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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)
##### Keywords:
weakly quasiregular maps; Ahlfors-Beurling operator
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##### References:
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