  In this paper, the boundedness properties of commutators generated by $b$ and
intrinsic square functions in the endpoint case are discussed, where $b\in
BMO(\mathbb R^n)$. We first establish the weighted weak $L\log L$-type
estimates for these commutator operators. Furthermore, we will prove endpoint
estimates of commutators generated by $BMO(\mathbb R^n)$functions and intrinsic
square functions in the weighted Morrey spaces $L^{1,\kappa}(w)$ for
$0<\kappa<1$ and $w\in A_1$, and in the generalized Morrey spaces
$L^{1,\Theta}$, where $\Theta$ is a growth function on $(0,+\infty)$ satisfying
the doubling condition.
