A Fefferman-Stein inequality for the Carleson operator
We provide a Fefferman-Stein type weighted inequality for maximally modulated Calderón-Zygmund operators that satisfy a priori weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of Pérez. Applying it to the Hilbert transform we obtain the corresponding inequality for the Carleson operator C, that is C : Lp(Mp+1w) → Lp(w) for any 1 < p < ∞ and any weight function w, with bound independent of w. We also provide a maximalmultiplier weighted theorem, a vector-valued extension, and more general two-weighted inequalities. Our proof builds on a recent work of Di Plinio and Lerner combined with some results on Orlicz spaces developed by Pérez.