Beurling-Ahlfors扩张伸缩商的上界估计

The Estimation of Upper-Bound of the Dilatation for Beurling-Ahlfors Extension

设h(x)是实轴上的保向同胚,满足h(±∞)=±∞.当h(x)的拟对称函数(,)()()()()x th x t h xρ=h x+?h?x?t(x∈R,t>0)被递减函数ρ(t)所控制时,h(x)的Beurling-Ahlfors扩张的伸缩商D(z)具有下述估计:21 1D≤ρ?+ρ??2,其中()2ρ?=ρy.

Let h be a homeomorphism of R onto itself with h（±∞） =±∞ ,when the quasisymmetrie function ρ（x, t） of h is controled by a decreasing function ρ（t） ,the dilatation D（z） obtained by the Beurling-Ahlfors extension of h is further estimated as follow： D≤2ρ^＊＋1/ρ^＊-1/2, where ρ^＊ =p（y/2）.