Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new estimate is established for the norm of the Ahlfors–Beurling transform $T\varphi (z):=\frac 1\pi \iint \frac {\varphi (\zeta ) dA(\zeta )}{(\zeta - z)^2}$ in $L^p(dA)$. Namely, it is proved that $\|T\|_{L^p\rightarrow L^p} \leq 2(p-1)$ 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 $1$).


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 42B20, 42C15, 42A50, 47B35

Retrieve articles in all journals with MSC (2000): 42B20, 42C15, 42A50, 47B35


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
MR Author ID: 233855
Email: fedja@math.msu.edu

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