A remark on bilinear Littlewood–Paley square functions

The aim of this note is to point out that the results proved in Bernicot and Shrivastava (Indiana Univ Math J 60(1):233–268, 2011) and Ratnakumar and Shrivastava (Proc Am Math Soc 140(12):4285–4293, 2012), for bilinear square functions associated with symbols of the form φ(x-n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (x-n)$$\end{document}, extend to a far more general family of bilinear multiplier symbols ϕn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _n$$\end{document}, supported in disjoint cubes Qn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_n$$\end{document}.