First, we give a new proof for the Beals commutator criterion for non-magnetic Weyl pseudo-differential operators based on classical Gabor tight frames. Second, by introducing a modified ‘magnetic’ Gabor tight frame, we naturally derive the magnetic analogue of the Beals criterion originally considered by Iftimie-Măntoiu-Purice.
Cornean, D. H., Helffer, B., & Purice, R. (2018). A Beals criterion for magnetic pseudo-differential operators proved with magnetic Gabor frames. Communications in Partial Differential Equations, 43(8), 1196-1204. https://doi.org/10.1080/03605302.2018.1499777