Weil–Petersson and little Teichmüller spaces on the real line

Abstract

We will prove that an increasing homeomorphism h in the Weil-Petersson class on the real line must be locally absolutely continuous such that log h belongs to the Sobolev class H 1 2 . We will also deal with the the pre-logarithmic derivative models of the little and Weil-Petersson Teichmller spaces in the half plane case.