On Burkholder function for orthogonal martingales and zeros of Legendre polynomials

Details On Burkholder function for orthogonal martingales and zeros of Legendre polynomials On Burkholder function for orthogonal martingales and zeros of Legendre polynomials

2010-02-11

arxiv.org/abs/1002.2314v3

Mathematics

Classical Analysis and ODEs

35 pages, to appear in Amer. J. Math, some misprints corrected

Scientific paper

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.

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