  Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a
singular or fractional integral operator with homogeneous kernel $\Omega$. In
this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced
and studied. In particular, the relationship between ${\rm CMO}_\alpha(\mathbb
R^n)$ and the Lipchitz space $Lip_\alpha(\mathbb R^n)$ is discussed. Moreover,
a necessary condition of restricted boundedness of the iterated commutator
$(T_{\Omega,\beta})^m_b$ on weighted Lebesgue spaces via functions in
$Lip_\alpha(\mathbb R^n)$, and an equivalent characterization of the
compactness for $(T_{\Omega,\beta})^m_b$ 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.
