A ROLE OF SYMBOLS OF MINIMUM TYPE IN EXPONENTIAL CALCULUS
Chang Hoon LEE · East Asian Mathematical Journal · 2016
Abstract. We introduce formal symbols of product type and of minimumtype and show that the formal power series representation for e p is a formalsymbol of product type, where p is a formal symbol of minimum type. 1. IntroductionPseudodi erential operators are essential for studying the theory of partialdi erential equations and itself have been studied in various respects. A pseudo-di erential operator can be represented as an equivalent class of some functionde ned on the cotangent bundle. We call this function a symbol of pseudodi er-ential operator. We can represent the operations of addition, scalar multiplica-tion, composition, formal adjoint and change of variables of pseudodi erentialoperators by using these symbols and so can treat pseudodi erential operatorsconcretely. Symbolic calculus means calculating pseudodi erential operatorsby using symbols. Exponential calculus means symbolic calculus on pseudodif-ferential operators expressed as symbols of exponential functions. A symbol ofexponential function means that the symbol can be expressed as a form of expo-nential function. We can analyze (pseudo) di erential operators of in nite orderor (pseudo) di erential equations of in nite order by studying the pseudodi er-ential operators expressed as symbols of exponential functions. In particular,Sato, M., T. Kawai and M. Kashiwara (ref. [16]) won epoch-making results inthe theory of transformation of the system of linear partial (pseudo) di eren-tial equations by using pseudodi erential operators of in nite order. T. Aokiaccomplished exponential calculus of analytic pseudodi erential operators( ref.[1] { [8]). The author considers the case of a kind of the direct product structure(ref. [12] { [15]). That is, we introduce and study calculus of analytic pseudo-di erential operators of product type and generalize the theory of T. Aoki insome sense. This study is deeply connected with exponential calculus of posi-tive de nite operators of in nite order which have deep relation to the energymethod in the hyperfunction theory(ref. [9] { [11]). In this article, we introduce