Almost Diagonalization Theorem and Global Wave Front Sets in Ultradifferentiable Classes

[EN] The main aim of this paper is to prove that the wave front set of aw (x, D)u, i.e. the action of the Weyl operator with symbol a on u, is contained in the wave front set of u and in the conic support of a in spaces of ω-tempered ultradistributions in the Beurling setting for adequate symbols of ultradifferentiable type. These symbols are not restricted to have order zero. To do so, we prove an almost diagonalization the- orem on Weyl operators. Furthermore, an almost diagonalization theorem involving time-frequency analysis leads to additional applications, such as invertibility of pseu- dodifferential operators or boundedness of them in modulation spaces with exponential growth. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The author was supported by the project GV PROMETEU/2021/070. Asensio López, V. (2024). Almost diagonalization theorem and global wave front sets in ultradifferentiable classes. Banach Journal of Mathematical Analysis. 18(4). https://doi.org/10.1007/s43037-024-00374-6