The authors prove sharp \(L^ p\)-estimates for Fourier integral operators. Mainly the local theory is used. Also there are regularity results for solutions for the initial value problems for strictly hyperbolic partial differential equations \[ \begin{cases} Lu(x,t)=0, & t\neq 0,\\ \partial^ j_ tu|_{t=0}=f_ j(x), & 0\leq j\leq m-1,\end{cases} \] where \(L(x,t,D_{x,t})=D^ m_ t+\Sigma^ m_{j=1}P_ j(x,t,D_ x)D^{m-1}_ t\) is an operator of order \(m\) on a compact manifold \(X\) of dimension \(n\).