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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A new proof of an inequality of Littlewood and Paley


Author: Daniel H. Luecking
Journal: Proc. Amer. Math. Soc. 103 (1988), 887-893
MSC: Primary 30D55
DOI: https://doi.org/10.1090/S0002-9939-1988-0947675-0
MathSciNet review: 947675
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Abstract: A fairly elementary new proof is presented of the inequality $ (p \geq 2)$:

$\displaystyle \int {{{\left\vert {h'} \right\vert}^p}{{\left( {1 - \left\vert z... ...ht)}^{p - 1}}dxdy \leq \left\Vert h \right\Vert _{{H^p}}^p} ,\quad f \in {H^p}.$

In addition, the inequality

$\displaystyle \int {{{\left\vert h \right\vert}^{p - s}}{{\left\vert {h'} \righ... ...ft\vert z \right\vert)}^{s - 1}}dxdy \leq \left\Vert h \right\Vert _{{H^p}}^p} $

is shown to hold for $ h \in {H^p},p > 0$, if and only if $ 2 \leq s < p + 2$, generalizing the known case $ s = 2$.

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DOI: https://doi.org/10.1090/S0002-9939-1988-0947675-0
Article copyright: © Copyright 1988 American Mathematical Society