Bilinear Spectral Multipliers on Heisenberg Groups

As we know, thus far, there has appeared no definition of bilinear spectral multipliers on Heisenberg groups. In this article, we present one reasonable definition of bilinear spectral multipliers on Heisenberg groups and investigate its boundedness. We find some restrained conditions to separately ensure its boundedness from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal C}_0}\left({{\mathbb{H}^n}} \right) \times {L^2}\left({{\mathbb{H}^n}} \right)\;{\rm{to}}\;{L^2}\left({{\mathbb{H}^n}} \right)$$\end{document}, from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^2}\left({{\mathbb{H}^n}} \right) \times {{\cal C}_0}\left({{\mathbb{H}^n}} \right)\;{\rm{to}}\;{L^2}\left({{\mathbb{H}^n}} \right)$$\end{document}, and from Lp × Lq to Lr with 2 < p, q < ∞, 2 ≤ r ≤ ∞.