Buerling-Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations

Abstract

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

24 pages; Submitted