On the Boundedness of the Carleson Operator near $L^1$

Abstract

Based on the tile discretization elaborated by the author in "The Polynomial Carleson Operator", we develop a Calderon-Zygmund type decomposition of the Carleson operator. As a consequence, through a unitary method that makes no use of extrapolation techniques, we recover the previously known results regarding the largest rearrangement invariant space of functions with almost everywhere convergent Fourier series.

23 pages, no figures. Some more details inserted in the proof of Lemma 2. The Calderon-Zygmund tile decomposition in Section 5.2. adapted to the behavior of the adjoint Carleson operator (minor errors have been corrected with respect to the previous version of the paper). Several typos further corrected. To appear in Revista Mat. Iberoamericana

Cited by in corpus (2)

• On weighted norm inequalities for the Carleson and Walsh-Carleson operators
• On the pointwise convergence of the sequence of partial Fourier Sums along lacunary subsequences