Weak and strong type estimates for the multilinear Littlewood-Paley operators

Let $S_{\alpha}$ be the multilinear square function defined by means of the cone in $\mathbb{R}_+^{n+1}$ with aperture $\alpha \geq 1$. In this paper we prove that the estimate \begin{equation*} \begin{split} ||S_{\alpha}(\vec{f})||_{L^{p}(\nu_{\vec{w}})} \le C_{n,m,\psi,\vec{p}} \alpha^{mn} [\vec{w}]_{A_{\vec{p}}}^{\max\{\frac{1}{2},\frac{p_1'}{p},\cdots,\frac{p_m'}{p}\}}\prod_{i=1}^m ||f_i||_{L^{p_i}(w_i)} \end{split} \end{equation*} is sharp in $\alpha$ for all $\vec{w} \in A_{\vec{p}}$, where $\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}$ with $1更多