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Littlewood-Paley inequality for arbitrary rectangles in \(\mathbb R^2\) for \(0<p\leq2\). (English. Russian original) Zbl 1219.42011

St. Petersbg. Math. J. 22, No. 2, 293-306 (2011); translation from Algebra Anal. 22, No. 2, 164-184 (2010).
The following result is proved in this paper:
Let \(\{ f_m\}\) be a sequence in \(L^1({\mathbb R}^2)\) such that \(\text{supp}\, \widehat f_m\subset \Delta_m\), where the \(\Delta_m\) are disjoint rectangles in \({\mathbb R}^2\) with sides parallel to the coordinates axes. Then, for \(0<p\leq 2\),
\[ \| \sum_m f_m\|_{L^p(\mathbb R^2)}\leq C_p \|\{f_m\}\|_{L^p(\mathbb R^2,\ell^2)}, \]
where \(C_p\) does not depend on the functions or the rectangles.
This result can be considered as an extension of (the dual of) the Littlewood-Paley inequality proved by Journé for \(1< p\leq 2\) (valid for \({\mathbb R}^n\)), for which F. Soria [J. Lond. Math. Soc., II. Ser. 36, No. 1–2, 137–142 (1987; Zbl 0592.42012)] gave a simpler proof if \(n=2\). The ideas of Soria’s proof are used in this dual situation.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B15 Multipliers for harmonic analysis in several variables

Citations:

Zbl 0592.42012
Full Text: DOI

References:

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