A Fefferman-Stein inequality for the Carleson operator

We provide a Fefferman-Stein type weighted inequality for maximally modulated Calder'on-Zygmund operators that satisfy $\mathit{\text{a}}\mathit{\text{priori}}$ weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of P'erez. Applying it to the Hilbert transform we obtain the corresponding Fefferman-Stein inequality for the Carleson operator $\mathcal{C}$, that is $\mathcal{C}:L^p(M^{\lfloor p\rfloor +1}w)\to L^p(w)$ for any $1<p<\mathrm{\infty }$ and any weight function $w$, with bound independent of $w$. We also provide a maximal-multiplier 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 P'erez.