拟圆盘与其它平面域的Schwartz等价性
On Schwartz equivalence of quasidiscs and other planar domains
摘要：
$\mathbb{R}^n$的两个开子集称为Schwartz等价的，如果它们之间存在一个同构，导致它们的Schwartz函数空间之间的Fréchet空间同构。本文利用拟共形几何的工具证明了几类平面域的Schwartz等价性。我们证明了所有拟圆盘都是Schwartz等价的，并且任何两个非单连通平面区域的边界都是拟圆，它们都是Schwartz等价的。我们分类的两个Schwartz等价类的域组成的整个平面减去一个拟弧，并证明了Koebe型定理，说明任何平面域的连通分支的边界是许多拟圆是Schwartz等价的一个圆形域。我们还证明了Schwartz等价的概念是严格精细的概念，$C^\infty$ -同构构造的例子是$\mathbb{R}^n$的开子集是$C^\infty$ -同构，而不是Schwartz等价。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
