On Schwartz equivalence of quasidiscs and other planar domains

Abstract

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. We also prove that any non-simply-connected planar domain whose boundary is a quasicircle is Schwartz equivalent to the complement of the closed unit disc. We classify the two Schwartz equivalence classes of domains that consist of the entire plane minus a quasiarc. We prove a Koebe-type theorem, stating that any planar domain whose connected components of its boundary are finitely many quasicircles, with at most one unbounded, 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.