# St. Petersburg Mathematical Journal

Published by the American Mathematical Society, the St. Petersburg Mathematical Journal (SPMJ) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.54.

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## Weighted Littlewood–Paley inequality for arbitrary rectangles in $\mathbb {R}^2$HTML articles powered by AMS MathViewer

by V. Borovitskiy
Translated by: the author
St. Petersburg Math. J. 32 (2021), 975-997 Request permission

## Abstract:

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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