论文标题
关于准轴和其他平面域的施瓦茨等效性
On Schwartz equivalence of quasidiscs and other planar domains
论文作者
论文摘要
$ \ mathbb {r}^n $的两个开放子集如果之间存在差异性，则称为schwartz等效，从而诱发了schwartz函数空间之间的fréchet空间的同构。在本文中，我们使用清单几何形状中的工具，以证明少数平面域家族的施瓦茨等效性。我们证明所有准轴都是施瓦茨等效的，并且任何两个非相互连接的平面域的边界是准圆，均为schwartz等效。我们对由整个平面组成的域的两个schwartz等效类别进行分类，并证明了koebe型定理，并指出其边界连接组件的任何平面域都是有限的，许多Quasicircles都是Schwartz等于圆形域。我们还证明，Schwartz等效性的概念严格比$ c^\ infty $ -diffeomormorphism的概念通过构建$ \ Mathbb {r}^n $的$ c^\ infty $ -Diffty $ -diffeomorphic and Schwartz等等的概念。
Two open subsets of $\mathbb{R}^n$ are called Schwartz equivalent if there exists a diffeomorphism between them that induces an isomorphism of Fréchet spaces between their spaces of Schwartz functions. In this paper we use tools from quasiconformal geometry in order to prove the Schwartz equivalence of a few families of planar domains. We prove that all quasidiscs are Schwartz equivalent and that any two non-simply-connected planar domains whose boundaries are quasicircles are Schwartz equivalent. We classify the two Schwartz equivalence classes of domains that consist of the entire plane minus a quasiarc and prove a Koebe type theorem, stating that any planar domain whose connected components of its boundary are finitely many quasicircles is Schwartz equivalent to a circle domain. We also prove that the notion of Schwartz equivalence is strictly finer than the notion of $C^\infty$-diffeomorphism by constructing examples of open subsets of $\mathbb{R}^n$ that are $C^\infty$-diffeomorphic and are not Schwartz equivalent.
