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Zap. Nauchn. Sem. POMI, 2010, Volume 376, Pages 88–115 (Mi znsl3620)  

This article is cited in 2 scientific papers (total in 2 papers)

One-sided Littlewood–Paley inequality in $\mathbb R^n$ for $0<p\le2$

N. N. Osipov

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We prove the one-sided Littlewood–Paley inequality for arbitrary collections of mutually disjoint rectangular parallelepipeds in $\mathbb R^n$ for the $L^p$-metric, $0<p\le2$. The paper supplements the author's earlier work, which dealt with the situation of $n=2$. That work was based on R. Fefferman's theory, which makes it possible to verify the boundedness of certain linear operators on two-parameter Hardy spaces (i.e., Hardy spaces on the product of two Euclidean spaces $H^p(\mathbb R^{d_1}\times\mathbb R^{d_2})$). However, Fefferman's results are not applicable in the situation where the number of Euclidean factors is greater than 2. Here we employ the more complicated Carbery–Seeger theory, which is a further development of Fefferman's ideas. It allows us to verify the boundedness of some singular integral operators on the multiparameter Hardy spaces $H^p(\mathbb R^{d_1}\times\cdots\times\mathbb R^{d_n})$, which leads eventually to the required inequality of Littlewood–Paley type. Bibl. – 13 titles.

Key words and phrases: Hardy space, atomic decomposition, Journé's lemma, Calderón–Zygmund operator.

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English version:
Journal of Mathematical Sciences (New York), 2011, 172:2, 229–242

UDC: 517.443+517.982.27
Received: 10.04.2010

Citation: N. N. Osipov, “One-sided Littlewood–Paley inequality in $\mathbb R^n$ for $0<p\le2$”, Investigations on linear operators and function theory. Part 38, Zap. Nauchn. Sem. POMI, 376, POMI, St. Petersburg, 2010, 88–115; J. Math. Sci. (N. Y.), 172:2 (2011), 229–242

Citation in format AMSBIB
\Bibitem{Osi10}
\by N.~N.~Osipov
\paper One-sided Littlewood--Paley inequality in $\mathbb R^n$ for $0<p\le2$
\inbook Investigations on linear operators and function theory. Part~38
\serial Zap. Nauchn. Sem. POMI
\yr 2010
\vol 376
\pages 88--115
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3620}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2011
\vol 172
\issue 2
\pages 229--242
\crossref{https://doi.org/10.1007/s10958-010-0195-4}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78651281560}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. N. Osipov, “The Littlewood-Paley-Rubio de Francia inequality in Morrey-Campanato spaces”, Sb. Math., 205:7 (2014), 1004–1023  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. S. N. Kudryavtsev, “An analogue of the Littlewood–Paley theorem for orthoprojectors onto wavelet subspaces”, Izv. Math., 80:3 (2016), 557–601  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Записки научных семинаров ПОМИ
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