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1 April 2013 Subordination by conformal martingales in Lp and zeros of Laguerre polynomials
Alexander Borichev, Prabhu Janakiraman, Alexander Volberg
Duke Math. J. 162(5): 889-924 (1 April 2013). DOI: 10.1215/00127094-2081372


Given martingales W and Z such that W is differentially subordinate to Z, Burkholder obtained the sharp inequality E|W|p(p1)pE|Z|p, where p=max {p,p/(p1)}. What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if p2 and W is a conformal martingale differentially subordinate to any martingale Z, then E|W|p[(p2p)/2]p/2E|Z|p. In this paper, we establish that if p2, Z is conformal, and W is any martingale subordinate to Z, then E|W|p[2(1zp)/zp]pE|Z|p, where zp is the smallest positive zero of a certain solution of the Laguerre ordinary differential equation. We also prove the sharpness of this estimate and an analogous one in the dual case for 1<p<2. Finally, we give an application of our results. Previous estimates on the Lp-norm of the Beurling–Ahlfors transform give at best Bp2p as p. We improve this to Bp1.3922p as p.


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Alexander Borichev. Prabhu Janakiraman. Alexander Volberg. "Subordination by conformal martingales in Lp and zeros of Laguerre polynomials." Duke Math. J. 162 (5) 889 - 924, 1 April 2013.


Published: 1 April 2013
First available in Project Euclid: 29 March 2013

zbMATH: 1266.32006
MathSciNet: MR3047469
Digital Object Identifier: 10.1215/00127094-2081372

Primary: 32A55
Secondary: 42A15 , 42B20 , 60G46

Rights: Copyright © 2013 Duke University Press


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Vol.162 • No. 5 • 1 April 2013
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