Subordination by conformal martingales in Lp and zeros of Laguerre polynomials

Abstract

Given martingales $W$ and $Z$ such that $W$ is differentially subordinate to $Z$, Burkholder obtained the sharp inequality $E|W{|}^p\le (p^{\ast }-1{)}^{p}E|Z{|}^p$, where $p^{\ast }=max\hspace{0.17em}\{p,p/(p-1\left)\right\}$. What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if $p\ge 2$ and $W$ is a conformal martingale differentially subordinate to any martingale $Z$, then $E|W{|}^p\le [(p^2-p)/2{]}^{p/2}E|Z{|}^p$. In this paper, we establish that if $p\ge 2$, $Z$ 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 $1<p<2$. Finally, we give an application of our results. Previous estimates on the $L^p$-norm of the Beurling–Ahlfors transform give at best $\Vert B{\Vert }_p\lesssim \sqrt{2}p$ as $p\to \infty$. We improve this to $\Vert B{\Vert }_p\lesssim 1.3922p$ as $p\to \infty$.