Given a Hilbert space valued martingale (Mₙ), let (M*ₙ) and (Sₙ(M)) denote its maximal function and square function, respectively. We prove that 𝔼|Mₙ| ≤ 2𝔼 Sₙ(M), n=0,1,2,..., 𝔼 M*ₙ ≤ 𝔼 |Mₙ| + 2𝔼 Sₙ(M), n=0,1,2,.... The first inequality is sharp, and it is strict in all nontrivial cases.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-4-9,
author = {Adam Os\k ekowski},
title = {Two Inequalities for the First Moments of a Martingale, its Square Function and its Maximal Function},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {53},
year = {2005},
pages = {441-449},
zbl = {1113.60045},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-4-9}
}
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) pp. 441-449. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-4-9/