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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Littlewood-Paley inequality for arbitrary rectangles in $ \mathbb{R}^2$ for  $ 0 < p \le 2$


Author: N. N. Osipov
Translated by: The author
Original publication: Algebra i Analiz, tom 22 (2010), nomer 2.
Journal: St. Petersburg Math. J. 22 (2011), 293-306
MSC (2010): Primary 42B25, 42B15
Published electronically: February 8, 2011
MathSciNet review: 2668127
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Abstract: The one-sided Littlewood-Paley inequality for pairwise disjoint rectangles in $ \mathbb{R}^2$ is proved for the $ L^p$-metric, $ 0 < p \le 2$. This result can be treated as an extension of Kislyakov and Parilov's result (they considered the one-dimensional situation) or as an extension of Journé's result (he considered disjoint parallelepipeds in $ \mathbb{R}^n$ but his approach is only suitable for $ p\in(1,2]$). We combine Kislyakov and Parilov's methods with methods ``dual'' to Journé's arguments.


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Additional Information

N. N. Osipov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, 27 Fontanka, St. Petersburg 191023, Russia
Email: nicknick@pdmi.ras.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01141-0
PII: S 1061-0022(2011)01141-0
Keywords: Littlewood–Paley inequality, Hardy class, atomic decomposition, Journé lemma, Calderón–Zygmund operator
Received by editor(s): September 11, 2009
Published electronically: February 8, 2011
Additional Notes: The author was supported by RFBR (grant no. 08-01-00358)
Article copyright: © Copyright 2011 American Mathematical Society