In the present paper the $\left(H_b^p, L^p\right)$-type and $\left(H_b^{p, \infty}, L^{p, \infty}\right)$-type boundedness for the commutators associated with the Littlewood-Paley operators and $b\in BMO(R^n)$ are obtained, where $H_b^p$ and $H_b^{p,\infty}$ are, respectively, variants of the standard Hardy spaces and weak Hardy spaces, and $n/(n+\varepsilon)
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