Notes on Regularity of Fourier integral operators with symbol in $S_{0,\delta }^m$

Abstract

Let $T_{a,\phi }$ be a Fourier integral operator defined with $a\in S_{0,\delta }^m(0\le \delta <1)$ and $\phi \in {\Phi }^2$ satisfying the strong non-degenerate condition. We demonstrate that when the order satisfies $$m\le -\frac{n}{2}-\frac{n}{p}\delta +\frac{n}{p},$$ the operator $T_{a,\phi }$ becomes bounded on $L^p({\mathbb{R}}^n)$ for $2<p<\infty$ and maps $L^{\infty }({\mathbb{R}}^n)$ to $BMO({\mathbb{R}}^n)$ when $p=\infty$. Furthermore, the derived bound on $m$ is sharp for $L^p$ estimates in the case $\delta =0$, and for $(L^{\infty },BMO)$ when $0\le \delta <1$.