Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group $e^{itleft( {m^2 I + left| Delta right|} right)^{1/2} }$ (m ?? 0) of the Klein-Gordon equation is bounded on the modulation space M _p,q ^s (? ⁿ ) for all 0 < p, q ? ?? and s ?? ?. Similarly, we prove that the wave semi-group $e^{itleft| Delta right|^{1/2} }$ is bounded on the Hardy type modulation spaces ?? _p,q ^s (? ⁿ ) for all 0 < p, q ? ??, and s ?? ?. All the bounds have an asymptotic factor t ^n|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 $e^{itleft( {m^2 I + left| Delta right|} right)^{1/2} }$ are obtained. Finally we discuss the optimum of the factor t ^n|1/p?1/2| and raise some unsolved problems.