## (Lobachevskii Journal of Mathematics, Vol.12, pp.63-71)

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)