Compactness of the bloom sparse operators and applications

Abstract We establish the characterization of compactness for the sparse operator (associated with symbol in weighted VMO space) in the two weight setting on the spaces of homogeneous type in the sense of Coifman and Weiss. As an application, we obtain the compactness characterization for the commutator of Calderón–Zygmund operators on the homogeneous spaces. Furthermore, our approach can be applied to proving compactness for sparse operators in the multi-linear setting.