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Abstract

We give an alternative proof of a theorem due to 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, and on Tao’s converse extrapolation theorem. 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, in contrast to what happens when $n=1$.