Estimates for generalized wave‐type Fourier multipliers on modulation spaces and applications

We consider the boundedness of the Fourier multipliers σt(ξ)=sin(tϕ(h(ξ)))ϕ(h(ξ))$\sigma _{t}(\xi ) = \frac{\sin (t\phi (h(\xi )))}{\phi (h(\xi ))}$ on modulation spaces, where h$h$ defined on Rd${\mathbb {R}}^{d}$ is a C∞(Rd∖0)$C^{\infty }({\mathbb {R}}^{d}\setminus \left\lbrace 0\right\rbrace )$ positive homogeneous function with degree λ>0$\lambda >0$ and ϕ:R+→R+$\phi : {\mathbb {R}}^{+} \rightarrow {\mathbb {R}}^{+}$ is a smooth function satisfying some decay conditions. We prove the boundedness of this kind of Fourier multipliers and obtain its asymptotic estimates as t$t$ goes to infinity. We remote the restriction λ>1/2$\lambda >1/2$ in Deng, Ding, and Sun's result in [Nonlinear Anal. 85 (2013), 78–92], and we consider the more general form of this multiplier. As applications, we obtain the grow‐up rate of the solutions for the Cauchy problems for the generalized wave and Klein–Gordon equations, and we obtain the local well‐posedness of nonlinear wave and Klein–Gordon equations in modulation spaces.