Geodesic
Two Inequalities for the First Moments of a Martingale, its Square Function and its Maximal Function
Adam Osękowski1
1 Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 4, pp. 441-449

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Given a Hilbert space valued martingale $(M_n)$, let $(M^*_n)$ and $(S_n(M))$ denote its maximal function and square function, respectively. We prove that $$\displaylines{ \mathbb{E}|M_n|\leq 2\mathbb{E}S_n(M), \quad\ n=0,1,2,\ldots,\cr \mathbb{E}M^*_n \leq \mathbb{E}|M_n|+2\mathbb{E}S_n(M), \quad\ n=0,1,2,\ldots. } $$ The first inequality is sharp, and it is strict in all nontrivial cases.
DOI : 10.4064/ba53-4-9
Keywords: given hilbert space valued martingale * denote its maximal function square function respectively prove displaylines mathbb leq mathbb m quad ldots mathbb * leq mathbb mathbb m quad ldots first inequality sharp strict nontrivial cases

Adam Osękowski 1

1 Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland 
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Adam Osękowski. Two Inequalities for the First Moments of a
Martingale, its Square Function
 and its Maximal Function. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 4, pp. 441-449. doi: 10.4064/ba53-4-9

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