Cauchy problem for hyperbolic \(\mathcal{D}\)-modules with regular singularities
The aim of the paper is to generalize classical results concerning the hyperbolic Cauchy problem for a linear partial differential operator with analytic coefficients [see \textit{M. Kashiwara} and \textit{T. Oshima}, Ann. Math., II. Ser. 106, 145-200 (1977; Zbl 0358.35073)] to the case of general systems of linear partial differential equations. More precisely, let \(\mathcal M\) be a coherent \(\mathcal D\)-module on the complexification of a real analytic manifold \(M.\) Assume that the characteristic variety of \(\mathcal M\) is hyperbolic with respect to a submanifold \(N \subset M\) and \(\mathcal M\) has regular singularities along a regular involutive submanifold of real type. The authors prove that under the above assumptions the Cauchy problem is well posed in the space of distributions. Thus, they generalize results of \textit{M. Kashiwara} and \textit{P. Schapira} [`Sheaves on manifolds', Springer, Berlin (1990; Zbl 0709.18001)], and others obtained in the case of Sato's hyperfunctions. In addition, they recover a classical result in the case when \(\mathcal M\) is induced by a single differential operator or by a normal square system whose characteristics have constant multiplicities since the above hypotheses correspond to Levi conditions [\textit{L. Hörmander}, `The analysis of linear partial differential operators. II', Springer (1983; Zbl 0521.35002)].
scientific article; zbMATH DE number 1186194