Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group e^it（m^2│△│）^1/2 （m≠ 0） of the Klein-Gordon equation is bounded on the modulation space M^8p,q（R^n） for all 0 〈 p, q ≤∞ and s ∈ R. Similarly, we prove that the wave semi-group e^it│△│^1/2 is bounded on the Hardy type modulation spaces μ^εp,q（R^n） for all 0 〈 p, q ≤ ∞, and s ∈R. 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 eit（m^2I＋│△│）1/2 are obtained. Finally we discuss the optimum of the factor t^n│1/p-1/2│ and raise some unsolved problems.