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

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.