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

TL;DR

This paper extends Rubio de Francia's Littlewood--Paley inequality to the Walsh system, providing a new inequality for this specific orthogonal system in harmonic analysis.

Contribution

The paper proves a Littlewood--Paley inequality for the Walsh system, which was previously established only for the classical Fourier setting.

Findings

Establishes the inequality for the Walsh system in L^p spaces.

Bridges a gap between classical harmonic analysis and Walsh analysis.

Provides tools for further analysis in Walsh function spaces.

Abstract

Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, such an inequality is proved for the Walsh system.