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Let p ? ( 1 , 8 ) $p\in (1, \infty )$ , ? ? (0,1) and w ? A p ( C ) . $w\in A_{p}({\mathbb C}).$ In this article, the authors obtain a boundedness (resp., compactness) characterization of the Beurling–Ahlfors commutator [ B , b ] $[\mathcal B, b]$ on the weighted Morrey space L w p , ? ( C ) $L_{w}^{p, \kappa }(\mathbb C)$ via BMO ( C ) $\text {BMO}({\mathbb C})$ [resp., CMO ( C ) $\text {CMO}({\mathbb C})$ ], where B $\mathcal B$ denotes the Beurling–Ahlfors transform and b ? BMO ( C ) $b\in \text {BMO}({\mathbb C})$ [resp., CMO ( C ) $\text {CMO}({\mathbb C})$ ]. Moreover, an application to the Beltrami equation is also given.

Beurling–Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations