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.
Categories: math.FA