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Micro-hyperbolic pseudo-differential operators

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Hyperfunctions and Theoretical Physics

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References

  1. Andersson, K. G.: Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat., 8 (1971) 277–302.

    Article  MathSciNet  MATH  Google Scholar 

  2. Atiyah, M. F., R. Bott and L. Gårding: Lacunas for hyperbolic differential operators with constant coefficients I., Acta Math., 124 (1970) 109–189.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bony, J. M. et P. Schapira: Problème de Cauchy, existence et prolongement pour les hyperfonctions solutions d'équations hyperboliques non strictes, C. R. Acad. Sci. Paris, 274 (1972) 188–191.

    MathSciNet  MATH  Google Scholar 

  4. __: Solutions hyperfonctions du problème de Cauchy, Hyperfunctions and Pseudo-differential Equations, Lecture Notes in Mathematics No. 287, Springer, Berlin-Heidelberg-New York, 1973, pp. 82–98.

    Book  Google Scholar 

  5. Chazarain, J.: Opérateurs hyperboliques à caracteristiques de multiplicité constante, to appear.

    Google Scholar 

  6. __: An article which will appear in Sém. Bourbaki (1972/1973).

    Google Scholar 

  7. Courant, R. u. D. Hilbert: Methoden der Mathematischen Physik, II, Springer, Berlin, 1937.

    Book  MATH  Google Scholar 

  8. __: Methods of Mathematical Physics, II, Interscience, New York, 1962.

    MATH  Google Scholar 

  9. Courant, R. and P. D. Lax: The propagation of discontinuities in wave motion, Proc. Nat. Acad. Sci. U.S.A., 42 (1956) 872–876.

    Article  MathSciNet  MATH  Google Scholar 

  10. Duistermaat, J. and L. Hörmander: Fourier integral operators II, Acta Math., 128 (1971) 183–269.

    Article  MathSciNet  MATH  Google Scholar 

  11. Egorov, Yu. V.: Conditions for the solvability of pseudo-differential operators, Dokl. Akad. Nauk USSR, 187 (1969) 1232–1234. (In Russian.)

    Google Scholar 

  12. __: On subelliptic pseudo-differential operators, Dokl. Akad. Nauk USSR, 188 (1969) 20–22. (In Russian.)

    Google Scholar 

  13. Friedrichs, K. O.: Symmetric hyperbolic system of linear differential equations, Comm. Pure Appl. Math., 7 (1954) 345–392.

    Article  MathSciNet  MATH  Google Scholar 

  14. Friedrichs, K. O. u. H. Lewy: Über die Eindeutigkeit und das Abhängigkeitsgebiet der Lösungen beim Anfangsproblem linearer hyperbolischer Differentialgleichungen, Math. Ann., 98 (1928) 177–195.

    MathSciNet  MATH  Google Scholar 

  15. Gårding, L.: Linear hyperbolic partial differential equations with constant coefficients, Acta Math., 85 (1950) 1–62.

    Article  MathSciNet  Google Scholar 

  16. __: Local hyperbolicity, Israel J. Math., 13 (1972) 65–81.

    Article  MathSciNet  Google Scholar 

  17. __: A note which will appear in Israel J. Math. as a supplement to Gårding [2].

    Google Scholar 

  18. Hadamard, J.: Lectures on Cauchy Problem in Linear Partial Differential Equations, Yale Univ. Press, 1923. Reprinted by Dover, New York.

    Google Scholar 

  19. Hörmander, L.: On the theory of general partial differential operators, Acta Math., 94 (1955) 161–184.

    Article  MathSciNet  MATH  Google Scholar 

  20. __: Linear Partial Differential Operators, Springer, Berlin-Heidelberg-New York, 1963.

    Book  MATH  Google Scholar 

  21. __: On the singularities of solutions of partial differential equations, Proc. Int. Conf. Functional Analysis and Related Topics, Univ. of Tokyo Press, Tokyo, 1970, pp.31–40.

    MATH  Google Scholar 

  22. Kashiwara, M. and T. Kawai: Micro-hyperbolic pseudo-differential operators I, to appear.

    Google Scholar 

  23. Kawai, T.: Construction of local elementary solutions for linear partial differential operators with real analytic coefficients (I)— The case with real principal symbols—, Publ. RIMS, Kyoto Univ., 7 (1971) 363–397.

    Article  MathSciNet  MATH  Google Scholar 

  24. __: On the global existence of real analytic solutions of linear differential equations (I), J. Math. Soc. Japan, 24 (1972) 481–517.

    Article  MathSciNet  MATH  Google Scholar 

  25. Lax, P. D.: Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957) 627–646.

    Article  MathSciNet  MATH  Google Scholar 

  26. Leray, J.: Hyperbolic Differential Equations, The Institute for Advanced Study, Princeton, 1952.

    Google Scholar 

  27. __: Un prolongement de la transformation de Laplace qui transforme la solution unitaire d'un opérateur hyperbolique en sa solution élémentaire, Bull. Soc. Math. Fr., 90 (1962) 39–156.

    MathSciNet  MATH  Google Scholar 

  28. Leray, J. and Y. Ohya: Èquations et systèmes non-linéaires, hyperboliques non-stricts, Math. Ann., 170 (1967) 167–205.

    Article  MathSciNet  MATH  Google Scholar 

  29. Ludwig, D.: Exact and asymptotic solutions of the Cauchy problem, Comm. Pure Appl. Math., 13 (1960) 473–508.

    Article  MathSciNet  MATH  Google Scholar 

  30. Mizohata, S.: Analyticity of solutions of hyperbolic systems with analytic coefficients, Comm. Pure Appl. Math., 14 (1961) 547–559.

    Article  MathSciNet  MATH  Google Scholar 

  31. __: Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 (1961) 109–127.

    MathSciNet  MATH  Google Scholar 

  32. __: Solutions nulles et solutions non-analytiques, ibid., 1 (1961) 271–302.

    MathSciNet  MATH  Google Scholar 

  33. Mizohata, S. and Y. Ohya: Sur la condition d'hyperbolicité pour les équations a characteristiques multiples, II, Japanese J. Math., 40 (1971) 63–104.

    MathSciNet  MATH  Google Scholar 

  34. Nirenberg, L. and F. Treves: On local solvability of linear partial differential equations — Part II. Sufficient conditions, Comm. Pure Appl. Math., 23 (1970) 459–510.

    Article  MathSciNet  MATH  Google Scholar 

  35. Petrowsky, I. G.: Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen, Mat. Sbornik, 44 (1937), 815–868.

    MATH  Google Scholar 

  36. Riemann, B.: Über die Fortpflanzung ebener Luftwellen von endlicher Schwingusweite, Mathematische Werke, Dover, New York, 1953, pp. 156–175.

    Google Scholar 

  37. Sato, M., T. Kawai and M. Kashiwara: (Refered to as S-K-K[1]) Microfunctions and pseudo-differential equations, Hyperfunctions and Pseudo-differential Equations, Lecture Notes in Mathematics. No. 287, Springer, Berlin-Heidelberg-New York, 1973, pp. 265–529.

    Google Scholar 

  38. __: On the structure of single linear pseudo-differential equations, Proc. Japan Acad., 48 (1972), 643–646.

    Article  MathSciNet  MATH  Google Scholar 

  39. Treves, F.: Ovcyannikov Theorem and Hyperdifferential Operators, I. M. P. A., Rio-de-Janeiro (Brasil), 1969.

    MATH  Google Scholar 

  40. __: Analytic-hypoelliptic partial differential equations of principal type, Comm. Pure Appl. Math., 24 (1971) 537–570.

    Article  MathSciNet  MATH  Google Scholar 

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Frédéric Pham

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© 1975 Springer-Verlag

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Kashiwara, M., Kawai, T. (1975). Micro-hyperbolic pseudo-differential operators. In: Pham, F. (eds) Hyperfunctions and Theoretical Physics. Lecture Notes in Mathematics, vol 449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062916

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  • DOI: https://doi.org/10.1007/BFb0062916

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  • Print ISBN: 978-3-540-07151-8

  • Online ISBN: 978-3-540-37454-1

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