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The Gabor wave front set in spaces of ultradifferentiable functions (1706.08413v1)
Published 26 Jun 2017 in math.FA
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 $WFG_\omega(u)$ 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.