BMO-Teichm\"uller spaces revisited

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.