Fourier multipliers for Sobolev spaces on the Heisenberg groupМультипликаторы Фурье для пространств Соболева на группе Гейэенберга

In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space Wk,p(Hn) coincides with the class of right Fourier multipliers for Lp(Hn) for k ∈ ℕ, 1 < p < ∞. Towards this end, it is shown that the operators Rj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar R $$\end{document}jℒ−1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar R $$\end{document}jRjℒ−1 are bounded on Lp(Hn), 1 < p < ∞, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_j = \frac{\partial } {{\partial z_j }} - \frac{i} {4}\bar z_j \frac{\partial } {{\partial t}}, \bar R_j = \frac{\partial } {{\partial \bar z_j }} + \frac{i} {4}z_j \frac{\partial } {{\partial t}} $$\end{document} and ℒ is the sublaplacian on Hn. This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space Wk,1(Hn) coincides with the dual space of the projective tensor product of two function spaces.