Microlocal properties of Shubin pseudodifferential and localization operators (Q262296)

The authors study the action of a global pseudodifferential operator \(P\) by \textit{M. A. Shubin} [Pseudodifferential operators and spectral theory. Transl. from the Russian by Stig I. Andersson. 2nd ed. Berlin: Springer. xii, 288 p. (2001; Zbl 0980.35180)] on the Gabor wave front set \(WF_G(u)\) of \(u\in S'(\mathbb{R}^n)\), defined as by \textit{L. Hörmander} [Microlocal analysis and applications, Lect. 2nd Sess. CIME, Montecatini Terme/Italy 1989, Lect. Notes Math. 1495, 118--160 (1991; Zbl 0761.35004)], \textit{L. Rodino} and \textit{P. Wahlberg} [Monatsh. Math. 173, No. 4, 625--655 (2014; Zbl 1366.42030)]. The novelty of the present paper is the anti-Wick representation of \(P\) and the use of a weighted version of the Gabor wave front set. Precise results of micro-locality and micro-hypoellipticity are given in this frame.