The real analytic structure of the Teichmüller space of circle diffeomorphisms with Zygmund continuous derivatives

Abstract: We apply the methods of simultaneous uniformization and composition operators on Besov spaces to the Teichmüller space $T<sup>Z$ 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 $T<sup>Z$ with the real Banach space of Zygmund continuous functions on the unit circle.