Systems of Microdifferential Equations

The purpose of the chapter is to present to the reader a series of applications of the theory developed so far to the study of systems of microdifferential equations. The plan of the chapter is as follows. In this chapter, the authors state and prove Sato’s Fundamental Theorem, for the case of linear differential operators of finite order. It discusses some special cases of differential equations, namely the wave equation and hyperbolic equations, which provide a first hint of what is the microlocal aspect of the study of differential equations. This chapter is devoted to the detailed proof of the so called weak form of Sato’s Fundamental Theorem, which states the invertibility for any finite order differential operator in the ring of microlocal operators. In this chapter, the author reviews some basic notions from contact geometry, they define the notion of contact transformations, and will finally microlocalize a notion.