VMO-Teichmüller space on the real line


  • Yuliang Shen Soochow University, Department of Mathematics


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


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.





How to Cite

Shen, Y. (2021). VMO-Teichmüller space on the real line. Annales Fennici Mathematici, 47(1), 57–82. https://doi.org/10.54330/afm.112456