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Transactions of the American Mathematical Society

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

 
 

 

Sharp square-function inequalities for conditionally symmetric martingales


Author: Gang Wang
Journal: Trans. Amer. Math. Soc. 328 (1991), 393-419
MSC: Primary 60G42
DOI: https://doi.org/10.1090/S0002-9947-1991-1018577-3
MathSciNet review: 1018577
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Abstract: Let $f$ be a conditionally symmetric martingale taking values in a Hilbert space $\mathbb {H}$ and let $S(f)$ be its square function. If ${\nu _p}$ is the smallest positive zero of the confluent hypergeometric function and ${\mu _p}$ is the largest positive zero of the parabolic cylinder function of parameter $p$, then the following inequalities are sharp: \[ \| f \|_{p} \leq \nu _{p}\| S(f)\|_{p}\qquad \text {if}\;0 < p \leq 2,\] \[ \|f \|_{p} \leq \mu _{p} \| S(f)\|_{p}\qquad \text {if}\;p \geq 3,\] \[ \nu _{p}\| S(f)\|_{p}\; \leq \; \|f\|_p \qquad \text {if}\; p \geq 2.\] Moreover, the constants $\nu _p$ and $\mu _p$ for the cases mentioned above are also best possible for the Marcinkiewicz-Paley inequalities for Haar functions.


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Keywords: Martingale, conditionally symmetric martingale, dyadic martingale, square-function inequality, confluent hypergeometric function, parabolic cylinder function, Brownian motion, Haar function
Article copyright: © Copyright 1991 American Mathematical Society