Boundedness and compactness of commutators associated with Lipschitz functions

Let $\alpha \in (0,1]$, $\beta \in [0,n)$ and $T_{\mathrm{\Omega },\beta }$ be a singular or fractional integral operator with homogeneous kernel $\mathrm{\Omega }$. In this article, a CMO type space ${\mathrm{C}\mathrm{M}\mathrm{O}}_{\alpha }({\mathbb{R}}^n)$ is introduced and studied. In particular, the relationship between ${\rm CMO}\alpha(\mathbb R^n)$andtheLipchitzspace$Lip\alpha(\mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{\mathrm{\Omega },\beta }{)}_b^m$ on weighted Lebesgue spaces via functions in $Li{p}_{\alpha }({\mathbb{R}}^n)$, and an equivalent characterization of the compactness for $(T_{\mathrm{\Omega },\beta }{)}_b^m$ via functions in ${\rm CMO}_\alpha(\mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators.