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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in $ L\sp p(0,1)$


Author: K. P. Choi
Journal: Trans. Amer. Math. Soc. 330 (1992), 509-529
MSC: Primary 60G42
DOI: https://doi.org/10.1090/S0002-9947-1992-1034661-3
MathSciNet review: 1034661
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Abstract: Let $ 1 < p < \infty $. Let $ d = ({d_1},{d_2}, \ldots)$ be a real-valued martingale difference sequence, $ \theta = ({\theta _1},{\theta _2}, \ldots)$ is a predictable sequence taking values in $ [0,1]$. We show that the best constant of the inequality,

$\displaystyle {\left\Vert {\sum\limits_{k = 1}^n {{\theta _k}{d_k}} } \right\Ve... ...p}{\left\Vert {\sum\limits_{k = 1}^n {{d_k}} } \right\Vert _p}, \quad n \geq 1,$

satisfies

$\displaystyle {c_p} = \frac{p}{2} + \frac{1}{2}\;\log \;\left({\frac{{1 + y}}{2}} \right) + \frac{{{\alpha _2}}}{p} + \cdots,$

where $ \gamma = {e^{ - 2}}$ and $ {\alpha _2} = {\left[ {\frac{1}{2}\;\log \;\frac{{1 + \gamma }}{2}} \right]^2}... ... \;\frac{{1 + \gamma }}{2} - 2{\left({\frac{\gamma }{{1 + \gamma }}} \right)^2}$. The best constant equals the unconditional basis constant of a monotone basis of $ {L^p}(0,1)$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1034661-3
Keywords: Martingale, martingale transform, zigzag martingale, stochastic integral, unconditional basis constant, Haar system, monotone basis, contractive projection, biconcave functions
Article copyright: © Copyright 1992 American Mathematical Society