Boundedness of Fourier integral operators on classical function spaces

TL;DR

This paper studies the boundedness of Fourier integral operators with general amplitudes and phase functions on various classical function spaces, providing sharp, endpoint results and applications to Klein-Gordon oscillatory integrals.

Contribution

It establishes sharp, endpoint boundedness results for Fourier integral operators on Besov-Lipschitz and Triebel-Lizorkin spaces, extending previous knowledge to more general classes.

Findings

Boundedness results are sharp and include endpoint cases.

Applications to regularity of Klein-Gordon-type oscillatory integrals.

Results cover a wide range of amplitude classes and phase function ranks.

Abstract

We investigate the global boundedness of Fourier integral operators with amplitudes in the general H\"ormander classes $S^{m}_{\rho, \delta}(\mathbb{R}^n)$, $\rho, \delta\in [0,1]$ and non-degenerate phase functions of arbitrary rank $\kappa\in \{0,1,\dots, n-1\}$ on Besov-Lipschitz $B^{s}_{p,q}(\mathbb{R}^n)$ and Triebel-Lizorkin $F^{s}_{p,q}(\mathbb{R}^n)$ of order $s$ and $0<p\leq\infty$, $0<q\leq\infty$. The results that are obtained are all up to the end-point and sharp and are also applied to the regularity of Klein-Gordon-type oscillatory integrals in the aforementioned function spaces.