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La transformation de Fourier pour les \({\mathcal D}\)-modules. (Fourier transform for \({\mathcal D}\)-modules). (French) Zbl 0963.35002
The starting point of the author’s work is the following result, proved by B. Malgrange [Semin. Bourbaki, 40eme Annee, Vol. 1987/88, Exp. No. 692, Asterisque 161-162, 133-150 (1988; Zbl 0687.35003)]: Let \(W_n\) be the Weyl algebra on \({\mathbb C}^n\) and let \(M\) be a monodromic finite type \(W_n\)-module. Then there is an isomorphism \(\text{ Sol}({\mathcal F}M)\simeq{\mathcal F}^+\text{Sol}(M),\) where \({\mathcal F}\) denotes the formal Fourier transform for \(W_n\)-modules, \({\mathcal F}^+\) denotes the sheaf-theoretic Fourier transform, and \(\text{ Sol} (\cdot)\) denotes the solution-functor for \(W_n\)-modules. Malgrange conjectured in the same paper that the same result should hold true as long as, at infinity, \(M\) satisfies suitable conditions. In particular, when \(M\) is holonomic and regular.
In this paper the author gives a positive answer to that conjecture. More precisely (recall that \({\mathbb D}^b({\mathcal D}_E)\) denotes the derived category (of bounded complexes) of \({\mathcal M}\text{od}({\mathcal D}_E)\)) he proves the following theorem: Let \(E\) be a finite-dimensional complex vector space, let \(E'\) be its dual. Consider the sheaf of algebraic (i.e. with polynomial coefficients) differential operators \({\mathcal D}_E\) on \(E,\) and let \({\mathcal M}\) be a bounded complex of left \({\mathcal D}_E\)-modules, with 1-specializable cohomology at infinity. Then in \({\mathbb D}^b({\mathbb C}_{E'})\) there exists a canonical isomorphism \({\mathcal F}^+\text{ Sol}({\mathcal M})\rightarrow\text{Sol}({\mathcal F}_* {\mathcal M})\). When \({\mathcal M}\) is regular on \(E\), the conditions of the theorem are fulfilled, and, by a result of M. Kashiwara and T. Kawai [Complex analysis, microlocal calculus and relativistic quantum theory, Proc. Colloq., Les Houches 1979, Lect. Notes Phys. 126, 21-76 (1980; Zbl 0458.46027)], the author gets that the result conjectured by B. Malgrange holds true for such \({\mathcal M}.\)

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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