  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 $n=1$.
