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

TL;DR

This paper establishes boundedness conditions for Fourier integral operators with symbols in $S^{m}_{0, ext{delta}}$, showing sharp bounds for $L^p$ and $BMO$ spaces depending on the order $m$ and parameters $p$ and $ ext{delta}$.

Contribution

It provides new sharp bounds for the regularity of Fourier integral operators with symbols in $S^{m}_{0, ext{delta}}$, extending previous results to a broader class of operators.

Findings

Operators are bounded on $L^p$ for certain $m$, $p$, and $ ext{delta}$.

Boundedness from $L^{ ext{infinity}}$ to $BMO$ is established for specific parameters.

The bounds on $m$ are shown to be sharp in particular cases.

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 $0\leq\delta<1$.