弦弧曲线空间在BMOA空间中是否连通是一个长期的、重要的开问题. 目前只知道拟共形延拓的复伸缩商在小Carleson范数情形是有效的. 我们用偏导数来代替复伸缩商研究弦弧曲线空间与W-P Teichmu空间. 首先利用奇异积分算子给出偏导数一些好的估计，再利用拟共形映射的表示理论给出其与黎曼映射的Schwarzian导数的关系，进而得到对数导数与拟对称同胚的刻画. . 我们用曲线弧长参数的切向量空间来恒等弦弧曲线空间. 利用奇异积分算子找出Beurling-Ahlfors型延拓的复伸缩商与拟对称映射的依赖关系，进而证明将曲线的切向量映到其对应的黎曼映射这个算子在W-P Teichmuller空间实解析性.. 利用Poission延拓与共形映射边界行为的一些估计来刻画Morrey区域的几何性质. 然后引进并研究Morrey-Teichmuller空间.

An important open problem for a long time is whether the Chord arc curve space is connected in BMOA space. We only know the case of small Carleson measure of complex dilatation of quasiconformal extension is effective. We will use partial derivative to study the chord arc curves of space and Weil-petersson Teichmuller space. First, we give some good estimations of partial derivative by using singular integral operator . Then we give some relationship between partial derivative and the Schwarzian derivative of Riemann mapping by using the representation theory of quasiconformal mappings. Finally, we get some characterizations of logarithmic derivative of Riemann mapping and quasiymmetric homeomorphisms.. We identify the chord arc curve space with the tangent vector space to the arc-length parameter curve. By using the singular integral operator，we find the dependence relationship between complex dilatation of Beurling-Ahlfors extension and quasisymmetric mapping , then prove the operator which seeds the tangent vector to the Riemann mapping is real analytic in the chord arc curve space and Weil-petersson Teichmuller space.. By means of Poission extension and the boundary behavior of conformal mapping, we give some geometric characterizations of Morrey domain. Then we introduce and investigate the Morrey-Teichmuller space.