The intrinsic square functions including the Lusin area function, Littlewood-Paley
g
g
-function and
g
λ
∗
g^\ast _\lambda
-function dominate pointwisely the classical Littlewood-Paley functions and can be used to characterize the weighted Hardy spaces and more general Musielak-Orlicz Hardy spaces etc. This paper shows that for
b
∈
B
M
O
(
R
n
)
b\in \mathrm {BMO(\mathbb {R}^n)}
, the commutators generated by these intrinsic square functions with
b
b
are bounded from
H
ω
p
(
R
n
)
H^p_\omega (\mathbb {R}^n)
to
L
ω
p
(
R
n
)
L^p_\omega (\mathbb {R}^n)
for some
0
>
p
≤
1
0>p\le 1
and
ω
∈
A
∞
\omega \in A_\infty
if and only if
b
∈
B
M
O
ω
,
p
(
R
n
)
b\in \mathcal {BMO}_{\omega ,p}(\mathbb {R}^n)
, which are a class of non-trivial subspaces of
B
M
O
(
R
n
)
\mathrm {BMO(\mathbb {R}^n)}
.