The Gabor wave front set

We define the Gabor wave front set $WF_G(u)$ of a tempered distribution $u$ in terms of rapid decay of its Gabor coefficients in a conic subset of the phase space. We show the inclusion $$WF_G(a^w(x,D) u) \subseteq WF_G(u), u \in \mathscr S'(\mathbb R^d), a \in S_{0,0}^0,$$ where $S_{0,0}^0$ denotes the H\"ormander symbol class of order zero and parameter values zero. We compare our definition with other definitions in the literature, namely the classical and the global wave front sets of H\"ormander, and the $\cS$-wave front set of Coriasco and Maniccia. In particular, we prove that the Gabor wave front set and the global wave front set of H\"ormander coincide.