Let $S_{alpha}$ be the multilinear square function defined on the cone with aperture $alpha geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_{alpha}$. We first obtain a sharp weighted estimate in terms of aperture $alpha$ and $vec{w} in A_{vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyers conjecture, for which a Coifman-Fefferman inequality with the precise $A_{infty}$ norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood-Paley $g^*_{lambda}$ function. Some results are new even in the linear case.
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