Sharp squarefunction inequalities for conditionally symmetric martingales
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 Trans. Amer. Math. Soc. 328 (1991), 393419 Request permission
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 MarcinkiewiczPaley inequalities for Haar functions.References

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Additional Information
 © Copyright 1991 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 328 (1991), 393419
 MSC: Primary 60G42
 DOI: https://doi.org/10.1090/S00029947199110185773
 MathSciNet review: 1018577