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

TL;DR

This paper offers a simplified approach to a key estimate in proving Astala's conjecture on quasiconformal distortion by applying non-homogeneous Harmonic Analysis to the Ahlfors--Beurling operator.

Contribution

It introduces a novel perspective using non-homogeneous Harmonic Analysis to streamline the proof of a critical estimate in Astala's conjecture.

Findings

Simplified the estimate of the Ahlfors--Beurling operator in weighted spaces.

Provided new insights into quasiconformal distortion via metric space analysis.

Demonstrated the effectiveness of non-homogeneous Harmonic Analysis in complex analysis proofs.

Abstract

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.