Global and local regularity of Fourier integral operators on weighted and unweighted spaces

TL;DR

This paper studies the global and local regularity of Fourier integral operators on weighted and unweighted L^p spaces, establishing new continuity and weighted norm inequalities under certain conditions.

Contribution

It introduces the analysis of Fourier integral operators' continuity on weighted L^p spaces with Muckenhoupt weights and derives weighted norm inequalities for operators with rough and smooth amplitudes.

Findings

Established weighted norm inequalities for Fourier integral operators.

Proved estimates for commutators with BMO functions.

Applied results to hyperbolic PDE solutions.

Abstract

We investigate the global continuity on $L^p$ spaces with $p\in [1,\infty]$ of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain non-degeneracy conditions. We initiate the investigation of the continuity of smooth and rough Fourier integral operators on weighted $L^{p}$ spaces, $L_{w}^p$ with $1< p < \infty$ and $w\in A_{p},$ (i.e. the Muckenhoput weights), and establish weighted norm inequalities for operators with rough and smooth amplitudes and phase functions satisfying a suitable rank condition. These results are then applied to prove weighted and unweighted estimates for the commutators of Fourier integral operators with functions of bounded mean oscillation BMO, then to some estimates on weighted Triebel-Lizorkin spaces, and finally to global unweighted and local weighted estimates for the solutions of the Cauchy problem for $m$-th and second order hyperbolic partial differential equations on $\mathbf{R}^n .$