On Schwartz equivalence of quasidiscs and other planar domains

Two open subsets of $${\mathbb {R}}^n$$ are called Schwartz equivalent if there exists a diffeomorphism between them that induces an isomorphism Fréchet spaces their functions. In this paper we use tools from quasiconformal geometry in order to prove the equivalence few families planar domains. We all quasidiscs equivalent. also any non-simply-connected domain whose boundary is quasicircle complement closed unit disc. classify two classes domains consist entire plane minus quasiarc. Koebe-type theorem, stating connected components its finitely many quasicircles, with at most one unbounded, circle domain. notion strictly finer than $$C^\infty $$ -diffeomorphism by constructing examples -diffeomorphic and not