Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Integral transforms with exponential kernels and Laplace transform

Authors: Masaki Kashiwara and Pierre Schapira
Journal: J. Amer. Math. Soc. 10 (1997), 939-972
MSC (1991): Primary 32C38, 14F10, 44A10
MathSciNet review: 1447834
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $X \underset{f}{\longleftarrow} Z \underset{g}{\longrightarrow} Y$ be a correspondence of complex manifolds. We study integral transforms associated to kernels $\exp(\varphi)$, with $\varphi$ meromorphic on $Z$, acting on formal or moderate cohomologies. Our main application is the Laplace transform. In this case, $X$ is the projective compactification of the vector space $V \simeq {\Bbb C}^n$, $Y$ is its dual space, $Z=X\times Y$ and $\varphi(z,w) =\langle z,w \rangle$. We obtain the isomorphisms:

\begin{align*}&F \mathop\otimes\limits^W{\cal {O}}_V \simeq F^\wedge[n] \mathop\otimes\limits^W{\cal {O}}_{V^*},\quad \operatorname{ THom}(F,{\cal {O}}_V) \simeq \operatorname{ THom}(F^\wedge[n],{\cal {O}}_{V^*}) \end{align*}

where $F$ is a conic and ${\Bbb R}$-constructible sheaf on $V$ and $F^\wedge$ is its Fourier-Sato transform. Some applications are discussed.

References [Enhancements On Off] (What's this?)

  • [A] E. Andronikof, Microlocalisation tempérée, Mém. Soc. Math. France, 57 (1994). MR 95e:58168
  • [B] J-E. Björk, Analytic ${\cal {D}}$-modules and Applications, Kluwer Academic Publisher, Dordrecht-Boston-London (1993). MR 95f:32014
  • [B-M-V] J-L. Brylinski, B. Malgrange and J-L. Verdier, Transformation de Fourier géométrique II, C. R. Acad. Sci., 303 (1986), 193-198. MR 88m:58176
  • [D'A-S1] A. D'Agnolo and P. Schapira, The Radon-Penrose transform for ${\cal {D}}-$modules, J. of Functional Analysis, 139 (1996), 349-382. CMP 96:16
  • [D'A-S2] A. D'Agnolo and P. Schapira, Leray's quantization of projective duality, Duke Math. J. 84 (1996), 453-496. CMP 96:17
  • [D] L. Daia, La transformation de Fourier pour les ${\cal {D}}$-modules, Thèse, Université de Grenoble (1995).
  • [F-G] J. Faraut and S. Gindikin, Private communication to P.S., (1995).
  • [H-K] R. Hotta and M. Kashiwara, The invariant holonomic systems on a semi-simple Lie algebra, Inventiones Math., 75 (1984), no.2, 327-358. MR 87i:22041
  • [K] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., 20 (1984), no.2, 319-365. MR 86j:58142
  • [K-S1] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Math. Wiss., Springer, 292 (1990). MR 92a:58132; MR 95g:58222
  • [K-S2] M. Kashiwara and P. Schapira, Moderate and formal cohomology associated with constructible sheaves, Mémoires Soc. Math. France, 64 (1996). CMP 97:04
  • [K-Sm] M. Kashiwara and W. Schmid, Quasi-equivariant ${\cal {D}}$-modules, equivariant derived category, and representations of reductive Lie groups, in Lie theory and Geometry in honor of Bertram Kostant, Progress of Mathematics, 123 (1994), 457-488. MR 96e:22031
  • [K-L] N. M. Katz and G. Laumon, Transformation de Fourier et majoration de sommes d'exponentielles, Publ. I.H.E.S., 62 (1985), 361-418. MR 87i:14017
  • [M] B. Malgrange, Transformation de Fourier géométrique, Séminaire Bourbaki, 692 (1987-88). MR 90c:58178
  • [Mr] A. Martineau, Distributions et valeurs au bord des fonctions holomorphes, in Proceedings International Summer, Institute Gulbenkian, Lisbon (1964), 195-226, Oeuvres de André Martineau, édition de CNRS (1977), 439-582. MR 36:2833
  • [S-K-K] M. Sato, T. Kawai and M. Kashiwara, Hyperfunctions and pseudo-differential equations, in Proceedings Katata 1971, Lecture Notes in Math., Springer-Verlag, 287 (1973), 265-529. MR 54:8747

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 32C38, 14F10, 44A10

Retrieve articles in all journals with MSC (1991): 32C38, 14F10, 44A10

Additional Information

Masaki Kashiwara
Affiliation: RIMS, Kyoto University, Kyoto 606-01, Japan

Pierre Schapira
Affiliation: Institut de Mathématiques, Université Paris VI, Case 82, 4 pl Jussieu, 75252 Paris, France

Received by editor(s): September 17, 1996
Received by editor(s) in revised form: May 23, 1997
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society