# The Gabor wave front set in spaces of ultradifferentiable functions

## TL;DR

This paper introduces the $ ext{Gabor}$ wave front set within ultradifferentiable function spaces, establishing their equivalence and exploring applications to differential and pseudo-differential operators.

## Contribution

It defines the $ ext{Gabor}$ wave front set in ultradifferentiable spaces and proves its equivalence with the $ ext{omega}$-wave front set, extending microlocal analysis tools.

## Key findings

- The $ ext{Gabor}$ and $ ext{omega}$-wave front sets coincide.
- The framework applies to differential and pseudo-differential operators.
- Extension of wave front set concepts to ultradifferentiable function spaces.

## Abstract

Given a non-quasianalytic subadditive weight function $\omega$ we consider the weighted Schwartz space $\mathcal{S}_\omega$ and the short-time Fourier transform on $\mathcal{S}_\omega$, $\mathcal{S}'_\omega$ and on the related modulation spaces with exponential weights. In this setting we define the $\omega$-wave front set $WF'_\omega(u)$ and the Gabor $\omega$-wave front set $WF^G_\omega(u)$ of $u\in\mathcal{S}'_{\omega}$, and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.