Fourier integral operators with weighted symbols

TL;DR

This paper surveys Fourier integral operators with weighted symbols, detailing their boundedness and compactness properties in L^2 spaces based on the behavior of the symbol's weight.

Contribution

It provides a comprehensive overview of the boundedness and compactness criteria for Fourier integral operators with tempered weighted symbols.

Findings

Operators are bounded in L^2 if the symbol's weight is bounded.

Operators are compact in L^2 if the symbol's weight tends to zero.

The paper summarizes key properties of these operators in relation to symbol weights.

Abstract

The paper contains a survey of a class of Fourier integral operators defined by symbols with tempered weight. These operators are bounded (respectively compact) in $L^2$ if the weight of the amplitude is bounded (respectively tends to $0$).