Boundedness of Fourier integral operators on classical function spaces

We investigate the global boundedness of Fourier integral operators with amplitudes in the general Hörmander classes $S_{\rho ,\delta }^m({\mathbb{R}}^n)$, $\rho ,\delta \in [0,1]$ and non-degenerate phase functions of arbitrary rank $\kappa \in \{0,1,\dots ,n-1\}$ on Besov-Lipschitz $B_{p,q}^s({\mathbb{R}}^n)$ and Triebel-Lizorkin $F_{p,q}^s({\mathbb{R}}^n)$ of order $s$ and $0<p\le \mathrm{\infty }$, $0<q\le \mathrm{\infty }$. The results that are obtained are all up to the end-point and sharp and are also applied to the regularity of Klein-Gordon-type oscillatory integrals in the aforementioned function spaces.