The Gabor wave front set in spaces of ultradifferentiable functions

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 ${\mathcal{S}}_{\omega }$, ${\mathcal{S}}_{\omega }^{\prime }$ and on the related modulation spaces with exponential weights. In this setting we define the $\omega$-wave front set $W{F}_{\omega }^{\prime }(u)$ and the Gabor $\omega$-wave front set $W{F}_{\omega }^G(u)$ of $u\in {\mathcal{S}}_{\omega }^{\prime }$, and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.

35 pages