Lp-Spectral Multipliers for the Hodge Laplacian Acting on 1-Forms on the Heisenberg Group

We prove that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Delta_{1}$$ \end{document} is the Hodge Laplacian acting on differential 1-forms on the (2n + 1)-dimensional Heisenberg group, and if m is a Mihlin–Hörmander multiplier on the positive half-line, with L2-order of smoothness greater than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$n + \frac{1}{2}$$ \end{document} , then m(Δ1) is Lp-bounded for 1 < p < ∞. Our approach leads to an explicit description of the spectral decomposition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Delta_{1}$$ \end{document} on the space of L2-forms in terms of the spectral analysis of the sub-Laplacian L and the central derivative T, acting on scalar-valued functions.