Abstract
For martingales $f \in L_p (2 \leqq p < \infty)$ the inequality $\|Mf\|_p \leqq (p + 1)\|Sf\|_p$ is proved, where $Mf = \sup_n |f_n|$ is the maximal function and $S^2 = \sum_n |f_n - f_{n-1}|^2$ the martingale square function. For integer $p$ the estimate becomes $\|Mf\|_p \leqq p\|Sf\|_p$.
Citation
G. Klincsek. "A Square Function Inequality." Ann. Probab. 5 (5) 823 - 825, October, 1977. https://doi.org/10.1214/aop/1176995727
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