Mapping properties of pseudodifferential and Fourier operators

TL;DR

This paper explores the spectral properties of Fourier operators, which are compositions of Fourier transforms with pseudodifferential operators, focusing on their compactness and mapping characteristics in function spaces.

Contribution

It advances the understanding of Fourier operators' spectral theory by analyzing their mapping properties and compactness in relation to pseudodifferential operators.

Findings

Fourier operators are shown to be compact in certain function spaces.

Spectral properties of these operators are characterized based on their mapping behavior.

The study extends previous work on Fourier transform mapping properties to include pseudodifferential operators.

Abstract

The composition of the Fourier transform in $\mathbb{R}^n$ with a suitable pseudodifferential operator is called a Fourier operator. It is compact in appropriate function spaces. The paper deals with its spectral theory. This is based on mapping properties of the Fourier transform as developed in a preceding paper and related assertions for pseudodifferential operators.