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arXiv · 1612.09573

Endpoint Mapping properties of the Littlewood-Paley square function

Odysseas Bakas
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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 $n=1$.

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Odysseas Bakas. 2016-12-30. Endpoint Mapping properties of the Littlewood-Paley square function. https://arxiv.org/abs/1612.09573

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