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.