  Let $T$ be a multilinear Calder\'on-Zygmund operator of type $\omega$ with
$\omega(t)$ being nondecreasing and satisfying a kind of Dini's type condition.
Let $T_{\Pi\vec{b}}$ be the iterated commutators of $T$ with $BMO$ functions.
The weighted strong and weak $L(\log{L})$-type endpoint estimates for
$T_{\Pi\vec{b}}$ with multiple weights are established. Some boundedness
properties on weighted variable exponent Lebesgue spaces are also obtained.
  As applications, multiple weighted estimates for iterated commutators of
paraproducts and bilinear pseudo-differential operators with mild regularity
are given.
