Hyperfunctions on hypo-analytic manifolds.

*(English)*Zbl 0817.32001
Annals of Mathematics Studies. 136. Princeton, NJ: Princeton Univ. Press. xx, 377 p. $ 65.00; £ 50.00/hbk; $ 29.95; £ 25.00/pbk (1994).

An analogy of hyperfunction theory is developped on hypoanalytic manifolds, of which the notion has been introduced as a generalization of CR manifolds by the second author in the book “Hypo-analytic structures: Local theory”, Princeton (1992; Zbl 0787.35003). In the present book the necessary materials related with hypoanalytic structure are repeated so that the readers need not go back to the former book. Here are the contents:

Preface: A fairly detailed summary of the book.

Chapter 0 (not explicitly nominated): Contains a short summary of sheaf theory and cohomology theory.

Chapter 1 (Hyperfunctions in a maximal hypoanalytic structure): Gives a review on analytic functionals and related tools in several complex variables, including integral representations. Then hyperfunctions on a maximally hypoanalytic manifold are introduced by patching the analytic functionals with compact supports, following the recipe of Martineau- Schapira.

Chapter 2 (Microlocal theory of hyperfunctions on a maximally real submanifold of a complex space): FBI transform is introduced to study the analytic wave-front sets of hyperfunctions. The relation with the representability of hyperfunctions via boundary values of hypoanalytic functions is discussed, and analogues of the edge of the wedge theorems are given.

Chapter 3 (Hyperfunction solutions in a hypoanalytic manifold): The case where the hypoanalytic structure is not maximal is studied in detail. The notion of “solutions” is introduced as cohomology groups of the relative Dolbeault-De Rham complex with mixed coefficients, hyperfunctions-smooth functions. Their coordinate invariance is established.

Chapter 4 (Transversal smoothness of hyperfunction solutions): Hyperfunctions with smooth parameters are discussed. The results prepared hitherto are applied to the classical problem of solvability or regularity of solutions of systems of vector fields.

Preface: A fairly detailed summary of the book.

Chapter 0 (not explicitly nominated): Contains a short summary of sheaf theory and cohomology theory.

Chapter 1 (Hyperfunctions in a maximal hypoanalytic structure): Gives a review on analytic functionals and related tools in several complex variables, including integral representations. Then hyperfunctions on a maximally hypoanalytic manifold are introduced by patching the analytic functionals with compact supports, following the recipe of Martineau- Schapira.

Chapter 2 (Microlocal theory of hyperfunctions on a maximally real submanifold of a complex space): FBI transform is introduced to study the analytic wave-front sets of hyperfunctions. The relation with the representability of hyperfunctions via boundary values of hypoanalytic functions is discussed, and analogues of the edge of the wedge theorems are given.

Chapter 3 (Hyperfunction solutions in a hypoanalytic manifold): The case where the hypoanalytic structure is not maximal is studied in detail. The notion of “solutions” is introduced as cohomology groups of the relative Dolbeault-De Rham complex with mixed coefficients, hyperfunctions-smooth functions. Their coordinate invariance is established.

Chapter 4 (Transversal smoothness of hyperfunction solutions): Hyperfunctions with smooth parameters are discussed. The results prepared hitherto are applied to the classical problem of solvability or regularity of solutions of systems of vector fields.

Reviewer: A.Kaneko (Komaba)

##### MSC:

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

58J15 | Relations of PDEs on manifolds with hyperfunctions |

32V99 | CR manifolds |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

32A45 | Hyperfunctions |

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |