Foundation of symbol theory for analytic pseudodifferential operators, I

Abstract

A new symbol theory for pseudodifferential operators in the complex analytic category is given. This theory provides a cohomological foundation of symbolic calculus.

The equivalence of pseudodifferential operators and their symbols via Čech-Dolbeault cohomology

“…holds and the surjectivity of ι z * has been proved. 4 The equivalence of E R X and S ∞ /N ∞ In this section X is assumed to be a complex vector space of dimension n. We identify X × X with T X by the map…”

“…For a global case, however, such coverings are hard to be found when we manipulate the cohomological expression of E R X . In recent years the first problem has been solved by Aoki, Honda and Yamazaki in [4]. They introduce a new space with one apparent parameter and construct the sheaf morphism on it.…”

Sato hyperfunctions via relative Dolbeault cohomology

“…In the course of our study, we introduce several fundamental methods and ideas, which become important ingredients for further works. In fact this kind of cohomology theory serves to establish the foundation of various topics in algebraic analysis such as the theory of Laplace hyperfunctions ( [21], [14] and [15]) and the symbol theory of analytic pseudodifferential operators ( [2], see also [3]). Recently, K. Umeta is working on the former in [38] and D. Komori is doing for the latter in [22], where our fundamental methods introduced in this paper play essential roles in their arguments.…”