Endpoint Mapping properties of the Littlewood-Paley square function

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 $p\to 1^{+}$. 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 $p\to 1^{+}$. Furthermore, we obtain sharp weak-type inequalities for the Littlewood-Paley square function on ${\mathbb{T}}^n$, but when $n\ge 2$ the weak-type endpoint estimate on the product Hardy space over the $n$-torus fails, contrary to what happens when $n=1$.

10 pages