Construction of Local Elementary Solutions for Linear Partial Differential Operators with Real Analytic Coefficients (I)

In this paper we construct local elementary solutions for linear differential operators P(x9 Dx) whose principal symbols are real and of simple characteristics and investigate their regularity properties using Sato's theory of the sheaf ^ (Sato DC^CAl). Throughout this paper we assume that the coefficients of differential operators are analytic. In §1 we prepare some theorems which extend the classical existence theorem of Cauchy-Kowalevsky in complex domain to cases of singular initial data. (Cf. Hamada [1].) In §2 we employ the results of §1 to construct local elementary solutions for Cauchy problems for (/-) hyperbolic operators. As an application of the method employed there, we construct a singular solution u(x) of P(x, Dx)u = Q whose singular support is "very small". (See Theorem 2.8 for the precise meaning of this statement.) In §3 we construct a local elementary solution for a linear partial differential operator of real principal symbols with simple characteristics and decide where its singularities locate to conclude that the result of §2 on the existence of singular solutions with small singularities is the best possible one. Throughout this paper we denote by M an ra-dimensional real analytic manifold which we identify with a domain in R containing the origin.