Regularity of oscillatory integral operators

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