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.