The Faber Operator Acting on BMOA, BMO-Teichmüller Space and Chord-arc Curves

After reviewing Grunsky operator and Faber operator acting on Dirichlet space, we discuss the boundedness of Faber operator on BMOA, a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space. In particular, we show that the Faber operator acts as a bounded operator on BMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space. Meanwhile, we obtain several results on quasiconformal mappings, BMO-Teichmüller space and chord-arc curves as well. As by-products, this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on a chord-arc curve near to the unit circle in the BMO-Teichmüller space.