Two weight commutators for Beurling--Ahlfors operator

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.