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

In the present paper the $(H_b^p,L^p)$-type and $(H_b^{p,\infty},L^{p,\infty})$-type boundedness for the commutators associated with the Littlewood-Paley operators and $b\in BMO(R^n)$ are obtained, where $H^p_b$ and $H_b^{p,\infty}$ are, respectively, variants of the standard Hardy spaces and weak Hardy spaces, and $n/(n+\varepsilon)<p\leq 1$.