On Burkholder function for orthogonal martingales and zeros of Legendre polynomials

Abstract

Burkholder obtained a sharp estimate of $\E|W|^p$ via $\E|Z|^p$, where $W$ is a martingale transform of $Z$, or, in other words, for martingales $W$ differentially subordinated to martingales $Z$. His result is that $\E|W|^p\le (p^*-1)^p\E|Z|^p$, where $p^* =\max (p, \frac{p}{p-1})$. What happens if the martingales have an extra property of being orthogonal martingales? This property is an analog (for martingales) of the Cauchy-Riemann equation for functions, and it naturally appears from a problem on singular integrals (see the references at the end of Section~1). We establish here that in this case the constant is quite different. Actually, $\E|W|^p\le (\frac{1+z_p}{1-z_p})^p\E|Z|^p$, $p\ge 2$, where $z_p$ is a specific zero of a certain solution of a Legendre ODE. We also prove the sharpness of this estimate. Asymptotically, $(1+z_p)/(1-z_p)=(4j^{-2}_0+o(1))p$, $p\to\infty$, where $j_0$ is the first positive zero of the Bessel function of zero order. This connection with zeros of special functions (and orthogonal polynomials for $p=n(n+1)$) is rather unexpected.