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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:

$\displaystyle \Vert f \Vert _{p} \leq \nu_{p}\Vert S(f)\Vert _{p}$   if$\displaystyle \;0 < p \leq 2,$

$\displaystyle \Vert f \Vert _{p} \leq \mu_{p} \Vert S(f)\Vert _{p}$   if$\displaystyle \;p \geq 3,$

$\displaystyle \nu_{p}\Vert S(f)\Vert _{p}\; \leq\; \Vert f\Vert _p$   if$\displaystyle \; p \geq 2.$

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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1018577-3
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

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