The Equivalence of Pseudodifferential Operators and Their Symbols via Čech–Dolbeault Cohomology

Abstract

In this paper we construct the sheaf morphism from the sheaf of pseudodifferential operators to its symbol class. Since it is hard to construct the morphism directly, we realize it with two original ideas as follows. Firstly, to calculate cohomologies we use the theory of Čech–Dolbeault cohomology introduced by Honda, Izawa and Suwa (J. Math. Soc. Japan 75 (2023), 229–290). Secondly, we construct a new symbol class, which is called the symbols of $C^{\infty }$-type. These ideas enable us to construct the sheaf morphism, which is actually an isomorphism of sheaves.