Cauchy problem for hyperbolic D-modules with regular singularities

Let M be a coherent D-module (e.g., an overdetermined system of partial differential equations) on the complexification of a real analytic manifold M. Assume that the characteristic variety of M is hyperbolic with respect to a submanifold N ⊂ M. Then, it is well-known that the Cauchy problem for M with data on N is well posed in the space of hyperfunctions. In this paper, under the additional assumption that M has regular singularities along a regular involutive submanifold of real type, we prove that the Cauchy problem is well posed in the space of distributions. When M is induced by a single differential operator (or by a normal square system) with characteristics of constant multiplicities, our hypotheses correspond to Levi conditions, and we recover a classical result.