Edge of the Wedge Theorem for Tempered Ultrahyperfunctions

Tempered ultra-hyperfunctions do not have the same type of localization properties as Schwartz distributions or Sato hyperfunctions; but the localization properties seem to play an important role in the proofs of the various versions of the edge of the wedge theorem. Thus, for tempered ultra hyper-functions, one finds a global form of this result in the literature, but no local version. In this paper we propose and prove a formulation of the edge of the wedge theorem for tempered ultra-hyperfunctions, both in global and local form. We explain our strategy first for the one variable case. We argue that in view of the cohomological definition of hyperfunctions and ultra-hyperfunctions, the global form of the edge of the wedge theorem is not surprising at all.