Astala's conjecture from the point of view of singular integrals on   metric spaces
  In the proof of Astala's conjecture on quasiconformal distortion obtained by Lacey--Sawyer--Uriarte-Tuero one of the key point is an estimate of the Ahlfors--Beurling operator in a certain weighted space. We show that the point of view of non-homogeneous Harmonic Analysis simplifies considerably this key point. 
Alexander Volberg
math.AP math.CV