Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Endpoint Mapping properties of the Littlewood-Paley square function   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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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg