Microlocal analysis in gevrey classes and in complex domains

  • Hikosaburo Komatsu
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1495)


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  1. [1]
    E. Albrecht-M. Neumann, Local operators between spaces of ultradifferentiable functions and ultradistributions, Manuscripta Math., 38 (1982), 131–161.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    T. Aoki, An invariant measuring the irregularity of a differential operator and a microdifferential operator, J. Math. Pures Appl., 61 (1982), 131–148.MATHMathSciNetGoogle Scholar
  3. [3]
    T. Aoki, Growth order of microdifferential operators of infinite order, J. Fac. Sci. Univ. Tokyo, Sec. IA, 29 (1982), 143–159.MATHGoogle Scholar
  4. [4]
    T. Aoki, Symbols and formal symbols of pseudodifferential operators, Group Representations and Systems of Differential Equations, Advanced Studies in Pure Math., 4 (1984), 181–208.Google Scholar
  5. [5]
    G. Bengel, Das Weyl'sche Lemma in der Theorie der Hyperfunktionen, Math. Z., 96 (1967), 373–392.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    G. Bengel-P. Schapira, Décomposition microlocale analytique des distributions, Ann. Inst. Fourier, Grenoble, 29–3 (1979), 101–124.CrossRefMathSciNetGoogle Scholar
  7. [7]
    G. Björck, Linear partial differential operators and generalized distributions, Ark. f. Mat., 6 (1966), 351–407.MATHCrossRefGoogle Scholar
  8. [8]
    J.-M. Bony, Propagation des singularités différentiables pour une classe d'opérateurs différentiels à coefficients analytiques, Astérisque, 34–35 (1976), 43–91.MathSciNetGoogle Scholar
  9. [9]
    J.-M. Bony-P. Schapira, Propagation des singularités analytiques pour les solutions des équations aux dérivées partielles, Ann. Inst. Fourier, Grenoble, 26 (1976), 81–140.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    L. Boutet de Monvel, Opérateurs pseudodifferentiels analytiques et opérateurs d'ordre infini, Ann. Inst. Fourier, Grenoble, 22–3 (1972), 229–268.CrossRefMathSciNetGoogle Scholar
  11. [11]
    J. Bros-D. Iagolnitzer, Causality and local analyticity: mathematical study, Ann. Inst. H. Poincaré, Sect. A, 18 (1973), 147–184.MATHMathSciNetGoogle Scholar
  12. [12]
    J. Bruna, An extension theorem of Whitney type for non-quasianalytic class of functions, J. London Math. Soc., (2), 22 (1980), 495–505.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A.P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16–36.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    J.W. De Roever, Hyperfunctional singular support of ultradistributions, J. Fac. Sci., Univ. Tokyo, Sec. IA, 31 (1984), 585–631.MATHGoogle Scholar
  15. [15]
    J.J. Duistermaat-L. Hörmander, Fourier integral operators II, Acta Math., 128 (1972), 183–269.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Yu.V. Egorov, On canonical transformations of pseudo-differential operators, Uspehi Mat. Nauk, 25 (1969), 235–236 (in Russian).Google Scholar
  17. [17]
    L. Ehrenpreis, A fundamental principle for systems of linear differential equations with constant coefficients and some of its applications, Proc. Internat. Symp. on Linear Spaces, Jerusalem, (1961), 161–174.Google Scholar
  18. [18]
    L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley-Interscience, New York-London-Sydney-Toronto, (1970).MATHGoogle Scholar
  19. [19]
    A. Eida, On microlocal decomposition of ultradistributions, Master's thesis, Univ. Tokyo, (1989), (in Japanese).Google Scholar
  20. [20]
    R. Godement, Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris, (1964).Google Scholar
  21. [21]
    H. Grauert, On Levi's problem and the imbedding of real analytic manifolds, Ann. of Math., 68 (1958), 460–472.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    A. Grothendieck, Sur les espaces de solutions d'une classe générale d'équations aux dérivées partielles, J. Analyse Math., 2 (1952–53), 243–280.CrossRefMathSciNetGoogle Scholar
  23. [23]
    A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasil. Math., 3 (1954), 57–122.MathSciNetGoogle Scholar
  24. [24]
    A. Grothendieck, Produits Tensoriels Topologiques et Espaces Nucéaires, Mem. Amer. Math. Soc., 16, Amer. Math. Soc., Providence, (1954).Google Scholar
  25. [25]
    A. Grothendieck, Local Cohomology, Lecture Notes in Math., 41, Springer, Berlin-Heidelberg-New York, (1967).MATHGoogle Scholar
  26. [26]
    R. Hartshorn, Residue and Duality, Lecture Notes in Math., 20, Springer, Berlin-Heidelberg-New York, (1966).Google Scholar
  27. [27]
    R. Harvey, Hyperfunctions and linear partial differential equations, Proc. Nat. Acad. Sci. U.S.A., 55 (1966), 1042–1046.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    L. Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math., 18 (1965), 501–517.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    L. Hörmander, An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, (1966).MATHGoogle Scholar
  30. [30]
    L. Hörmander, Fourier integral operators I, Acta Math., 127 (1971), 79–183.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math., 24 (1971), 671–704.MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, Berlin-Heidelberg-New York-Tokyo, (1983).MATHCrossRefGoogle Scholar
  33. [33]
    A. Kaneko, Introduction to Hyperfunctions, Kluwer Academic, Dordrecht-Boston-London, (1988).MATHGoogle Scholar
  34. [34]
    M. Kashiwara-T. Kawai, Pseudo-differential operators in the theory of hyperfunctions, Proc. Japan Acad., 46 (1970), 1130–1134.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    M. Kashiwara-T. Kawai-T. Kimura, Foundations of Algebraic Analysis, Princeton Univ. Press, Princeton, (1986).MATHGoogle Scholar
  36. [36]
    K. Kataoka, On the theory of Radon transformations of hyperfunctions, J. Fac. Sci., Univ. Tokyo, Sec. IA, 28 (1981), 331–413.MATHMathSciNetGoogle Scholar
  37. [37]
    C.O. Kiselman, Prolongement des solutions d'une équation aux dérivées partielles à coefficients constants, Bull. Soc. Math. France, 97 (1969), 329–396.MATHMathSciNetGoogle Scholar
  38. [38]
    J.J. Kohn-L. Nirenberg, On the algebra of pseudo-differential operators, Comm. Pure Appl. Math., 18 (1965), 269–305.MATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    H. Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 19 (1967), 366–383.MATHCrossRefMathSciNetGoogle Scholar
  40. [40]
    H. Komatsu, Resolutions by hyperfunctions of sheaves of solutions of differential equations with constant coefficients, Math. Ann., 176 (1968), 77–86.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    H. Komatsu On the Alexander-Pontrjagin duality theorem, Proc. Japan Acad., 44 (1968), 489–490.MATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    H. Komatsu, Boundary values for solutions of elliptic equations, Proc. Internat. Conf. on Functional Analysis and Related Topics, 1969, Univ. of Tokyo Press, (1970), 107–121.Google Scholar
  43. [43]
    H. Komatsu, Relative cohomology of sheaves of solutions of differential equations, Hyperfunctions and Pseudo-Differential Equations, Lecture Notes in Math., 287, Springer, Berlin-Heidelberg-New York, (1972), 192–259.CrossRefGoogle Scholar
  44. [44]
    H. Komatsu, A local version of Bochner's tube theorem, J. Fac. Sci. Univ. Tokyo, Sec. IA, 19 (1972), 201–214.MATHMathSciNetGoogle Scholar
  45. [45]
    H. Komatsu, Ultradistributions, I, Structure theorems and a characterization, J. Fac. Sci., Univ. Tokyo, Sec. IA, 20 (1973), 25–105.MATHMathSciNetGoogle Scholar
  46. [46]
    H. Komatsu, Theory of Locally Convex Spaces, Dept. of Math., Univ. Tokyo, (1974).Google Scholar
  47. [47]
    H. Komatsu, Irregularity of characteristic elements and construction of nullsolutions, J. Fac. Sci., Univ. Tokyo, Sec. IA, 23 (1976), 297–342.MATHMathSciNetGoogle Scholar
  48. [48]
    H. Komatsu, Ultradistributions, II, The kernel theorem and ultradistributions with support in a submanifold, J. Fac. Sci., Univ. Tokyo, Sec. IA, 24 (1977), 607–628.MATHMathSciNetGoogle Scholar
  49. [49]
    H. Komatsu, An Introduction to the Theory of Generalized Functions, Iwanami, Tokyo, (1978), (in Japanese).Google Scholar
  50. [50]
    H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad., Ser. A, 55 (1979), 69–72.MATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    H. Komatsu, Linear hyperbolic equations with Gevrey coefficients, J. Math. Pures Appl., 59 (1980), 145–185.MATHMathSciNetGoogle Scholar
  52. [52]
    H. Komatsu, Ultradifferentiability of solutions of ordinary differential equations, Proc. Japan Acad., Ser. A, 56 (1980), 137–142.MATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    H. Komatsu, Ultradistributions, III, Vector valued ultradistributions and the theory of kernels, J. Fac. Sci., Univ. Tokyo, Sec. IA, 29 (1982), 653–717.MATHMathSciNetGoogle Scholar
  54. [54]
    H. Komatsu, Irregularity of hyperbolic operators, Hyperbolic Equations and Related Topics, 1984, Kinokuniya, Tokyo, (1986), 155–179.Google Scholar
  55. [55]
    H. Komatsu-T. Kawai, Boundary values of hyperfunction solutions of linear partial differential equations, Publ. RIMS, Kyoto Univ., 7 (1971–72), 95–104.MATHCrossRefMathSciNetGoogle Scholar
  56. [56]
    G. Köthe, Dualität in der Funktionentheorie, J. Reine Angew. Math., 191 (1953), 30–49.MATHMathSciNetGoogle Scholar
  57. [57]
    P. Laubin, Analyse microlocale des singularités analytiques, Bull. Soc. Roy. Sci. Liège, 52-2 (1983), 103–212.MATHMathSciNetGoogle Scholar
  58. [58]
    G. Lebeau, Fonctions harmoniques et spectre singulier, Ann. Sci. Ecole Norm Sup. (4), 13 (1980), 269–291.MATHMathSciNetGoogle Scholar
  59. [59]
    B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, Grenoble, 6 (1955–56), 271–355.CrossRefMathSciNetGoogle Scholar
  60. [60]
    B. Malgrange, Faisceaux sur des variétés analytiques réelles, Bull. Soc. Math. France, 85 (1957), 231–237.MATHMathSciNetGoogle Scholar
  61. [61]
    F. Mantovani-S. Spagnolo, Funzionali analitici reali e funzioni armoniche, Ann. Scuola Norm. Sup. Pisa, (6), 18 (1964), 475–513.MATHMathSciNetGoogle Scholar
  62. [62]
    A. Martineau, Les hyperfonctions de M.Sato, Séminaire Bourbaki, 13 (1960–61), No. 214.Google Scholar
  63. [63]
    A. Martineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, J. Analyse Math., 9 (1963), 1–164.CrossRefMathSciNetGoogle Scholar
  64. [64]
    A. Martineau, Distributions et valeurs au bord des fonctions holomorphes, Theory of Distributions, Inst. Gulbenkian de Ciencia, Lisboa, (1964), 193–326.Google Scholar
  65. [65]
    A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Math. Ann., 163 (1966), 62–88.MATHCrossRefMathSciNetGoogle Scholar
  66. [66]
    A. Martineau, Le “edge of the wedge theorem” en théorie des hyperfonctions de Sato, Proc. Internat. Conf. on Functional Analysis and Related Topics, 1969, Univ. of Tokyo Press, (1970), 95–106.Google Scholar
  67. [67]
    R. Meise-B.A. Taylor, Whitney's extension theorem for ultradifferentiable functions of Beurling type, Ark. f. Mat., 26 (1988), 265–287.MATHCrossRefMathSciNetGoogle Scholar
  68. [68]
    S. Mizohata, Note sur le traitement par les opérateurs d'integrale singulière du problème de Cauchy, J. Math. Soc. Japan, 11 (1959), 234–240.MATHCrossRefMathSciNetGoogle Scholar
  69. [69]
    M. Morimoto, Sur la décomposition du faisceau des germs de singularités d'hyperfonctions, J. Fac. Sci., Univ. Tokyo, Sec. I, 17 (1970), 215–239.MATHMathSciNetGoogle Scholar
  70. [70]
    K. Nishiwada, On local characterization of wave front sets in terms of boundary values of holomorphic functions, Publ. RIMS, Kyoto Univ., 14 (1978), 309–320.MATHCrossRefMathSciNetGoogle Scholar
  71. [71]
    J. Peetre, Une caractérisation abstraite des opérateurs différentielles, Math. Scand., 7 (1959), 211–218; 8 (1960), 116–120.MATHMathSciNetGoogle Scholar
  72. [72]
    H.-J. Petzsche-D. Vogt, Almost analytic extension of ultradifferentiable functions and the boundary values of holomorphic functions, Math. Ann., 267 (1984), 17–35.MATHCrossRefMathSciNetGoogle Scholar
  73. [73]
    D.A. Raikov, On two classes of locally convex spaces important in applications, Trudy Sem. Funkcional Anal., Voronez, 5 (1957), 22–34 (in Russian).Google Scholar
  74. [74]
    C. Roumieu, Sur quelques extensions de la notion de distribution, Ann. Ecole Norm. Sup., 77 (1960), 41–121.MATHMathSciNetGoogle Scholar
  75. [75]
    C. Roumieu, Ultra-distributions définies sur R n et sur variétés différentiables, J. Analyse Math., 10 (1962–63), 153–192.MATHCrossRefMathSciNetGoogle Scholar
  76. [76]
    M. Sato, Theory of hyperfunctions, I, J. Fac. Sci., Univ. Tokyo, Sec I, 8 (1959), 139–193.MATHGoogle Scholar
  77. [77]
    M. Sato, Theory of hyperfunctions, II, J. Fac. Sci., Univ. Tokyo, Sec I, 8 (1960), 387–436.MATHGoogle Scholar
  78. [78]
    M. Sato, Hyperfunctions and partial differential equations, Proc. Internat. Conf. on Functional Analysis and Related Topics, 1969, Univ. of Tokyo Press, (1970), 91–94.Google Scholar
  79. [79]
    M. Sato-T. Kawai-M. Kashiwara, Microfunctions and pseudodifferential equations, Hyperfunctions and Pseudo-Differential Equations, Lecture Notes in Math., 287, Springer, Berlin-Heidelberg-New York, (1973), 265–529.Google Scholar
  80. [80]
    P. Schapira, Problème de Dirichlet et solutions hyperfonctions des equations elliptiques, Boll. U.M.I. (4), 3 (1969), 367–372.MathSciNetGoogle Scholar
  81. [81]
    L. Schwartz, Théorie des Distributions, Hermann, Paris, (1950–51).MATHGoogle Scholar
  82. [82]
    L. Schwartz, Espaces de fonctions différentiables à valeurs vectorielles, J. Analyse Math., 4 (1954–55), 88–148.CrossRefMathSciNetGoogle Scholar
  83. [83]
    L. Schwartz, Théorie des distributions à valeurs vectorielles, Ann. Inst. Fourier, Grenoble, 7 (1957), 1–141; 8 (1958), 1–209.MATHCrossRefMathSciNetGoogle Scholar
  84. [84]
    J. Sebastião e Silva, Su certe classi di spazi localmente convessi importanti, per le applicazioni, Rend. Mat. e Appl. Univ. Roma, Ser. V, 14 (1955), 388–410.MATHGoogle Scholar
  85. [85]
    J. Siciak, Holomorphic continuation of harmonic functions, Ann. Polon. Math., 29 (1974), 67–73.MATHMathSciNetGoogle Scholar
  86. [86]
    J. Sjöstrand, Singularités analytiques microlocales, Astérisque, 95 (1982), 1–166.MATHGoogle Scholar
  87. [87]
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, (1970).Google Scholar
  88. [88]
    E.M. Stein-G. Weiss, Introduction to Fourier Analysis on Eucledean Spaces, Princeton Univ. Press, (1971).Google Scholar
  89. [89]
    H.G. Tillmann, Dualität in der Potentialtheorie, Portugal. Math., 13 (1954), 55–86.MathSciNetGoogle Scholar
  90. [90]
    K. Uchikoshi, Microlocal analysis of partial differential operators with irregular singularities, J. Fac. Sci. Univ. Tokyo, Sec. IA, 30 (1983), 299–332.MATHMathSciNetGoogle Scholar

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  • Hikosaburo Komatsu

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