VMO-Teichmüller space on the real line

Yuliang Shen¹

¹Soochow University, Department of Mathematics

Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 57-82

Résumé

An increasing homeomorphism $h$ on the real line $\mathbb{R}$ is said to be strongly symmetric if it can be extended to a quasiconformal homeomorphism of the upper half plane $\mathbb{U}$ onto itself whose Beltrami coefficient $\mu$ induces a vanishing Carleson measure $|\mu(z)|^2/y\,dx\,dy$ on $\mathbb{U}$. We will deal with the class of strongly symmetric homeomorphisms on the real line and its Teichmüller space, which we call the VMO-Teichmüller space. In particular, we will show that if $h$ is strongly symmetric on the real line, then it is strongly quasisymmetric such that $\log h'$ is a VMO function. This improves some classical results of Carleson (1967) and Anderson-Becker-Lesley (1988) on the problem about the local absolute continuity of a quasisymmetric homeomorphism in terms of the Beltrami coefficient of a quasiconformal extension. We will also discuss various models of the VMO-Teichmüller space and endow it with a complex Banach manifold structure via the standard Bers embedding.  

Détail

DOI : 10.54330/afm.112456

Keywords: Universal Teichmüller space, quasiconformal mapping, quasisymmetric homeomorphism, Beltrami coefficient, strongly symmetric homeomorphism, Carleson measure, vanishing Carleson measure, BMOA, VMOA