Boundedness for commutators of Littlewood-Paley operators on some Hardy spaces

$(H_b^p$, $L^p)$-type and $(H_b^{p,\infty }$ , $L^{p,\infty })$-type boundedness for the commutators associated with the Littlewood-Paley operators and $b\in \mathrm{BMO}\left(R^n\right)$ are obtained, where $H_b^p,$ and $H_b^{p,\infty }$ are, respectively, variants of the standard Hardy spaces and weak Hardy spaces, and $\frac{n}{(n+\epsilon )}<p\le 1$

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