Weighted inequalities for the martingale square and maximal functions

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• 作者：Adam Osȩkowski ^{ados@mimuw.edu.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60G44 ; 60G42
• 刊名：Statistics & Probability Letters
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：120
• 期：Complete
• 页码：95-100
• 全文大小：396 K

文摘

Let W$W$ be a weight, i.e., a uniformly integrable, continuous-path martingale, and let 878a2048c147717e53186451554b7427" title="Click to view the MathML source">W^∗$W^{\ast }$ denote the associated maximal function. We show that if b016af446456854381a" title="Click to view the MathML source">X$X$ is an arbitrary càdlàg martingale and X^∗$X^{\ast }$, ae884e430e636017d2a8ac53045926e" title="Click to view the MathML source">[X]$\left[X\right]$ denote its maximal and square functions, then where The estimate is sharp for 95b3367bdc47c0d44f6e8ba53ff56" title="Click to view the MathML source">p∈{1,2}$p\in \{1,2\}$. Furthermore, it is proved that if ae3ea24e" title="Click to view the MathML source">p>2$p>2$, then the above weighted inequality does not hold with any finite constant 8def7d81036a169f6a87fbd0f17dbe" title="Click to view the MathML source">γ_p${\gamma }_p$ depending only on e6e0f799fa227293c04ecc8d606cc" title="Click to view the MathML source">p$p$.