具有无界梯度的拟共形调和映照

Quasiconformal harmonic mappings with unbounded gradient

设f=h+g为单位圆盘U到凸区域上的调和映照,其中h和g为U上的解析函数且满足g(0)=0.本文首先给出f的梯度Λ_f具有控制增长函数1/(1-|x|)~α(其中z∈U,0≤α≤1)时的一个等价刻画,进而得到了v-Bloch调和映照成为拟共形映照的条件.特别地,当v=0时,f即为双向Lipschitz映照.进一步地,本文还给出了当f(U)为一般区域(未必是凸)而h为凸映照时f成为拟共形映照的充分必要条件.

Let f = h＋g be a harmonic mapping of the unit disk U onto a convex domain, where h and g are analytic in U and g（0） = 0. In this paper, we first find the equivalent description for Λ_f, the gradient of f, which has the control growth function 1/（1-｜z｜）~α,where z ∈U and 0≤α≤1. Then, we obtain the sufficient conditions for a v-Bloch harmonic mapping to be quasiconformal. Moreover, if f（U） is a general domain（which may not be convex） and h is convex, then we give the necessary and sufficient conditions for f to be quasiconformal.