VMO-Teichmüller space on the real line

Abstract

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 $|\mu (z){|}^2/ydxdy$ 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 $\log h^{\prime }$ 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.