Duke Mathematical Journal

Subordination by conformal martingales in Lp and zeros of Laguerre polynomials

Alexander Borichev, Prabhu Janakiraman, and Alexander Volberg

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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.

Article information

Duke Math. J., Volume 162, Number 5 (2013), 889-924.

First available in Project Euclid: 29 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32A55: Singular integrals
Secondary: 60G46: Martingales and classical analysis 42A15: Trigonometric interpolation 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)


Borichev, Alexander; Janakiraman, Prabhu; Volberg, Alexander. Subordination by conformal martingales in $L^{p}$ and zeros of Laguerre polynomials. Duke Math. J. 162 (2013), no. 5, 889--924. doi:10.1215/00127094-2081372. https://projecteuclid.org/euclid.dmj/1364562914

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