关于正交鞅的Burkholder函数与勒让德多项式的零点
On Burkholder function for orthogonal martingales and zeros of Legendre polynomials
摘要：
Burkholder通过$\E|Z|^p$得到了$\E|W|^p$的精确估计，其中$W$是$Z$的鞅变换，或者换句话说，对于鞅$W$微分从属于鞅$Z$。他的结果是$\E|W|^p\le (p^*-1)^p\E|Z|^p$，其中$p^* =\max (p, \frac{p}{p-1})$。如果鞅有一个额外的性质是正交鞅，会发生什么？这一性质类似于函数的柯西-黎曼方程（对于鞅），它自然地出现在奇异积分的问题中（见第1节末尾的参考文献）。我们在这里确定，在这种情况下，常数是完全不同的。实际上，是$\E|W|^p\le (\frac{1+z_p}{1-z_p})^p\E|Z|^p$，$p\ge 2$，其中$z_p$是Legendre常微分方程的某个解的特定零点。我们还证明了这一估计的锐度。渐近地，$(1+z_p)/(1-z_p)=(4j^{-2}_0+o(1))p$，$p\to\infty$，其中$j_0$是零阶贝塞尔函数的第一个正零点。与特殊函数的零点（以及$p=n(n+1)$的正交多项式）的这种联系是相当出乎意料的。
英文摘要：
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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