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In the proof of Astala's conjecture on quasiconformal distortion obtained byLacey--Sawyer--Uriarte-Tuero one of the key point is an estimate of theAhlfors--Beurling operator in a certain weighted space. We show that the pointof view of non-homogeneous Harmonic Analysis simplifies considerably this keypoint.

Astala's conjecture from the point of view of singular integrals on metric spaces