A local-to-global boundedness argument and Fourier integral operators

TL;DR

This paper introduces a new criterion for extending local boundedness of integral operators to global boundedness, with applications to Fourier integral operators and their $L^p$-boundedness.

Contribution

It provides a novel local-to-global boundedness criterion and establishes new sufficient conditions for the global $L^p$-boundedness of Fourier integral operators.

Findings

Established a criterion for global boundedness from local boundedness.

Identified sufficient conditions for $L^p$-boundedness of Fourier integral operators.

Extended known local results to global contexts for integral operators.

Abstract

We give a criterion for the global boundedness of integral operators which are known to be locally bounded. As an application, we discuss the global $L^p$-boundedness for a class of Fourier integral operators. While the local $L^p$-boundedness of Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p({\mathbb R}^n)$. We give several natural sufficient conditions for them.