The real analytic structure of the Teichm\"uller space of circle diffeomorphisms with Zygmund continuous derivatives

TL;DR

This paper establishes the real analytic structure of the Teichmüller space of circle diffeomorphisms with Zygmund continuous derivatives, linking it to Besov spaces and providing new proofs of key correspondences.

Contribution

It introduces a novel real-analytic framework for the Teichmüller space of Zygmund continuous diffeomorphisms, connecting it with Besov spaces and quasisymmetric extensions.

Findings

Proves the correspondence between quasiconformal maps and Zygmund regularity.

Shows the Teichmüller space is real-analytically equivalent to a Banach space.

Provides new proofs of classical Teichmüller theory results.

Abstract

We apply the methods of simultaneous uniformization and composition operators on Besov spaces to the Teichm\"uller space $T^Z$ of circle diffeomorphisms with Zygmund continuous derivatives. As consequences, we obtain the following: (1) a new proof of the correspondence between quasiconformal self-homeomorphisms of the unit disk with complex dilatations of linear decay order and their quasisymmetric extensions to the unit circle with regularity in the Zygmund continuously differentiable class; (2) a real-analytic equivalence of $T^Z$ with the real Banach space of Zygmund continuous functions on the unit circle.