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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ L^p$-bounds for the Beurling-Ahlfors transform


Authors: Rodrigo Bañuelos and Prabhu Janakiraman
Journal: Trans. Amer. Math. Soc. 360 (2008), 3603-3612
MSC (2000): Primary 42B20, 60H05
Published electronically: February 13, 2008
MathSciNet review: 2386238
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ B$ denote the Beurling-Ahlfors transform defined on $ L^p(\mathbb{C})$, $ 1<p<\infty$. The celebrated conjecture of T. Iwaniec states that its $ L^p$ norm $ \Vert B\Vert _p=p^*-1$ where $ p^*= \max\{p,\frac{p}{p-1}\}$. In this paper the new upper estimate

$\displaystyle \Vert B\Vert _p\leq 1.575\,(p^*-1), \hspace{3mm} 1<p<\infty,$

is found.


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Additional Information

Rodrigo Bañuelos
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395
Email: banuelos@math.purdue.edu

Prabhu Janakiraman
Affiliation: Department of Mathematics, University of Illinois, Urbana-Champaign, Illinois 61801
Email: pjanakir@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04537-6
PII: S 0002-9947(08)04537-6
Keywords: Singular integrals, stochastic integrals
Received by editor(s): November 15, 2005
Received by editor(s) in revised form: April 26, 2006
Published electronically: February 13, 2008
Additional Notes: The first author was supported in part by NSF grant #0603701-DMS
The second author was supported in part by an NSF VIGRE postdoctoral fellowship
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.