Weighted Bounds for Multilinear Square Functions

Let P→=(p1,…,pm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vec {P}=(p_{1},\dotsc ,p_{m})$\end{document} with 1 < p1, …, pm < ∞, 1/p1+⋯+1/pm=1/p and w→=(w1,…,wm)∈AP→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vec {w}=(w_{1},\dotsc ,w_{m})\in A_{\vec {P}}$\end{document}. In this paper, we investigate the weighted bounds with dependence on aperture α for multilinear square functions Sα,ψ(f→)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{\alpha ,\psi }(\vec {f})$\end{document}. We show that