A note on commutators on weighted Morrey spaces on spaces of homogeneous type

TL;DR

This paper characterizes the boundedness and compactness of commutators of Calderón-Zygmund operators on weighted Morrey spaces over spaces of homogeneous type, linking these properties to BMO and VMO function spaces.

Contribution

It provides necessary and sufficient conditions for the boundedness and compactness of commutators on weighted Morrey spaces in the setting of spaces of homogeneous type.

Findings

Boundedness of commutators characterized by BMO functions.

Compactness of commutators characterized by VMO functions.

Results extend to weighted Morrey spaces on spaces of homogeneous type.

Abstract

In this paper we study the boundedness and compactness characterizations of the commutator of Calder\'{o}n-Zygmund operators $T$ on spaces of homogeneous type $(X,d,\mu)$ in the sense of Coifman and Weiss. More precisely, We show that the commutator $[b, T]$ is bounded on weighted Morrey space $L_{\omega}^{p,\kappa}(X)$ ($\kappa\in(0,1), \omega\in A_{p}(X), 1<p<\infty$) if and only if $b$ is in the BMO space. Moreover, the commutator $[b, T]$ is compact on weighted Morrey space $L_{\omega}^{p,\kappa}(X)$ ($\kappa\in(0,1), \omega\in A_{p}(X), 1<p<\infty$) if and only if $b$ is in the VMO space.