Weighted mixed endpoint estimates of Fefferman-Stein type for commutators of singular integral operators

TL;DR

This paper establishes weighted mixed endpoint estimates of Fefferman-Stein type for higher order commutators of Calderón-Zygmund operators with BMO symbols, extending previous results to less regular kernels and broader contexts.

Contribution

It introduces new weighted mixed endpoint estimates for commutators with less regular kernels, generalizing prior Fefferman-Stein inequalities and classical weak endpoint estimates.

Findings

Fefferman-Stein inequalities for higher order commutators

Extension to operators with less regular kernels

Inclusion of previous estimates as special cases

Abstract

We deal with mixed weak estimates of Fefferman-Stein type for higher order commutators of Calder\'on-Zygmund operators with BMO symbol. The results obtained are Fefferman-Stein inequalities that include the estimates proved in \cite{BCP22(JMS)} for the case of singular integral operators, as well as the classical weak endpoint estimate for commutators given in \cite{PP01}. We also consider commutators of operators involving less regular kernels satisfying an $L^{\Phi}$--H\"ormander condition. Particularly, the obtained results contain some previous estimates proved in \cite{BCP22(JMS)} and \cite{Lorente-Martell-Perez-Riveros}.