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

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

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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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
2. S. N. Kudryavtsev, “An analogue of the Littlewood–Paley theorem for orthoprojectors onto wavelet subspaces”, Izv. Math., 80:3 (2016), 557–601
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