On Schwartz equivalence of quasidiscs and other planar domains

TL;DR

This paper demonstrates that various planar domains, including quasidiscs and those bounded by quasicircles, are Schwartz equivalent, revealing a finer classification than smooth diffeomorphism using tools from quasiconformal geometry.

Contribution

It establishes Schwartz equivalence among quasidiscs and certain planar domains, providing a new classification framework that refines smooth diffeomorphism distinctions.

Findings

All quasidiscs are Schwartz equivalent.

Domains with boundaries as quasicircles are Schwartz equivalent.

Schwartz equivalence is strictly finer than $C^ abla$-diffeomorphism.

Abstract

Two open subsets of $\mathbb{R}^n$ are called Schwartz equivalent if there exists a diffeomorphism between them that induces an isomorphism of Fr\'echet 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.