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

Author(s): Rodrigo Bañuelos; Prabhu Janakiraman
Journal: Trans. Amer. Math. Soc. 360 (2008), 3603-3612.
MSC (2000): Primary 42B20, 60H05
Posted: February 13, 2008
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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: 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
Posted: 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
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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