1 April 2002 Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular
Stefanie Petermichl, Alexander Volberg
Duke Math. J. 112(2): 281-305 (1 April 2002). DOI: 10.1215/S0012-9074-02-11223-X


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].


Download Citation

Stefanie Petermichl. Alexander Volberg. "Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular." Duke Math. J. 112 (2) 281 - 305, 1 April 2002. https://doi.org/10.1215/S0012-9074-02-11223-X


Published: 1 April 2002
First available in Project Euclid: 18 June 2004

zbMATH: 1025.30018
MathSciNet: MR1894362
Digital Object Identifier: 10.1215/S0012-9074-02-11223-X

Primary: 42B20
Secondary: 30C62 , 35K05 , 42C15 , 47B35 , 47B38

Rights: Copyright © 2002 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.112 • No. 2 • 1 April 2002
Back to Top