Global regularity properties for a class of Fourier integral operators

TL;DR

This paper establishes sufficient conditions for the global $L^p$-boundedness of a broad class of Fourier integral operators, extending local results to global contexts and applying these to hyperbolic PDEs.

Contribution

It provides a new criterion for global $L^p$-boundedness of Fourier integral operators, including many natural examples, and offers a method to derive global results from local ones.

Findings

Established sufficient conditions for global $L^p$-boundedness.

Provided a construction to extend local to global boundedness.

Applied results to hyperbolic PDE Cauchy problems.

Abstract

While the local $L^p$-boundedness of nondegeneral Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p(\mathbb R^n)$. In this paper we give a sufficient condition for the global $L^p$-boundedness for a class of Fourier integral operators which includes many natural examples. We also describe a construction that can be used to deduce global results from the local ones. An application is given to obtain global $L^p$-estimates for solutions to Cauchy problems for hyperbolic partial differential equations.