# On the compactness of oscillation and variation of commutators

## TL;DR

This paper investigates the compactness properties of oscillation and variation in commutators of singular integral operators, providing new characterizations and weighted compactness results in harmonic analysis.

## Contribution

It introduces a novel characterization of CMO spaces through the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.

## Key findings

- Weighted compactness results for oscillation and variation of commutators
- New characterization of CMO(R^n) via compactness properties
- Extension of compactness results to weighted Lebesgue spaces

## Abstract

In this paper, we first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, we establish a new $CMO(\mathbb{R}^n)$ characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.