On the global boundedness of Fourier integral operators

TL;DR

This paper establishes sharp continuity results for a class of Fourier integral operators on modulation spaces, revealing minimal derivative loss and decay, and provides examples of unboundedness on L^p spaces.

Contribution

It introduces a global boundedness analysis of Fourier integral operators on modulation spaces with precise loss estimates, extending previous local results.

Findings

Proves sharp continuity of Fourier integral operators on $M^p$ spaces.

Identifies minimal derivative loss as $d|1/2-1/p|$.

Provides examples of unboundedness on $L^p$ spaces.

Abstract

We consider a class of Fourier integral operators, globally defined on $\mathbb{R}^{d}$, with symbols and phases satisfying product type estimates (the so-called $SG$ or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces $M^p$. The minimal loss of derivatives is shown to be $d|1/2-1/p|$. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on $L^p$ spaces are presented.