Some notes on the vector-valued extension of Littlewood--Paley--Rubio de Francia inequality for Walsh functions

TL;DR

This paper explores the extension of Littlewood--Paley--Rubio de Francia inequalities to vector-valued functions with Walsh functions, analyzing properties of Banach spaces where these inequalities hold.

Contribution

It investigates conditions on Banach spaces that allow the Walsh function inequality to extend to vector-valued functions, advancing understanding of vector-valued harmonic analysis.

Findings

Identifies properties of Banach spaces supporting the inequality

Extends scalar inequalities to vector-valued functions in Walsh analysis

Provides insights into the structure of Banach spaces related to harmonic analysis

Abstract

J. L. Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2\le p<\infty$ and later N. N. Osipov proved the similar inequality for Walsh functions. In this paper we investigate some properties of Banach spaces $X$ such that the latter inequality holds for $X$-valued functions.