SG-pseudodifferential operators and Gelfand-Shilov spaces

In this paper, we study a class of pseudo-differential operators of SG type in the functional setting of the Gelfand-Shilov spaces $S^{\theta}_{\theta}(R^n), \theta >1$. As an application we prove a result of hypoellipticity in the same classes. In the last of the paper, we define a notion of wave front set for tempered ultradistributions which allows to describe both the local regularity and the behaviour at infinity of the elements of the dual space $(S^{\theta}_{\theta})'(R^n)$.