BMO-Teichm\"uller spaces revisited

TL;DR

This paper revisits BMO-Teichmüller spaces linked to Fuchsian groups, proving the equivalence of definitions are biholomorphisms and enhancing the understanding of the Douady-Earle extension operator's differentiability.

Contribution

It demonstrates that the previously established equivalences are biholomorphic mappings and proves Gâteaux-differentiability of the Douady-Earle extension at the origin.

Findings

Equivalence among three BMO-Teichmüller space definitions are biholomorphisms.

Douady-Earle extension operator is Gâteaux-differentiable at the origin.

Enhanced understanding of the structure of BMO-Teichmüller spaces.

Abstract

In a paper of Cui and Zinsmeister the equivalence among three definitions of BMO-Teichm\"uller spaces associated with a Fuchsian group was proven using the Douady-Earle extension operator. In this paper, we show that these equivalences are actually biholomorphisms. It was further shown in the above quoted paper that the Douady-Earle extension operator is continuous at the origin. We improve this result by showing G\^ateaux-differentiability at this point.