In the first two chapters of this book, the reader will find a complete
and systematic exposition of the theory of hyperfunctions on totally
real submanifolds of multidimensional complex space, in particular of
hyperfunction theory in real space. The book provides precise
definitions of the hypo-analytic wave-front set and of the
Fourier-Bros-Iagolnitzer transform of a hyperfunction. These are used to
prove a very general version of the famed Theorem of the Edge of the
Wedge. The last two chapters define the hyperfunction solutions on a
general (smooth) hypo-analytic manifold, of which particular examples
are the real analytic manifolds and the embedded CR manifolds. The main
results here are the invariance of the spaces of hyperfunction solutions
and the transversal smoothness of every hyperfunction solution. From
this follows the uniqueness of solutions in the Cauchy problem with
initial data on a maximally real submanifold, and the fact that the
support of any solution is the union of orbits of the structure.