Vaisala's quasiconformal extension f : R-n -> R-n of an arbitrary quasisymmetric mapping g : Rn-p -> Rn-p is shown to preserve Q(alpha()R(n)) for any (alpha, p) is an element of(0, 1) x [2, n) boolean OR (0, 1/2) x {1}]. Moreover, Q(alpha) (R-n) is shown to be invariant under inversions for all 0 < alpha < 1.
- 出版日期2017
- 单位北京航空航天大学
