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

Abstract

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