Endpoint Mapping properties of the Littlewood-Paley square function

TL;DR

This paper provides an alternative proof for the growth of constants in Littlewood-Paley inequalities on tori as p approaches 1, using endpoint mapping properties and extrapolation theorems, and explores weak-type inequalities and their limitations.

Contribution

It introduces a new proof approach based on endpoint properties and extends results to higher dimensions, also analyzing weak-type inequalities and endpoint failures.

Findings

Growth of Littlewood-Paley constants as p→1+ on torus

Sharp weak-type inequalities for Littlewood-Paley square function

Failure of endpoint estimates on product Hardy spaces for n  2

Abstract

In this note we give an alternative proof of a theorem due to Bourgain \cite{Bourgain} concerning the growth of the constant in the Littlewood-Paley inequality on $\mathbb{T}$ as $p \rightarrow 1^+$. Our argument is based on the endpoint mapping properties of Marcinkiewicz multiplier operators, obtained by Tao and Wright in \cite{TW}, and on Tao's converse extrapolation theorem \cite{Tao}. Our method also establishes the growth of the constant in the Littlewood-Paley inequality on $\mathbb{T}^n$ as $p \rightarrow 1^+$. Furthermore, we obtain sharp weak-type inequalities for the Littlewood-Paley square function on $\mathbb{T}^n$, but when $n \geq 2$ the weak-type endpoint estimate on the product Hardy space over the $n$-torus fails, contrary to what happens when $n=1$.