Littlewood--Paley--Rubio de Francia inequality for the two-parameter Walsh system

V. Borovitskiy

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 48, Tome 491 (2020), pp. 27-42

Résumé

We prove a version of Littlewood–Paley–Rubio de Francia inequality for the two-parameter Walsh system: for any family of disjoint rectangles $I_k = I_k^1 \times I_k^2$ in ${\mathbb{Z}_+ \times \mathbb{Z}_+}$ and a family of functions $f_k$ with Walsh spectrum inside $I_k$ the following is true $$ \left\|\sum\limits_k f_k\right\|_{L^p} \leq C_p \left\|\left(\sum\limits_{k = 1}^\infty |f_k|^2\right)^{1/2}\right\|_{L^p} , 1 p \leq 2, $$ where $C_p$ does not depend on the choice of rectangles $\{I_k\}$ or functions $\{f_k\}$. The arguments are based on the atomic theory of two-parameter martingale Hardy spaces. In the course of the proof, we formulate a two-parametric version of the Gundy theorem on the boundedness of operators taking martingales to measurable functions, which might be of independent interest.