  In this paper, we establish the global boundedness of oscillatory integral
operators on Besov-Lipschitz and Triebel-Lizorkin spaces, with amplitudes in
general $S^m_{\rho,\delta}(\mathbb{R}^n)$-classes and non-degenerate phase
functions in the class $\textart F^k$. Our results hold for a wide range of
parameters $0\leq\rho\leq1$, $0\leq\delta<1$, $0<p\leq\infty$, $0<q\leq\infty$
and $k>0$. We also provide a sufficient condition for the boundedness of
operators with amplitudes in the forbidden class $S^m_{1,1}(\mathbb{R}^n)$ in
Triebel-Lizorkin spaces.
