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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

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Weighted Littlewood–Paley inequality for arbitrary rectangles in $\mathbb {R}^2$
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by V. Borovitskiy
Translated by: the author
St. Petersburg Math. J. 32 (2021), 975-997
Published electronically: October 20, 2021


Weighted counterparts of the one-sided Littlewood–Paley inequalities for arbitrary rectangles in $\mathbb {R}^2$ are proved.

For a partition $\mathcal {I}$ of the plane $\mathbb {R}^2$ into rectangles with sides parallel to coordinate axes and a weight $w(\,\cdot \,, \,\cdot \,)$ satisfying the two-parameter Muckenhoupt condition $A_{p/2}$ for $2 < p < \infty$, the following inequality holds: \begin{equation*} c_{p, w}\lVert \{M_I f\}_{I \in \mathcal {I}} \rVert _{L^p_w(l^2)} \leq \lVert f \rVert _{L_w^p} , \end{equation*} where the symbols $\widehat {M_I f} = \widehat {f} \chi _{I}$ denote the corresponding Fourier multipliers.

For $\mathcal {I}$ as above, $p$ in the range $0 < p < 2$, and weights $w$ satisfying a dual condition $\alpha _{r(p)}$, the following inequality holds \begin{equation*} \Big \|{\sum }_{I \in \mathcal {I}} f_I\Big \|_{L^p_w} \leq C_{p, w} \big \| \left \{ f_I \right \}_{I \in \mathcal {I}} \big \|_{L^p_w(l^2)} , \text { where } \operatorname {supp}{\widehat {f_I}} \subseteq I \text { for } I \in \mathcal {I}. \end{equation*} The proof is based on the theory of two-parameter singular integral operators on Hardy spaces developed by R. Fefferman and some of its more recent weighted generalizations. The former and the latter inequalities are extensions to the weighted setting, respectively, for Journe’s result of 1985 and Osipov’s result of 2010.

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Bibliographic Information
  • V. Borovitskiy
  • Affiliation: St. Petersburg Department of Steklov Mathematical Institute and St. Petersburg State University
  • Email:
  • Received by editor(s): December 14, 2019
  • Published electronically: October 20, 2021
  • Additional Notes: This research was supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1620), and by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”
  • © Copyright 2021 American Mathematical Society
  • Journal: St. Petersburg Math. J. 32 (2021), 975-997
  • MSC (2020): Primary 42B25; Secondary 42B30
  • DOI:
  • MathSciNet review: 4219490