Regularity of oscillatory integral operators

Abstract: 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.