VMO-Teichmüller space on the real line

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.