微分从属鞅的尖锐弱型不等式
Sharp weak-type inequalities for differentially subordinated martingales
摘要：
设$M,N$是实值鞅，使得$N$微分从属于$M$ .本文证明了下列弱型不等式：（i）若$M\geq0$和$0<p\leq1$，则\[\Vert N\Vert_{p,\infty}\leq2\Vert M\Vert_p\]和常数是最佳可能的. (ii)如果$M\geq0$和$p\geq2$，则\[\Vert N\Vert_{p,\infty}\leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_p\]和常数是最好的可能。(iii)如果$1\leq p\leq2$和$M$和$N$正交，则\[\Vert N\Vert_{p,\infty}\leq K_p\Vert M\Vert_p,\]其中\[K_p^p=\frac{1}{Γ(p+1)}\cdot\biggl(\fracπ{2}\biggr)^{p-1}\cdot\frac{1+1/3^2+1/5^2+1/7^2+...}{1-1/3^{p+1}+1/5^ {p+1}-1/7^{p+1}+...}.\]常数是最佳可能。我们还提供了相关的估计调和函数的欧几里德域。
英文摘要：
Let $M,N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities: (i) If $M\geq0$ and $0<p\leq1$, then \[\Vert N\Vert_{p,\infty}\leq2\Vert M\Vert_p\] and the constant is the best possible. (ii) If $M\geq0$ and $p\geq2$, then \[\Vert N\Vert_{p,\infty}\leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_p\] and the constant is the best possible. (iii) If $1\leq p\leq2$ and $M$ and $N$ are orthogonal, then \[\Vert N\Vert_{p,\infty}\leq K_p\Vert M\Vert_p,\] where \[K_p^p=\frac{1}{Γ(p+1)}\cdot\biggl(\fracπ{2}\biggr)^{p-1}\cdot\frac{1+1/3^2+1/5^2+1/7^2+...}{1-1/3^{p+1}+1/5^ {p+1}-1/7^{p+1}+...}.\] The constant is the best possible. We also provide related estimates for harmonic functions on Euclidean domains.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
