Kernel calculus and extension of contact transformations to D-modules

  • Andrea D’Agnolo
  • Pierre Schapira


There is an important literature dealing with integral transformations. In our papers [6], [7] we proposed a general framework to the study of such transforms in the language of sheaves and D-modules. In particular, we showed that there are two natural adjunction formulas which split many difficulties into two totally different kind of problems: one of analytical nature, the calculation of the transform of a D-module, the other one topological, the calculation of the transform of a constructible sheaf. Similar adjunction formulas for temperate and formal cohomology are obtained by M. Kashiwara and P. S. in [15], and allow one to treat C -functions and distributions in this framework.


Complex Manifold Natural Isomorphism Holomorphic Vector Bundle Dualizing Complex Contact Transformation 
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  1. 1.
    E. Andronikof, Microlocalisation tempérée, Mém. Soc. Math. France 57 (1994), Supl. au Bull. de la Soc. Math. France 122 (2).Google Scholar
  2. 2.
    R. J. Baston and M. G. Eastwood, The Penrose transform: its interaction with representation theory, Oxford Univ. Press, 1989.Google Scholar
  3. 3.
    J-E. Björk, Analytic D-modules and Applications. Kluwer Academic Publisher, Dordrecht-Boston-London, 1993.Google Scholar
  4. 4.
    J. L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque 140-141 (1986), 3–134.MathSciNetGoogle Scholar
  5. 5.
    A. D’Agnolo, Nonlocal Differentials, Radon Transform, and Cavalieri Condition: a Cohomological Approach, Prépublication Univ. Paris VI and Paris VII (1996), and article to appear.Google Scholar
  6. 6.
    A. D’Agnolo and P. Schapira, Radon-Penrose transform for D-modules, to appear in J. of Functional Analysis; see also: La transformée de Radon-Penrose des D-modules, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 461–466.MathSciNetMATHGoogle Scholar
  7. 7.
    A. D’Agnolo and P. Schapira, Leray’s quantization of projective duality, Duke Math. J. 84 (1996), 453–496.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    M. G. Eastwood, R. Penrose, and R. O. Jr. Wells. Cohomology and massless fields. Comm. Math. Phys. 78 (1981), 305–351.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    I. M. Gelfand, S. G. Gindikin and M. I. Graev, Integral geometry in affine and projective spaces, Journal of Soviet Math. 18 (1982), 39–167.CrossRefGoogle Scholar
  10. 10.
    S. Helgason, The Radon Transform, Progress in Math. 5, Birkhäuser, 1980.Google Scholar
  11. 11.
    M. Kashiwara, Systems of microdifferential equations, Progress in Math. 34, Birkhäuser, 1983.Google Scholar
  12. 12.
    M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. RIMS, Kyoto Univ. 20(20) (1984), 319–365.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    M. Kashiwara and T. Kawai, On holonomic systems of micro-differential equations III — systems with regular singularities, Publ. RIMS, Kyoto Univ. 17 (1981), 813–979.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Math. Wiss. 292, Springer, 1990.Google Scholar
  15. 15.
    M. Kashiwara and P. Schapira, Moderate and formal cohomology associated with constructible sheaves, Mém. Soc. Math. France (N.S.) 64 (1996).Google Scholar
  16. 16.
    J. Leray, Le calcul différentiel et intégral sur une variété analytique complexe, Bull. Soc. Math. France 87 (1959), 81–180.MathSciNetMATHGoogle Scholar
  17. 17.
    C. Marastoni, Grassmann duality and D-modules, (in preparation).Google Scholar
  18. 18.
    A. Martineau, Indicatrice des fonctions analytiques et inversion de la transformation de Fourier-Borel par la transormation de Laplace, C. R. Acad. Sci. Paris Sér. I Math. 255 (1962), 2888–2890.MathSciNetMATHGoogle Scholar
  19. 19.
    M. Sato, T. Kawai, and M. Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (H Komatsu, ed.), Lecture Notes in Math. 287, Springer, 1973, Proceedings Katata 1971, pp. 265–529.Google Scholar
  20. 20.
    P. Schapira, Microdifferential systems in the complex domain, Grundlehren der Math. Wiss. 269, Springer, 1985.Google Scholar
  21. 21.
    P. Schapira and J.-P. Schneiders, Index theorems for elliptic pairs, Astérisque 224, 1994.Google Scholar
  22. 22.
    J.-P. Schneiders, Introduction to D-modules, Bull. Soc. Roy. Sci. Liège, 63(3-4) (1994).Google Scholar

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© Springer-Verlag Tokyo 1997

Authors and Affiliations

  • Andrea D’Agnolo
    • 1
  • Pierre Schapira
    • 1
  1. 1.Institut de Mathématiques; Analyse AlgébriqueUniversité Pierre et Marie CurieParis Cedex 05France

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