Revisiting mixed weak inequalities of Fefferman-Stein type for commutators of Calderón-Zygmund operators: an improvement

TL;DR

This work tackles mixed weak Fefferman-Stein inequalities for Calderón-Zygmund operators and their commutators in a weighted setting. By developing a strong Fefferman-Stein-type inequality with respect to a weight measure and employing a Calderón-Zygmund decomposition relative to $d\mu(x)=v(x)\,dx$, the authors derive improved two-weight mixed inequalities for both $T$ and $T_b^m$ that generalize classical results. The key contributions include sharp inequalities for $\int |T_b^m f|^p w v^{1-p}$ in terms of Luxemburg maximal operators $M_{\Phi_{m+\varepsilon},v^{1-q}}$, with precise weight and Young-function conditions, extending results of Pérez, Sawyer, Pérez–Pérez, and BCP/BPR frameworks. The findings advance weighted harmonic analysis by providing unified, sharper bounds for CZOs and their higher-order commutators under broad two-weight hypotheses, with potential implications for related PDE estimates.

Abstract

In this paper we establish mixed weak inequalities of Fefferman-Stein type for Calderón-Zygmund operators and their commutators, improving some previous results known in the literature. The main estimates also generalize the classical weighted weak Fefferman-Stein inequalities proved in [19] and [22]. In order to obtain the main results, our approach is to give a strong Fefferman-Stein type inequality for the operators involved with respect to an adequate measure.

Theorems & Definitions (36)

• Remark 1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Remark 2
• Proposition 2.2
• proof
• Proposition 2.3
• proof