Magnetic Pseudo-differential Operators with H\"ormander Symbols Dominated by Tempered Weights

TL;DR

This paper extends the matrix representation of magnetic pseudo-differential operators to more general symbols dominated by tempered weights, providing new insights into their calculus and Schatten-class properties.

Contribution

It introduces a generalized matrix representation for magnetic pseudo-differential operators with tempered weight symbols, expanding previous frameworks.

Findings

Extended matrix representation to asymmetrical quantizations

Derived new symbol calculus results for tempered weight symbols

Established Schatten-class properties for these operators

Abstract

We extend the matrix representation of magnetic pseudo-differential operators in a tight Gabor frame from [arXiv:1804.05220, arXiv:2212.12229] to asymmetrical quantizations and smooth symbols dominated by a tempered weight (and not just decay/growth properties in the momentum variables). This leads to new results regarding the symbol calculus of such operators and their Schatten-class properties.