Let \(S_{\alpha }\) be the multilinear square function defined on the cone with aperture \(\alpha \ge 1\). 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.

让\(S_{\alpha }\)是定义在具有孔径\(\alpha \ge 1\)的锥体上的多线性平方函数。在本文中，我们研究了\(S_{\alpha }\) 的几种加权范数不等式。我们首先在孔径\(\alpha \)和\(\vec {w} \in A_{\vec {p}}\)方面获得清晰的加权估计。通过一些逐点估计，我们还建立了两权重不等式，包括凹凸和熵凹凸估计，以及具有任意权重的 Fefferman-Stein 不等式。除此之外，我们考虑对应于 Sawyer 猜想的混合弱类型估计，为此 Coifman-Fefferman 不等式具有精确的\(A_{\infty }\)范数得到证明。最后，我们分别使用外推技术和二元分析来呈现局部衰减估计。上述所有结论都适用于 Littlewood–Paley \(g^*_{\lambda }\)函数。即使在线性情况下，一些结果也是新的。