A unified approach to the boundedness of a class of multilinear operators and their iterated commutators

In this paper, we systematically investigate the boundedness of a wide class of multilinear operators and their iterated commutators, under the sole assumption that they satisfy a certain type of sparse domination condition. This class of operators not only includes the classical multilinear Calderón–Zygmund operators but also encompasses a variety of non-convolution-type operators that go beyond the classical framework. More specifically, we first establish a quantitative weighted version of the Coifman–Fefferman inequality, based on which we prove the weighted restricted weak-type inequality for multilinear operators and its two-weight version. Second, we derive a multi-weights version of the Sawyer-type inequality in Lorentz spaces. Finally, we establish a quantitative weighted version of the two-weight bump estimate and obtain the Fefferman–Stein inequality for arbitrary weights. As applications, we demonstrate that several prominent classes of classical operators, including multilinear [Formula: see text]-Calderón–Zygmund operators, multilinear pseudo-differential operators, Calderón commutators, multilinear Stein’s square functions, and Marcinkiewicz integral operators, all satisfy the aforementioned results.