Regularity of oscillatory integral operators

TL;DR

This paper proves the boundedness of oscillatory integral operators on advanced function spaces with broad parameter ranges, extending understanding of their regularity and providing conditions for more challenging amplitude classes.

Contribution

It establishes new boundedness results for oscillatory integral operators on Besov-Lipschitz and Triebel-Lizorkin spaces with general amplitude and phase function classes.

Findings

Global boundedness on Besov-Lipschitz and Triebel-Lizorkin spaces.

Boundedness conditions for amplitudes in $S^m_{1,1}$ class.

Applicable to a wide range of parameters and phase functions.

Abstract

In this paper, we establish the global boundedness of oscillatory integral operators on Besov-Lipschitz and Triebel-Lizorkin spaces, with amplitudes in general $S^m_{\rho,\delta}(\mathbb{R}^n)$-classes and non-degenerate phase functions in the class $\textart F^k$. Our results hold for a wide range of parameters $0\leq\rho\leq1$, $0\leq\delta<1$, $0<p\leq\infty$, $0<q\leq\infty$ and $k>0$. We also provide a sufficient condition for the boundedness of operators with amplitudes in the forbidden class $S^m_{1,1}(\mathbb{R}^n)$ in Triebel-Lizorkin spaces.