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Problème de Cauchy pour les hyperfonctions à croissance. (Cauchy problem for hyperfunctions with growth order). (French) Zbl 0686.35006

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1989, Exp. No. 1, 11 p. (1989).
This is a detailed announcement of the results of a forthcoming article of the author with Z. Mebkhout given in the 1989 PDE Colloquium in St. Jean-De-Monts. The author formerly gave a generalization of the Cauchy- Kowalewsky theorem à la Kashiwara for D-modules as follows: Let M be a coherent \(D_ X\)-module and Y,Z\(\subset X\) two complex analytic hypersurfaces transversal to each other. If Z is “non r- microcharacteristic” for M along Y, then \[ Ext^ k_{D_ X}(M,B_{Y| X}\{r\})=Ext^ k_{D_{Z| Z\cap Y}\{r\}}(M_ Z\{r\},B_{Y\cap Z| Z| Z}\{r\}) \] holds for any k. Here \(B_{Y| X}\{r\}\) is a subsheaf of holomorphic microfunctions with growth order, and \(M_ Z\{r\}\) denotes the induced system in the corresponding enlarged ring of formal differential operators of infinite order with the growth condition. In the article it is shown that if M is a holonomic \(D_ X\)-module and the non microcharacteristic condition holds for all r, then the right-hand side does not depend on r and can be replaced by \(Ext^ k_{D_ Z(M_ Z,B_{Y\cap Z| Z}\{r\})}\).
Reviewer: A.Kaneko

MSC:

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35A10 Cauchy-Kovalevskaya theorems
35G05 Linear higher-order PDEs
35J45 Systems of elliptic equations, general (MSC2000)
32A45 Hyperfunctions
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