In this paper, we investigate a Fourier integral operator \(T_{\phi ,a}\) which is defined by \(\begin{aligned}&T_{\phi ,a} f(x)=\int _{\mathbb {R}^n}e^{i\phi (x,\xi )}a(x,\xi )\widehat{f}(\xi )d\xi . \end{aligned}\) We assume that the phase \(\phi \) belongs to the class \(\Phi ^2\) , satisfying the strong non-degeneracy condition and define \(\begin{aligned} m_{n,p,\rho ,\delta }=(\rho -n)(\frac{1}{2}-\frac{1}{p})+\min \{0,\frac{n}{p}(\rho -\delta )\}. \end{aligned}\) Firstly, for \(0\le \rho ,\delta \le 1,s>0,2\le p\le \infty ,1\le q\le \infty \) and the symbol a belongs to the Hörmander class \(S^{m_{n,p,\rho ,\delta }}_{\rho ,\delta }\) , we demonstrate that \(T_{\phi ,a}\) is bounded on the Besov space \(B^s_{p,q}\) . Consequently, as a corollary, if \(0\le \rho ,\delta \le 1,s\ge 0,2\le p\le \infty ,0<q\le \infty \) and the symbol a belongs to the Hörmander class \(S^{m}_{\rho ,\delta }\) with \(m<m_{n,p,\rho ,\delta }\) , then \(T_{\phi ,a}\) is bounded on the Triebel-Lizorkin space \(F^s_{p,q}\) . Lastly, when \(0\le \rho \le 1,0\le \delta<1,2\le q\le p<\infty \) , s is a positive integer and the symbol a belongs to the Hörmander class \(S^{m_{n,p,\rho ,\delta }}_{\rho ,\delta }\) , we prove that \(T_{\phi ,a}\) is bounded on the Triebel-Lizorkin space \(F^s_{p,q}\) . These results are extensions of the respective theorems.
