On the boundedness of the Carleson operator near $L^1$

Abstract

Based on the tile discretization elaborated in [14], we develop a Calderón–Zygmund type decomposition of the Carleson operator. As a consequence, through a unitary method that makes no use of extrapolation techniques, we recover previously known results regarding the largest rearrangement invariant space of functions with almost everywhere convergent Fourier series.