Boundedness and compactness of commutators associated with Lipschitz functions

Let [Formula: see text], text] and be a singular or fractional integral operator with homogeneous kernel text]. In this paper, CMO type space is introduced studied. particular, the relationship between Lipchitz discussed. Moreover, necessary condition of restricted boundedness iterated commutator on weighted Lebesgue spaces via functions in an equivalent characterization compactness for are obtained. Some results new even unweighted setting first-order commutators.