Littlewood-Paley-Rubio de Francia Inequality for the Two-parameter Walsh System

TL;DR

This paper proves a Littlewood-Paley-Rubio de Francia inequality for the two-parameter Walsh system, extending harmonic analysis tools to a multi-parameter setting using martingale Hardy space theory.

Contribution

It introduces a two-parameter version of the inequality and develops a related two-parameter Gundy's theorem, advancing the analysis of Walsh systems.

Findings

Established the inequality for the two-parameter Walsh system.

Developed a two-parameter Gundy's theorem for martingale operators.

Provided bounds independent of the choice of rectangles and functions.

Abstract

A version of Littlewood-Paley-Rubio de Francia inequality for the two-parameter Walsh system is proved: 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 \[\left\lVert\sum\limits_k f_k\right\rVert_{L^p} \leq C_p \left\lVert\left(\sum\limits_k \left|f_k\right|^2\right)^{1/2}\right\rVert_{L^p}, \qquad 1 < p \leq 2,\] where $C_p$ does not depend on the choice of rectangles $\left\{I_k\right\}$ or functions $\left\{f_k\right\}$. The arguments are based on the atomic theory of two-parameter martingale Hardy spaces. In the course of the proof, a two-parameter version of Gundy's theorem on the boundedness of operators taking martingales to measurable functions is formulated, which might be of independent interest.