  In this paper, we consider the Cauchy problem for the nonlinear Klein-Gordon
equation whose nonlinearity is $|u|^{k}u$ in the modulation space, where $k$ is
not an integer. Our method can be applied to other equations whose nonlinear
parts have regularity estimates. We also study the global solution with small
initial value for the Klein-Gordon-Hartree equation. By this we can show some
advantages of modulation spaces both in high and low regularity cases.
