Duke Mathematical Journal

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

Stefanie Petermichl and Alexander Volberg

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We establish borderline regularity for solutions of the Beltrami equation $f\sb z-\mu f\sb {\overline {z}}=0$ on the plane, where $\mu$ is a bounded measurable function, $\parallel\mu\parallel\sb \infty=k<1$. What is the minimal requirement of the type $f\in W \sp {1,q}\sb {{\rm loc}}$ which guarantees that any solution of the Beltrami equation with any $\parallel\mu\parallel\sb \infty=k<1$ is a continuous function? A deep result of K. Astala says that $f\in W \sp {1,1+k+\varepsilon}\sb {{\rm loc}}$ suffices if $\varepsilon>0$. On the other hand, O. Lehto and T. Iwaniec showed that $q<1+k$ is not sufficient. In [2], the following question was asked: What happens for the borderline case $q=1+k$? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia's extrapolation technique and two-weight estimates for the [26].

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Duke Math. J., Volume 112, Number 2 (2002), 281-305.

First available in Project Euclid: 18 June 2004

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Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 30C62: Quasiconformal mappings in the plane 35K05: Heat equation 42C15: General harmonic expansions, frames 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47B38: Operators on function spaces (general)


Petermichl, Stefanie; Volberg, Alexander. Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular. Duke Math. J. 112 (2002), no. 2, 281--305. doi:10.1215/S0012-9074-02-11223-X. https://projecteuclid.org/euclid.dmj/1087575153

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