## Duke Mathematical Journal

### Subordination by conformal martingales in $L^{p}$ and zeros of Laguerre polynomials

#### Abstract

Given martingales $W$ and $Z$ such that $W$ is differentially subordinate to $Z$, Burkholder obtained the sharp inequality $E|W|^{p}\le(p^{*}-1)^{p}E|Z|^{p}$, where $p^{*}=\max\{p,p/(p-1)\}$. What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if $p\geq2$ and $W$ is a conformal martingale differentially subordinate to any martingale $Z$, then $E|W|^{p}\leq[(p^{2}-p)/2]^{p/2}E|Z|^{p}$. In this paper, we establish that if $p\geq2$, $Z$ is conformal, and $W$ is any martingale subordinate to $Z$, then $\mathbb{E}|W|^{p}\le[\sqrt{2}(1-z_{p})/z_{p}]^{p}\mathbb{E}|Z|^{p}$, where $z_{p}$ 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 $1\lt p\lt 2$. Finally, we give an application of our results. Previous estimates on the $L^{p}$-norm of the Beurling–Ahlfors transform give at best $\|B\|_{p}\lesssim\sqrt{2}p$ as $p\rightarrow\infty$. We improve this to $\|B\|_{p}\lesssim1.3922p$ as $p\rightarrow\infty$.

#### Article information

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

Dates
First available in Project Euclid: 29 March 2013

https://projecteuclid.org/euclid.dmj/1364562914

Digital Object Identifier
doi:10.1215/00127094-2081372

Mathematical Reviews number (MathSciNet)
MR3047469

Zentralblatt MATH identifier
1266.32006

#### Citation

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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