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Inversion formulas for complex radon transform on projective varieties and boundary value problems for systems of linear PDEs

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Abstract

Let G ⊂ ℂP n be a linearly convex compact set with smooth boundary, D = ℂPn \ G, and let D* ⊂ (ℂPn)* be the dual domain. Then for an algebraic, not necessarily reduced, complete intersection subvariety V of dimension d we construct an explicit inversion formula for the complex Radon transform R V : H d,d−1(V ∩ D) → H 1,0(D*) and explicit formulas for solutions of an appropriate boundary value problem for the corresponding system of differential equations with constant coefficients on D*.

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References

  1. A. Andreotti and F. Norguet, “La convexité holomorphe dans l’espace analytique des cycles d’une variété algébrique,” Ann. Scuola Norm. Super. Pisa, Sci. Fis. Mat., III Ser. 21, 31–82 (1967).

  2. A. Andreotti and F. Norguet, “Cycles of Algebraic Manifolds and ∂ \(\bar \partial \)-Cohomology,” Ann. Scuola Norm. Super. Pisa, Sci. Fis. Mat., III Ser. 25, 59–114 (1971).

    Google Scholar 

  3. B. Berndtsson and M. Passare, “Integral Formulas and an Explicit Version of the Fundamental Principle,” J. Funct. Anal. 84, 358–372 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Cartan, “Faisceaux analytiques sur les variétés de Stein,” in Fonctions analytiques de plusieurs variables complexes (Secr. Math., Paris, 1955), Exp. 18, 19, Sémin. H. Cartan Éc. Norm. Supér. 4: 1951/1952.

    Google Scholar 

  5. N. R. Coleff and M. E. Herrera, Les courants résiduels associés à une forme méromorphe (Springer, Berlin, 1978), Lect. Notes Math. 633.

    MATH  Google Scholar 

  6. A. Dickenstein and C. Sessa, “Canonical Representatives in Moderate Cohomology,” Invent. Math. 80, 417–434 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. G. Eastwood, R. Penrose, and R. O. Wells, Jr., “Cohomology and Massless Fields,” Commun. Math. Phys. 78, 305–351 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Fantappiè, “L’indicatrice proiettiva dei funzionali lineari e i predotti funzionali proiettivi,” Ann. Mat. Pura Appl., IV Ser. 22, 181–289 (1943).

  9. L. Fantappiè, “Sur les méthodes nouvelles d’intégration des équations aux dérivées partielles au moyen des fonctionnelles analytiques,” Colloq. Int. C. N. R. S. 71, 47–62 (1956).

    Google Scholar 

  10. A. S. Fokas, “On the Integrability of Linear and Nonlinear Partial Differential Equations,” J. Math. Phys. 41(6), 4188–4237 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  11. S. G. Gindikin and G. M. Khenkin, “Integral Geometry for \(\bar \partial \)-Cohomology in q-Linear Concave Domains in C P n,” Funkts. Anal. Prilozh. 12(4), 6–23 (1978) [Funct. Anal. Appl. 12, 247–261 (1979)].

    MathSciNet  MATH  Google Scholar 

  12. R. Hartshorne, Algebraic Geometry (Springer, New York, 1977).

    MATH  Google Scholar 

  13. G. M. Khenkin, “The Method of Integral Representations in Complex Analysis,” in Several Complex Variables, I: Introduction to Complex Analysis (Springer, Berlin, 1990), Encycl. Math. Sci. 7, pp. 19–116.

    Google Scholar 

  14. G. M. Henkin, “The Abel-Radon Transform and Several Complex Variables,” in Modern Methods in Complex Analysis (Princeton Univ. Press, Princeton, NJ, 1995), Ann. Math. Stud. 137, pp. 223–275; Preprint Univ. Paris VI, 1993.

    Google Scholar 

  15. G. M. Henkin and P. L. Polyakov, “Residual \(\bar \partial \)-Cohomology and the Complex Radon Transform on Subvarieties of ℂP n,” Math. Ann. 354, 497–527 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Hironaka, “Resolution of Singularities of an Algebraic Variety over a Field of Characteristic Zero. I, II,” Ann. Math. 79, 109–203, 205–326 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Leray, Hyperbolic Differential Equations (Inst. Adv. Study, Princeton, NJ, 1953).

    Google Scholar 

  18. J. Leray, “Le calcul differéntiel et intégral sur une variété analytique complexe,” Bull. Soc. Math. France 87, 81–180 (1959).

    MathSciNet  MATH  Google Scholar 

  19. B. Malgrange, “Systèmes différentiels à coefficients constants,” in Séminaire Bourbaki, 15e année: 1962/1963 (W.A. Benjamin, New York, 1966), Exp. 246.

    Google Scholar 

  20. A. Martineau, “Indicatrices des fonctionnelles analytiques et inversion de la transformée de Fourier-Borel par la transformation de Laplace,” C. R. Acad. Sci. Paris 255, 1845–1847 (1962); “Indicatrices des fonctions analytiques et inversion de la transformation de Fourier-Borel par la transformation de Laplace,” C. R. Acad. Sci. Paris 255, 2888–2890 (1962).

    MathSciNet  MATH  Google Scholar 

  21. A. Martineau, “Équations différentielles d’ordre infini,” Bull. Soc. Math. France 95, 109–154 (1967).

    MathSciNet  MATH  Google Scholar 

  22. F. Norguet, “Problèmes sur les formes différentielles et les courants,” Ann. Inst. Fourier 11, 1–82 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Passare, “Residues, Currents, and Their Relation to Ideals of Holomorphic Functions,” Math. Scand. 62, 75–152 (1988).

    MathSciNet  MATH  Google Scholar 

  24. R. Penrose, “Massless Fields and Sheaf Cohomology,” Twistor Newsletter, Oxford, No. 5, pp. 9–13 (July 1977).

    Google Scholar 

  25. P. L. Polyakov and G. M. Khenkin, “Homotopy Formulas for the \(\bar \partial \)-Operator on C P n and the Radon-Penrose Transform,” Izv. Akad. Nauk SSSR, Ser. Mat. 50(3), 566–597 (1986) [Math. USSR, Izv. 28, 555–587 (1987)].

    MathSciNet  MATH  Google Scholar 

  26. S. Rigat, “Application of the Fundamental Principle to Complex Cauchy Problem,” Ark. Mat. 38(2), 355–380 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  27. J.-P. Serre, “Un théorème de dualité,” Comment. Math. Helv. 29, 9–26 (1955).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. G. Vitushkin, “Remarkable Facts of Complex Analysis,” in Several Complex Variables, I: Introduction to Complex Analysis (Springer, Berlin, 1990), Encycl. Math. Sci. 7, pp. 1–17.

    Google Scholar 

  29. A. Weil, “L’intégrale de Cauchy et les fonctions de plusieurs variables,” Math. Ann. 111, 178–182 (1935).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques, Université Pierre et Marie Curie, Case 247, 4 place Jussieu, 75252, Paris, France

    Gennadi M. Henkin

  2. Central Economics and Mathematics Institute, Russian Academy of Sciences, Nakhimovskii pr. 47, Moscow, 117418, Russia

    Gennadi M. Henkin

  3. Department of Mathematics, University of Wyoming, Laramie, WY, 82071, USA

    Peter L. Polyakov

Authors
  1. Gennadi M. Henkin
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  2. Peter L. Polyakov
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Correspondence to Gennadi M. Henkin.

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Henkin, G.M., Polyakov, P.L. Inversion formulas for complex radon transform on projective varieties and boundary value problems for systems of linear PDEs. Proc. Steklov Inst. Math. 279, 230–244 (2012). https://doi.org/10.1134/S0081543812080160

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