A New Class of FBI Transforms and Applications

We introduce a class of FBI transforms using weight functions (which includes the subclass of Sjöstrand’s FBI transforms used by Christ in (Commun Partial Differ Equ 22(3–4):359–379, 1997)) that is well suited when dealing with ultradifferentiable functions (see Definition 2.3) and ultradistributions (see Definition 2.15) defined by weight functions in the sense of Braun, Meise and Taylor (BMT). We show how to characterize local regularity of BMT ultradistributions using this wider class of FBI transform and, as an application, we characterize the BMT vectors (see Definition 1.2) and prove a relation between BMT local regularity and BMT vectors.