A Fefferman-Stein inequality for the Carleson operator

TL;DR

This paper establishes a Fefferman-Stein type weighted inequality for the Carleson operator, extending classical results to a weighted setting and providing new bounds and extensions for related operators.

Contribution

It introduces a novel weighted inequality for the Carleson operator, generalizing previous unweighted results and including vector-valued and two-weighted inequalities.

Findings

Fefferman-Stein inequality for the Carleson operator established

Bound independent of weight function w

Extended results to maximal-multiplier and vector-valued cases

Abstract

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.