On a class of $h$-Fourier integral operators

TL;DR

This paper investigates the boundedness and compactness properties of a specific class of $h$-Fourier integral operators on $L^2$ spaces, establishing conditions based on the amplitude's weight.

Contribution

It provides new criteria linking the amplitude's weight to the $L^2$-boundedness and compactness of $h$-Fourier integral operators.

Findings

Operators are bounded if the amplitude's weight is bounded.

Operators are compact if the amplitude's weight tends to zero.

The results clarify the relationship between amplitude decay and operator properties.

Abstract

In this paper, we study the $L^{2}$-boundedness and $L^{2}$-compactness of a class of $h$-Fourier integral operators. These operators are bounded (respectively compact) if the weight of the amplitude is bounded (respectively tends to $0)$.