An excursion to multiplications and convolutions on modulation spaces

TL;DR

This paper introduces modulation spaces of ultradistributions, reviews their multiplication and convolution properties, and explores the Gabor product's role in phase-space formulations of nonlinear Schrödinger equations.

Contribution

It provides a self-contained introduction to modulation spaces and analyzes the boundedness of multiplications and convolutions, applying these to the Gabor product in nonlinear PDEs.

Findings

Boundedness results for multiplications and convolutions in modulation spaces.

Application of Gabor product to phase-space formulation of nonlinear Schrödinger equation.

Insight into the structure of ultradistribution modulation spaces.

Abstract

We give a self-contained introduction to (quasi-)Banach modulation spaces of ultradistributions, and review results on boundedness for multiplications and convolutions for elements in such spaces. Furthermore, we use these results to study the Gabor product. As an example, we show how it appears in a phase-space formulation of the nonlinear cubic Schr\"odinger equation.