The equivalence of pseudodifferential operators and their symbols via \v{C}ech-Dolbeault cohomology

Abstract

In this paper we construct the sheaf morphism from the sheaf of pseudodifferential operators to its symbol class. Since the map is hard to construct directly, we realize it with two original ideas as follows. First, to calculate cohomologies we use the theory of \v{C}ech-Dolbeault cohomology introduced by Honda, Izawa and Suwa. 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.