Weighted Littlewood–Paley inequality for arbitrary rectangles in $\mathbb {R}^2$

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.