Cauchy problem for hyperbolic 𝒟-modules with regular singularities

Andrea D’Agnolo and Francesco Tonin

Vol. 184 (1998), No. 1, 1–22

DOI: 10.2140/pjm.1998.184.1

Abstract

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.

Milestones

Received: 1 August 1996

Revised: 16 June 1997

Published: 1 May 1998