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

TL;DR

This paper characterizes the boundedness and compactness of Buerling-Ahlfors commutators on weighted Morrey spaces using BMO and CMO spaces, and applies these results to solve Beltrami equations.

Contribution

It provides new boundedness and compactness criteria for commutators on weighted Morrey spaces and connects these to solutions of Beltrami equations.

Findings

Characterization of boundedness via BMO spaces.

Characterization of compactness via CMO spaces.

Application to solving 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.