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Heating of the Ahlfors-Beurling operator, and estimates of its norm


Authors: A. Volberg and F. Nazarov
Translated by: the authors
Original publication: Algebra i Analiz, tom 15 (2003), nomer 4.
Journal: St. Petersburg Math. J. 15 (2004), 563-573
MSC (2000): Primary 42B20, 42C15, 42A50, 47B35
DOI: https://doi.org/10.1090/S1061-0022-04-00822-2
Published electronically: July 6, 2004
MathSciNet review: 2068982
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Abstract | References | Similar Articles | Additional Information

Abstract: A new estimate is established for the norm of the Ahlfors-Beurling transform $T\varphi(z):=\frac1\pi\iint\frac{\varphi(\zeta)\, dA(\zeta)}{(\zeta - z)^2}$ in $L^p(dA)$. Namely, it is proved that $\Vert T\Vert _{L^p\rightarrow L^p} \leq 2(p-1)$ for all $p\geq 2$. The method of Bellman function is used; however, the exact Bellman function of the problem has not been found. Instead, a certain approximation to the Bellman function is employed, which leads to the factor 2 on the right (in place of the conjectural $1$).


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Additional Information

A. Volberg
Affiliation: Michigan State University, East Lansing, Michigan, USA, and Equipe d’Analyse Université Paris VI, 4 Place Jussieu, 75 252 Paris cédex 05, France
Email: volberg@math.msu.edu

F. Nazarov
Affiliation: Michigan State University, East Lansing, Michigan, USA
Email: fedja@math.msu.edu

DOI: https://doi.org/10.1090/S1061-0022-04-00822-2
Received by editor(s): December 20, 2002
Published electronically: July 6, 2004
Additional Notes: Partially supported by the NSF grant DMS 0200713
Article copyright: © Copyright 2004 American Mathematical Society

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