Global wave front sets in ultradifferentiable classes (Q2114963)

The paper concerns global wave front sets of the type of those first considered by \textit{L. Hörmander} [Lect. Notes Math. 1495, 118--160 (1991; Zbl 0761.35004)] and subsequently in terms of short-time Fourier transform by \textit{L. Rodino} and \textit{P. Wahlberg} [Monatsh. Math. 173, No. 4, 625--655 (2014; Zbl 1366.42030)], under the name of Gabor wave front sets. Here attention is fixed on the global ultra-differentiable classes, defined as for example in [\textit{G. Björck}, Ark. Mat. 6, 351--407 (1966; Zbl 0166.36501)]. The corresponding microlocal regularity was already considered by \textit{C. Boiti} et al. [Monatsh. Math. 188, No. 2, 199--246 (2019; Zbl 1407.35003)]. In the present paper, taking advantage from the pseudo-differential calculus of \textit{V. Asensio} and \textit{D. Jornet} [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113, No. 4, 3477--3512 (2019; Zbl 1439.46028)], the authors give an equivalent definition and prove in this frame a result of micro-elliptic regularity. Note that the global ultra-differentiable setting involves severe technical difficulties and, on the other hand, it is very general including the Gelfand-Shilov spaces as a particular case.