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

Abstract

Let $T_{a,\varphi}$ be a Fourier integral operator defined with $a\in S^{m}_{0,\delta}(0\leq\delta<1)$ and $\varphi\in \Phi^{2}$ satisfying the strong non-degenerate condition. We demonstrate that when the order satisfies $$m\leq-\frac{n}{2}-\frac{n}{p}\delta+\frac{n}{p},$$ the operator $T_{a,\varphi}$ 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 $0\leq\delta<1$.