  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.
