Two inequalities for commutators of singular integral operators satisfying Hörmander conditions of Young type

In this paper, we systematically study the Fefferman-Stein inequality and Coifman-Fefferman inequality for the general commutators of singular integral operators that satisfy Hörmander 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 $\epsilon$, which essentially improved the results of Pérez and Rivera-Ríos (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 $L^r$-Hörmander operators, $\omega$-Calderón-Zygmund operators with $\omega$ satisfying a Dini condition, Calderón commutators, homogeneous singular integral operators and Fourier multipliers.