Two inequalities for commutators of singular integral operators satisfying H\"{o}rmander conditions of Young type

TL;DR

This paper establishes new inequalities for commutators of singular integral operators satisfying Hörmander conditions of Young type, using sparse domination and dyadic analysis, with applications to various classical operators.

Contribution

It introduces pointwise sparse domination and improves bounds relating the number of commutators and the index, extending previous results to a broader class of operators.

Findings

Proved Fefferman-Stein and Coifman-Fefferman inequalities for these commutators.

Decoupled the relationship between the number of commutators and the index .

Applied results to various operators including Calderf3n-Zygmund and Fourier multipliers.

Abstract

In this paper, we systematically study the Fefferman-Stein inequality and Coifman-Fefferman inequality for the general commutators of singular integral operators that satisfy H\"{o}rmander conditions of Young type. Specifically, we first establish the pointwise sparse domination for these operators. Then, relying on the dyadic analysis, the Fefferman-Stein inequality with respect to arbitrary weights and the quantitative weighted Coifman-Fefferman inequality are demonstrated. We decouple the relationship between the number of commutators and the index $\varepsilon$, which essentially improved the results of P\'{e}rez and Rivera-R\'{\i}os (Israel J. Math., 2017). As applications, it is shown that all the aforementioned results can be applied to a wide range of operators, such as singular integral operators satisfying the $L^r$-H\"{o}rmander operators, $\omega$-Calder\'{o}n-Zygmund operators with $\omega$ satisfying a Dini condition, Calder\'{o}n commutators, homogeneous singular integral operators and Fourier multipliers.