Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The weak-type Carleson theorem via wave packet estimates   We prove that the weak-$L^{p}$ norms, and in fact the sparse $(p,1)$-norms,
of the Carleson maximal partial Fourier sum operator are $\lesssim (p-1)^{-1}$
as $p\to 1^+$. This is an improvement on the Carleson-Hunt theorem, where the
same upper bound on the growth order is obtained for the restricted weak-$L^p$
type norm, and which was the strongest quantitative bound prior to our result.
Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at
the endpoint $p=1$. In particular, we obtain that the Fourier series of
functions from the weighted Arias de Reyna space $ QA_{\infty}(w) $, which
contains the weighted Antonov space $L\log L\log\log\log L(\mathbb T; w)$,
converge almost everywhere whenever $w\in A_1$. This is an extension of the
results of Antonov and Arias De Reyna, where $w$ must be Lebesgue measure.
  The backbone of our treatment is a new, sharply quantified near-$L^1$
Carleson embedding theorem for the modulation-invariant wave packet transform.
The proof of the Carleson embedding relies on a newly developed smooth
multi-frequency decomposition which, near the endpoint $p=1$, outperforms the
abstract Hilbert space approach of past works, including the seminal one by
Nazarov, Oberlin and Thiele. As a further example of application, we obtain a
quantified version of the family of sparse bounds for the bilinear Hilbert
transforms due to Culiuc, Ou and the first author.

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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas