论文标题
$π$ - 经营者在某些共同平坦的歧管和上半部空间上
The $Π$-operator on Some Conformally Flat Manifolds and the Upper Half Space
论文作者
论文摘要
$π$ - 经营者，也称为Ahlfors-beburling Transform，在解决Beltrami方程的局部准文献解决方案中起着重要作用。在本文中，我们首先在一般的Clifford-Hilbert模块上构造了$π$的操作员。这个$π$ - 操作员也是$ l^2 $等轴测图。此外，当希尔伯特空间是测量空间的$ l^2 $空间时，它也可用于求解某些Beltrami方程。然后，我们表明该技术可用于在复杂平面中构造经典的$π$ - 操作器，以及在某些保单平坦的流形上通过$ u/γ$构造的其他一些示例，其中$ u $是简单连接的子域的$ \ \ mathbb {r}^n} $或$ Mathbbbb; is $ is $ is $ as} $}克莱尼人集团在$ u $上不连续。这些流形上的$π$ - 操作器还将等距属性保留在某些$ l^2 $空间中，其$ l^p $规范受$ \π$ - 操作器的$ l^p $规范的界限，$ \ m \ m athbb {r}^{n}^{n} $或$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ \ \ $ \ $ {还讨论了$π$ - 手术器在Beltrami方程解决方案对这些共同平坦的歧管上的应用。最后，我们使用双曲线指标研究了上半空间中的$π$运营商理论。
The $Π$-operator, also known as Ahlfors-Beurling transform, plays an important role in solving the existence of locally quasiconformal solutions of Beltrami equations. In this paper, we first construct the $Π$-operator on a general Clifford-Hilbert module. This $Π$-operator is also an $L^2$ isometry. Further, it can also be used for solving certain Beltrami equations when the Hilbert space is the $L^2$ space of a measure space. Then, we show that this technique can be applied to construct the classical $Π$-operator in the complex plane and some other examples on some conformally flat manifolds, which are constructed by $U/Γ$, where $U$ is a simply connected subdomain of either $\mathbb{R}^{n}$ or $\mathbb{S}^{n}$, and $Γ$ is a Kleinian group acting discontinuously on $U$. The $Π$-operators on those manifolds also preserve the isometry property in certain $L^2$ spaces, and their $L^p$ norms are bounded by the $L^p$ norms of the $Π$-operators on $\mathbb{R}^{n}$ or $\mathbb{S}^{n}$, depending on where $U$ lies. The applications of the $Π$-operator to solutions of the Beltrami equations on those conformally flat manifolds are also discussed. At the end, we investigate the $Π$-operator theory in the upper-half space with the hyperbolic metric.
