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On Burkholder function for orthogonal martingales and zeros of Legendre polynomials
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 1, February 2013
- pp. 207-236
- 10.1353/ajm.2013.0004
- Article
- Additional Information
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Burkholder obtained a sharp estimate of ${\bf E}|W|^p$ via ${\bf E}|Z|^p$, for martingales $W$
differentially subordinated to martingales $Z$. His result is that ${\bf E}|W|^p\le (p^*-1)^p{\bf E}|Z|^p$,
where $p^* =\max \big(p, {p\over p-1}\big)$. 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 in 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, ${\bf E}|W|^p\le \big({1+z_p\over 1-z_p}\big)^p{\bf E}|Z|^p$, $p\ge 2$, where $z_p$ is a specific zero of a certain solution of the 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.