Asymptotic estimates for Klein-Gordon equation on α-modulation space

Recently, asymptotic estimates for the unimodular Fourier multipliers e i μ ( D ) have been studied for the function α -modulation space. In this paper, using the almost orthogonality of projections and some techniques on oscillating integrals, we obtain asymptotic estimates for the unimodular Fourier multiplier e i t ( I − Δ ) β 2 on the α -modulation space. For an application, we give the asymptotic estimate of the solution for the Klein-Gordon equation with initial data in a α -modulation space. We also obtain a quantitative form about the solution to the nonlinear Klein-Gordon equation.