Littlewood--Paley--Rubio de Francia inequality for unbounded Vilenkin systems

TL;DR

This paper extends the Littlewood--Paley--Rubio de Francia inequality to unbounded Vilenkin systems, broadening its applicability to functions on infinite products of cyclic groups without restrictions on group orders.

Contribution

It proves a Littlewood--Paley--Rubio de Francia inequality for unbounded Vilenkin systems, generalizing previous results to more complex group structures.

Findings

Established inequality for arbitrary Vilenkin systems.

No restrictions on the orders of cyclic groups.

Extended the inequality to infinite product groups.

Abstract

Rubio de Francia proved the one-sided version of Littlewood--Paley inequality for arbitrary intervals. In this paper, we prove the similar inequality in the context of arbitrary Vilenkin systems (that is, for functions on infinite products of cyclic groups). There are no assumptions on the orders of these groups.