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

Abstract

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 $x\in \mathbb{T}$.

11 pages, with a 5 pages appendix. An abridged version (without appendix) of this article has been accepted for publication on Comptes Rendus Mathematiques

References in corpus (2)

• Lacunary Fourier and Walsh-Fourier series near L^1
• On the pointwise convergence of the sequence of partial Fourier Sums along lacunary subsequences

Cited by in corpus (4)

• On weighted norm inequalities for the Carleson and Walsh-Carleson operators
• A modulation invariant Carleson embedding theorem outside local $L^2$
• The pointwise convergence of Fourier Series (I). On a conjecture of Konyagin
• Endpoint bounds for the bilinear Hilbert transform