Notes on Regularity of Fourier integral operators with symbol in $S^{m}_{0,δ}$
Guangqing Wang, Suixin He
TL;DR
This work establishes $L^{p}$-regularity for Fourier integral operators with symbols in $S^{m}_{0,\delta}$ and strongly non-degenerate phases $\varphi\in\Phi^{2}$, extending endpoint results beyond the classical $L^{2}$ theory. By a dyadic microlocal decomposition and precise control of localized amplitudes $a_j^{\nu}$, the authors prove that if $m \le -\frac{n}{2}-\frac{n}{p}\delta+\frac{n}{p}$, then $T_{a,\varphi}$ is bounded on $L^{p}$ for $2<p<\infty$ and maps $L^{\infty}$ to $BMO$ at $p=\infty$, with sharpness at the endpoint cases. The proof adapts pseudo-differential operator techniques to the FIO setting, circumventing limitations from lack of off-diagonal kernel regularity when $\varrho=0$, and suggests potential extensions to weighted and Morrey-type spaces for FIOs. Overall, the results unify and sharpen endpoint regularity for a broad class of FIOs and highlight the role of the $S^{m}_{0,\delta}$ symbol class in achieving optimal $L^{p}$-bounds.
Abstract
Let $T_{a,\varphi}$ be a Fourier integral operator defined with $a\in S^{m}_{0,δ}(0\leqδ<1)$ and $\varphi\in Φ^{2}$ satisfying the strong non-degenerate condition. We demonstrate that when the order satisfies $$m\leq-\frac{n}{2}-\frac{n}{p}δ+\frac{n}{p},$$ 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 $δ=0$, and for $(L^{\infty},BMO)$ when $0\leqδ<1$.