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Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular. (English) Zbl 1025.30018
In a recent paper by K. Astala, T. Iwaniec and 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.

MSC:
30C62 Quasiconformal mappings in the complex plane
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
35K05 Heat equation
42C15 General harmonic expansions, frames
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B38 Linear operators on function spaces (general)
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