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 $n=1$.
Categories: math.CA