Weak-L^p bounds for the Carleson and Walsh-Carleson operators

We prove a weak-$L^p$ bound for the Walsh-Carleson operator for $p$ near 1, improving on a theorem of Sjolin. We relate our result to the conjectures that the Walsh-Fourier and Fourier series of a function $f\in L\log L(\mathbb T)$ converge for almost every $x \in \mathbb T$.

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