Two weight commutators for Beurling--Ahlfors operator

TL;DR

This paper characterizes weighted BMO spaces on the complex plane using two-weight commutators of the Beurling--Ahlfors operator, leveraging explicit kernels and Muckenhoupt weights.

Contribution

It provides a new characterization of weighted BMO spaces via commutators with the Beurling--Ahlfors operator, connecting harmonic analysis and complex analysis.

Findings

Characterization of weighted BMO via two-weight commutators

Utilization of explicit kernel of Beurling--Ahlfors operator

Application of Muckenhoupt weight properties

Abstract

We establish the equivalent characterisation of the weighted BMO space on the complex plane $\mathbb{C}$ via the two weight commutator of the Beurling--Ahlfors operator with a BMO function. Our method of proofs relies on the explicit kernel of the Beurling--Ahlfors operator and the properties of Muckenhoupt weight class.