Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$e^{it\left( {m^2 I + \left| \Delta \right|} \right)^{1/2} }$$ \end{document} (m ≠ 0) of the Klein-Gordon equation is bounded on the modulation space Mp,qs (ℝn) for all 0 < p, q ⩽ ∞ and s ∈ ℝ. Similarly, we prove that the wave semi-group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$e^{it\left| \Delta \right|^{1/2} }$$ \end{document} is bounded on the Hardy type modulation spaces µp,qs (ℝn) for all 0 < p, q ⩽ ∞, and s ∈ ℝ. All the bounds have an asymptotic factor tn|1/p−1/2| as t goes to the infinity. These results extend some known results for the case of p ⩾ 1. Also, some applications for the Cauchy problems related to the semi-group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$e^{it\left( {m^2 I + \left| \Delta \right|} \right)^{1/2} }$$ \end{document} are obtained. Finally we discuss the optimum of the factor tn|1/p−1/2| and raise some unsolved problems.