Universal Teichmüller space and BMO

We first give some new characterizations on BMOA–Teichmüller space and various characterizations on VMOA–Teichmüller space as well. In particular, we prove that a quasisymmetric conformal welding h$h$ corresponds to an asymptotically smooth curve in the sense of Pommerenke (1978) [32] precisely when h$h$ is absolutely continuous with logh^′∈VMO$log{h}^{\prime }\in \text{VMO}$. We then show that these BMO–Teichmüller spaces have natural complex structures.