Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular (Q1847913)

In a recent paper by \textit{K. Astala}, \textit{T. Iwaniec} and \textit{E. Saksman} [Duke Math. J. 107, 27-56 (2002; Zbl 1009.30015)] it is shown that any solution \(f\in W^{1,q}_{\text{loc}}\) of the Beltrami equation with \(|\mu|_\infty= k<1\) is continuous, thus quasiregular, if \(q>k+1\) and that \(q<k+1\) is not sufficient for this result. The authors show that \(q=k+1\) is sufficient. The proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator.