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An introduction to Sato’s hyperfunctions. Transl. from the Japanese by Mitsuo Morimoto. (English) Zbl 0811.46034
Translations of Mathematical Monographs. 129. Providence, RI: American Mathematical Society (AMS). vii, 273 p. (1993).
The book is a translation of the author’s book on hyperfunctions which appeared in Japanese in 1976. As the title indicates it gives an introduction to the theory of hyperfunctions. The first two chapters recall some facts about complex function theory and analytic functionals, especially the Köthe duality theorem. Hyperfunctions of one variable on a locally closed subset $$A$$ of $$\mathbb{R}$$ are defined in the next chapter as elements of $$H_ A^ 1 (W,{\mathcal O})= {\mathcal O}(W \setminus A)/{\mathcal O} (W)$$. The usual operations of analysis are defined and the hyperfunctions with compact support are identified with analytic functionals via the duality theorem. The necessary tools from sheaf theory and cohomology theory with coefficients in $${\mathcal O}$$ are treated in the next two chapters and in chapter six the Martineau duality theorem is proved. This gives in the $$n$$-dimensional case the identification of hyperfunctions with compact support with analytic functionals. The space of hyperfunctions in $$n$$ dimensions is defined, as in Satos original paper, as relative cohomology group with coefficients in $${\mathcal O}$$ and it is proved that the sheaf of hyperfunctions is flabby. The representation as classes of holomorphic functions is given by Čech-cohomology and the operator of taking boundary values for holomorphic functions is defined. The last two chapters give an introduction to the theory of microfunctions and a proof of Satos regularity theorem and of the edge- of-the-wedge theorem are given as applications. These two chapters can serve as an introduction to microlocal analysis and motivate the more complicated treatment in more recent books by e.g. M. Kashiwara and P. Schapira, ‘Sheaves on manifolds’, Berlin (1990; Zbl 0709.18001). Many concrete examples illustrate the theory and make the book a valuable text for beginners.

##### MSC:
 46F15 Hyperfunctions, analytic functionals 46F20 Distributions and ultradistributions as boundary values of analytic functions 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis 32A45 Hyperfunctions