Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

TL;DR

This paper extends the theory of Banach algebras of pseudodifferential operators to quasi-Banach algebras of Fourier integral operators, called generalized metaplectic operators, with applications to evolution equations like the Schrödinger equation.

Contribution

It introduces quasi-Banach algebras of symbol classes for Fourier integral operators, generalizing Wiener properties to a broader class including pseudodifferential operators.

Findings

Established quasi-Banach algebra structures for generalized metaplectic operators.

Extended Wiener algebra concepts to Fourier integral operators.

Applied the framework to evolution equations such as the Schrödinger equation.

Abstract

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gr\"ochenig and Rzeszotnik in [24] to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators [11], which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schr\"odinger equation with bounded perturbations, cf. [7].