上半平面调和拟共形同胚的延拓定理

Extension Theorem for Quasiconformal Harmonic Homeomorphism of a Half-plane

设φ(x)为定义在实轴R上的保向同胚映照,本文证明:如果ess infφ'(x)>0,ess supφ'(x)<+∞且满足Dini条件∫+∞0∣φ'(x+t)-φ'(x-t)∣/tdt+∞,对于任意的x∈R,则φ(x)可以被延拓成上半平面到自身上的调和拟共形映照.

Let φ（x） be an orientation preserving homeomorphism defined in the real axis R.In this paper,we prove that if essinf φ＇（x）0,esssup φ＇（x）＋∞and Dini condition ∫0^＋∞∣φ＇（x＋t）-φ＇（x-t）∣/tdt＋∞ holds for ever x ∈R,then φ（x） can be extension into a quasiconformal harmonic mapping of a half-plane.