  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 Sawyer's 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.
