chapter  Chapter 1
90 Pages

Hyperfunctions

ByGoro Kato, Daniele C. Struppa

In this chapter we will introduce the first object necessary for the study of Microlocal Analysis, as developed by the Japanese school, namely the sheaf of hyperfunctions. As it is known since Schwartz’s introduction of the notion of distribution [210], there is no hope of dealing with the subtle issues posed by the theory of partial differential equations unless one resorts to some kind of generalized functions. In the case of Schwartz, the notion which was developed is that of distribution, i.e. of a continuous linear functional on the topological space of infinitely differentiable functions with compact support. Other different spaces have been studied and developed for a variety of different reasons, see e.g. [14], [17], [37], but the choice of the Japanese school led by M. Sato has been to employ a space of functions which can be defined on any analytic manifold and which somehow generalizes the space of distributions itself.