Compactness of the Bloom sparse operators and applications

TL;DR

This paper characterizes the compactness of Bloom sparse operators and related commutators on homogeneous spaces, extending the understanding of operator compactness in weighted VMO contexts and their multilinear generalizations.

Contribution

It provides a new characterization of compactness for Bloom sparse operators and their commutators in weighted VMO spaces on homogeneous spaces, including multilinear cases.

Findings

Characterization of compactness for Bloom sparse operators.

Compactness criteria for maximal commutators with weighted VMO functions.

Extension of compactness results to multilinear Bloom operators.

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 a direct application we obtain the compactness characterization for the maximal commutators with respect to the weighted VMO functions and the commutator of Calder\'on-Zygmund operators on the homogeneous spaces. Furthermore, our approach can be applied to compactness characterization for operators in the multilinear Bloom setting.