×

zbMATH — the first resource for mathematics

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 \({\mathcal M}.\)

MSC:
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] B. ABDEL GADIR, Analyse microlocale des systèmes différentiels holonomes, Thèse, Univ. de Grenoble 1, 1992.
[2] J.E. BJÖRK, Rings of differential operators, North-Holland, Amsterdam, 1979. · Zbl 0499.13009
[3] A. BOREL et al., Algebraic D-modules, Persp. in Math., n° 2, Academic Press, 1987. · Zbl 0642.32001
[4] L. BOUTET DE MONVEL, P. KRÉE, Pseudo-differential operators and Gevrey classes, Ann. Inst. Fourier, 17-1 (1967), 295-323. · Zbl 0195.14403
[5] L. BOUTET DE MONVEL, M. LEJEUNE, B. MALGRANGE, Opérateurs différentiels et pseudo-différentiels.
[6] J.-L. BRYLINSKY, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque, n° 140-141 (1986), 3-134. · Zbl 0624.32009
[7] J.-L. BRYLINSKY, B. MALGRANGE, J.-L. VERDIER, Transformation de Fourier géométrique I, C. R. Acad. Sci. Paris, 294 (1983), 55-58. · Zbl 0553.14005
[8] J.-L. BRYLINSKY, B. MALGRANGE, J.-L. VERDIER, Transformation de Fourier géométrique II, C. R. Acad. Sci. Paris, 303 (1986), 193-198. · Zbl 0601.32010
[9] P. DELIGNE, Le formalisme des cycles évanescents, Lecture Notes in Math., n° 340, Springer Verlag, Berlin (1973, SGA 7 II, exp. 13,14.). · Zbl 0266.14008
[10] R. HARTSHORNE, Residues and duality, Lecture Notes in Math., n° 20, Springer Verlag, Berlin, 1966. · Zbl 0212.26101
[11] L. HÖRMANDER, An introduction to complex analysis in several variables, D. van Nostrand Comp., Princeton, 1966. · Zbl 0138.06203
[12] R. HOTTA, M. KASHIWARA, The invariant holonomic system on a semisimple Lie algebra, Inv. Math., 75 (1984), 327-358. · Zbl 0538.22013
[13] M. KASHIWARA, On the maximally overdetermined systems of linear differential equations I, Publ. RIMS Kyoto, 10 (1975), 563-579. · Zbl 0313.58019
[14] M. KASHIWARA, B-functions and holonomic systems, Inv. Math., 38 (1976), 33-53. · Zbl 0354.35082
[15] M. KASHIWARA, On the holonomic systems of linear differential equations II, Inv. Math., 49 (1978), 121-135. · Zbl 0401.32005
[16] M. KASHIWARA, Vanishing cycle sheaves and holonomic systems of differential equation, Lecture Notes in Math., n° 1016, Springer Verlag, Berlin, 1983, 134-142. · Zbl 0566.32022
[17] M. KASHIWARA, The Riemann-Hilbert problem for holonomic systems, Publ. RIMS Kyoto, 20 (1984), 319-365. · Zbl 0566.32023
[18] M. KASHIWARA, T. KAWAÏ, Second microlocalization and asymptotic expansions, Lecture Notes in Physics, n° 126, Springer Verlag, Berlin, 1980, 21-76. · Zbl 0458.46027
[19] M. KASHIWARA, T. KAWAÏ, On holonomic systems of micro-differential equations III — systems with regular singularities, Publ. RIMS Kyoto, 17 (1981), 813-979. · Zbl 0505.58033
[20] M. KASHIWARA, T. KAWAÏ, Microlocal analysis, Publ. RIMS Kyoto, 19 (1983), 1003-1032. · Zbl 0536.58030
[21] M. KASHIWARA, T. KAWAÏ, T. KIMURA, Foundations of algebraic analysis, Princeton Univ. Press, Princeton, 1986. · Zbl 0605.35001
[22] M. KASHIWARA, P. SCHAPIRA, Sheaves on manifolds, Springer Verlag, Berlin, 1991. · Zbl 0709.18001
[23] N.M. KATZ, G. LAUMON, Transformation de Fourier et majoration de sommes exponentielles, Publ. Math. IHES, 62 (1986), 361-418. · Zbl 0603.14015
[24] G. LAUMON, Transformation de Fourier géométrique, Prépubl. IHES M/52, 1985. · Zbl 0759.14014
[25] Y. LAURENT, Théorie de la deuxième microlocalisation dans le domaine complexe, Progress in Math., n° 53, Birkhäuser, Boston, 1985. · Zbl 0561.32013
[26] Y. LAURENT, Polygone de Newton et b-fonctions pour LES modules microdifférentiels, Ann. École Norm. Sup., 4e série, 20 (1987), 391-441. · Zbl 0646.58021
[27] Y. LAURENT, Vanishing cycle sheaves of D-modules, Inv. Math., 112 (1993), 491-539. · Zbl 0799.32031
[28] Y. LAURENT, B. MALGRANGE, Cycles proches, spécialisation et D-modules, Prépubl. Inst. Fourier, n° 275, 1994. · Zbl 0837.35006
[29] B. MALGRANGE, Remarques sur LES équations différentielles à points singuliers irréguliers, Lecture Notes in Math., n° 712, Springer Verlag, 1979, 77-86. · Zbl 0423.32014
[30] B. MALGRANGE, Transformation de Fourier géométrique, Sém. Bourbaki, 692 (1987-1988). · Zbl 0687.35003
[31] B. MALGRANGE, Fourier transforms and differential equations, Proc. of the Summer School on Mathematical Physics, Brasov, 1989. · Zbl 0733.34010
[32] B. MALGRANGE, Equations différentielles à coefficients polynomiaux, Progress in Math., n° 96, Birkhäuser, Berlin, 1991. · Zbl 0764.32001
[33] B. MALGRANGE, Connexions méromorphes, in ‘Singularities Lille 1991’, J.-P. Brasselet, éd., London Math. Soc. Lecture Notes, n° 201, Cambridge Univ. Press, 1994, 251-261. · Zbl 0816.32001
[34] B. MALGRANGE, Filtrations des modules holonomes, in ‘Analyse algébrique des perturbations singulières II’, L. Boutet de Monvel, éd., Hermann, 1994, 35-41. · Zbl 0843.58115
[35] Z. MEBKHOUT, Le formalisme des six opérations de Grothendieck pour LES D-modules cohérents, Travaux en cours, n° 35, Hermann, Paris, 1989. · Zbl 0686.14020
[36] T. ODA, Introduction to algebraic analysis on complex manifolds, Adv. Stud. in Pure Math., n° 1, 1983. · Zbl 0512.14008
[37] F. PHAM, Singularités des systèmes différentiels de Gauss-Manin, Progress in Math., n° 2, Birkhäuser, Boston, 1979. · Zbl 0524.32015
[38] J.-P. RAMIS, À propos du théorème de Borel-Ritt à plusieurs variables, Lecture Notes in Math., n° 712, Springer Verlag, 1979, 289-292. · Zbl 0455.35036
[39] C. SABBAH, D-modules et cycles évanescents, Travaux en cours, n° 24, Hermann, Paris, 1987, 53-98. · Zbl 0623.32013
[40] M. SATO, M. KASHIWARA, T. KAWAÏ, Microfunctions and pseudo-differential equations, Lecture Notes in Math., n° 287, Springer Verlag, Berlin, 1973, 265-529. · Zbl 0277.46039
[41] P. SCHAPIRA, Microdifferential systems in the complex domain, Springer Verlag, Berlin, 1985. · Zbl 0554.32022
[42] J.-P. SCHNEIDERS, Dualité pour LES modules différentielles, Thèse, Univ. de Liège, 1986-1987.
[43] J.-C. TOUGERON, An introduction to the theory of Gevrey expansions and to the Borel-Laplace transforms, with some applications, notes de cours, 1989-1990.
[44] J.-L. VERDIER, Dualité pour LES espaces localement compacts, Sém. Bourbaki, 300 (1966). · Zbl 0268.55006
[45] J.-L. VERDIER, Géométrie microlocale, Lecture Notes in Math., n° 1016, Springer Verlag, Berlin, 1983, 127-133. · Zbl 0576.35096
[46] C. HUYGUE, Transformation de Fourier des D✝X,ℚ(∞)-modules, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 759-762. · Zbl 0872.14011
[47] M. KASHIWARA, P. SHAPIRA, Laplace transforms and Fourier-Sato transforms, Algebraic analysis methods in microlocal analysis (Japanese), Sūrikaisekiken-kyūusho Kōkyūroku, 983 (1997), 33-35. · Zbl 0932.32015
[48] L. RAMERO, Fourier transforms in geometry and arithmetic, number theory, II (Rome, 1995), Rend. Sem. Math. Univ. Politec Torino, 53-4 (1995), 419-436. · Zbl 0872.14012
[49] C. SABBAH, Monodromy at infinity and Fourier transform, Publ. Res. Inst. Math. Sci., 33-4 (1997), 643-685. · Zbl 0920.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.