We provide a Fefferman-Stein type weighted inequality for maximally modulated Calder\'on-Zygmund operators that satisfy \textit{ a 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 $1<p<\infty$ 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.
Categories: math.CA