On Paley-Wiener and Hardy theorems for NA groups

Let N be a H-type group and let S=NA be an one dimensional solvable extension of N. For the Helgason Fourier transform on S we prove the following analogue of Hardy’s theorem. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat f$\end{document}(λ,Y,Z) stand for the Helgason Fourier transform of f and let hα denote the heat kernel associated to the Laplace-Beltrami operator. Suppose a function f on S satisfies the conditions |f(x)| ≤ chα(x) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\int\limits_N |\hat f (\lambda, Y,Z)|^2 (1+|Z|^2)^\gamma dY dZ \leq c e^{{-2\beta \lambda^2}}}}$$\end{document} for all xS,λℝ where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\gamma > \frac{{k-1}}{{2}}, k}}$\end{document} being the dimension of the centre of N. Then f=0 or f=chα depending on whether α<β or α=β. We also establish a stronger version of Hardy’s theorem and a Paley-Wiener theorem. These are generalisations of the corresponding results for rank one symmetric spaces of noncompact type.