On the L2 boundedness for the multiple Littlewood-Paley functions with rough kernels

In this paper, we give a characterization of the L2 bounds for the Littlewood-Paley functions on product domains. As application, a new size condition, which implies the L2 boundedness of the multiple parametric Marcinkiewicz integrals with rough kernels and is weaker than the previous ones, is given.