# Convolutions for localization operators

## TL;DR

This paper explores the mathematical framework of localization operators through quantum harmonic analysis, utilizing convolution, Fourier transforms, and uncertainty principles to extend existing results and establish new inequalities.

## Contribution

It introduces a unified framework linking quantum harmonic analysis with localization operators, extending prior results using noncommutative Tauberian theorems and Fourier-Wigner transforms.

## Key findings

- Established a sharp Hausdorff-Young inequality for the Fourier-Wigner transform.
- Extended results on localization operators using noncommutative Tauberian theorems.
- Linked quantum harmonic analysis with the theory of Banach modules and Arveson spectrum.

## Abstract

Quantum harmonic analysis on phase space is shown to be linked with localization operators. The convolution between operators and the convolution between a function and an operator provide a conceptual framework for the theory of localization operators which is complemented by an appropriate Fourier transform, the Fourier-Wigner transform. We use Lieb's uncertainty principle to establish a sharp Hausdorff-Young inequality for the Fourier-Wigner transform. Noncommutative Tauberian theorems due to Werner allow us to extend results of Bayer and Gr\"ochenig on localization operators. Furthermore we show that the Arveson spectrum and the theory of Banach modules provide the abstract setting of quantum harmonic analysis.