The FBI transforms and their use in microlocal analysis

Using a more general class of FBI transforms introduced by S. Berhanu and J. Hounie in [12] we completely characterize regularity and microregularity in Denjoy-Carleman (non quasi analytic) classes, which includes the Gevrey classes and M. Christ version of the FBI transform defined in [22] as examples. We also exhibit a result on microlocal regularity for solutions of first order partial differential equations in these classes, that do not seem possible to prove using the classical FBI transform.