Complex Analysis in the Future Tube

  • A. G. Sergeev
  • V. S. Vladimirov
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 8)


The future tube in ℂ n +1 is the unbounded domain τ+ = z = (z o,...,z n ) ∈ ℂ n +1 : (Im z o)2 > (Imz 1)2 +...+ (Im z n )2 Im z o > 0. In other words, τ+ is a tube domain over the future cone V + = y ∈ ℝ n +1 : y 2 0 > y 2 1 +...+ y 2 n , y 0 > 0. The domain τ+ is biholomorphically equivalent to a classical Cartan domain of the IVth type, hence to a bounded symmetric domain in ℂ n +1. The future tube τ+ in ℂ4 (n = 3) is important in mathematical physics, especially in axiomatic quantum field theory, being the natural domain of definition of holomorphic relativistic fields. These specific features of the future tube motivated its investigation by mathematicians and physicists. Beginning with Elie Cartan’s classification of bounded symmetric domains, these domains were examined in many papers where the complex structure of their boundaries, integral representations, boundary values of holomorphic functions and so on were considered. The proof of the “edge-of-the-wedge” theorem by N.N. Bogolubov generated the rapid development of applications of the theory of several complex variables to axiomatic quantum field theory. During this period the “C-convex hull” and “finite covariance” theorems were proved, the Jost-Lehmann-Dyson representation was found et cetera. Recently R. Penrose has proposed a transformation connecting holomorphic solutions of the basic equations of field theory with analytic sheaf cohomologies of domains in ℝℙ3. These two directions developed, to a large extent, independently from each other, and some important results obtained in axiomatic quantum field theory still remain unknown to specialists in several complex variables and differential geometry. One of the goals of this paper is to give a unified presentation of advances in complex analysis in the future tube and related domains achieved in both of these directions.


Tube Cone Pseudoconvex Domain Symmetric Domain Distinguished Boundary Levi Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aizenberg, L.A., Dautov, Sh.A. (1976): Holomorphic functions of several complex variables with nonnegative real part. Traces of holomorphic and plurisubharmonic functions on the Shilov boundary. Mat. Sb., Nov. Ser. 99, No. 3, 342–355. Engl. transl.: Math. USSR, Sb. 28, 301-313 (1978), Zbl.341.32002.MathSciNetGoogle Scholar
  2. Aizenberg, L.A., Yuzhakov, A.P. (1979): Integral Representations and Residues in Multidimensional Complex Analysis. Novosibirsk: Nauka. 335 pp. Engl. transl.: transl. Math. Monogr. Vol. 58, Providence, 283 pp. (1983), Zbl.445.32002.Google Scholar
  3. Aleksandrov, A.B. (1983): On the boundary values of functions holomorphic in the ball. Dokl. Akad. Nauk SSSR 271, No. 4, 777–779. Engl. transl.: Sov. Math., Dokl. 28, 134-137 (1983), Zbl.543.32002.MathSciNetGoogle Scholar
  4. Aleksandrov, A.B. (1984): Inner functions on compact spaces. Funkts. Anal. Prilozh. 18, No. 2, 1–13. Engl. transl.: Funct. Anal. Appl. 18, 87-98 (1984), Zbl.574.32006.Google Scholar
  5. Araki, H. (1963): A generalization of Borchers’ theorem. Helv. Phys. Acta 36, No. 1, 132–139, Zbl. 112, 432.MathSciNetzbMATHGoogle Scholar
  6. Atiyah, M.F. (1979): Geometry of Yang-Mills Fields. Pisa: Scuola Normale Superiore. 99 pp., Zbl.435.58001.zbMATHGoogle Scholar
  7. Bekolle, D. (1984): Le dual de l’espace des fonctions holomorphes intégrables dans des domaines de Siegel. Ann. Inst. Fourier 34, No. 3, 125–154, Zbl.513.32032.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Bell, S. (1982): Proper holomorphic mappings between circular domains. Comment. Math. Helv. 57, No. 3, 532–538, Zbl.511.32013.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Bell, S. (1985): Proper holomorphic correspondences between circular domains. Math. Ann. 270 No. 3, 393–400, Zbl.554.32019.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Berline, N., Vergne, M. (1981): Equations de Hua et noyau de Poisson. Lect. Notes Math. 880, Berlin, Heidelberg, New York: Springer-Verlag, 1–51, Zbl.521.32024.Google Scholar
  11. Beurling, A. (1972): Analytic continuation across a linear boundary. Acta Math. 128, No. 3, 153–182, Zbl.235.30003.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Bochner, S. (1944): Group invariance of Cauchy’s formula in several variables. Ann. Math., II, Ser. 45, No. 4, 686–707, Zbl.60, 243.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Bogolubov, N.N., Vladimirov, V.S. (1958): A theorem on analytic continuation of generalized functions. Nauchn. Dokl. Vyssh. Shkoly, Fiz.-Mat. Nauki 1958, No. 3, 26–35 (Russian), Zbl.116, 85.Google Scholar
  14. Bogolubov, N.N., Vladimirov, V.S. (1971): Representation of n-point functions. Tr. Mat. Inst. Steklova 112, 5–21. Engl. transl.: Proc Steklov Inst. Math. 112, 1-18 (1973), Zbl.254.32015.Google Scholar
  15. Bogolubov, N.N., Medvedev, B.V., Polyvanov, M.K. (1958): Problems of the Theory of Dispersion Relations. Moscow: Fizmatgiz. 203 pp. (Russian), Zbl.83, 435.Google Scholar
  16. Bony, J.M. (1976): Propagation des singularités différentiables pour une classe d’opérateurs différentiels à coefficients analytiques. Astérisque, 34-35, 43–91, Zbl.344.35075.MathSciNetGoogle Scholar
  17. Borchers, H.J. (1961): Über die Vollständigkeit lorentzinvarianter Felder in einer zeitartigen Röhre. Nuovo Cimento 19, No. 4, 787–793, Zbl.111, 432.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Bros, J. (1977): Analytic completion and decomposability properties in tuboid domains. Publ. Res. Inst. Math. Sci. 12, Suppl., 19–37, Zbl.372.32002.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Bros, J., Epstein, H., Glaser, V. (1967): On the connection between analyticity and Lorentz covariance of Wightman functions. Commun. Math. Phys. 6, No. 1, 77–100, Zbl. 155, 323.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Bros, J., Iagolnitzer, D. (1975): Tuboides et structure analytique des distributions. Sémin. Goulaouic-Lions-Schwartz, 1974-1975, Exposé 16, 19 pp., Zbl.333.46028.Google Scholar
  21. Bros, J., Iagolnitzer, D. (1976): Tuboides dans ℂn et géneralisation d’un théorème de Cartan et Grauert. Ann. Inst. Fourier 26, No. 3, 49–72, Zbl.336.32003.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Bros, J., Itzykson, C., Pham, F. (1966): Représentations intégrales de fonctions analytiques et formule de Jost-Lehmann-Dyson. Ann. Inst. Henri Poincaré, New. Ser., Sect. A5, No. 1, 1–35, Zbl. 163, 225.MathSciNetGoogle Scholar
  23. Bros, J., Messiah, A., Stora, R. (1961): A problem of analytic completion related to the Jost-Lehmann-Dyson formula. J. Math. Phys. 2, No. 4, 639–651, Zbl. 131, 441.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Burns, D., Stout, E.L. (1976): Extending functions from submanifolds of the boundary. Duke Math. J. 43, No. 5, 391–404, Zbl.328.32013.MathSciNetzbMATHCrossRefGoogle Scholar
  25. Cartan, E. (1935): Sur les domaines bornés homogènes de l’espace de n variables complexes. Abh. Math. Semin. Univ. Hamb. 11, No. 1–2, 116–162, Zbl.11, 123.zbMATHCrossRefGoogle Scholar
  26. Chern, S.S. (1956): Complex Manifolds. Chicago: Univ. of Chicago, 181 pp., Zbl.88, 378.Google Scholar
  27. Chirka, E.M. (1973): Theorems of Lindelöf and Fatou in ℂn. Mat. Sb., Nov. Ser. 92, No. 4, 622–644. Engl. transl.: Math. USSR, Sb. 21, 619-639 (1975), Zbl.297.32001.Google Scholar
  28. Chirka, E.M., Khenkin, G.M. (1975): Boundary properties of holomorphic functions of several complex variables. Itogi Nauki Tekh., Ser. Sovrem. Probl. Math. 4, 13–142. Engl. transl.: J. Sov. Math. 5, 612-687 (1976), Zbl.375.32005.Google Scholar
  29. Dadok, J., Yang, P. (1985): Automorphisms of tube domains and spherical hypersurfaces. Am. J. Math. 107, No. 4, 999–1013, Zbl.586.32035.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Danilov, L.I. (1985): On regularity of proper cones in R n. Sib. Math. Zh. 26, No. 2, 198–201, Zbl.581.32002.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Drozhzhinov, Yu.N. (1982): Multidimensional Tauberian theorems for holomorphic functions of a bounded argument and quasiasymptotics of passive systems. Mat. Sb., Nov. Ser. 117, No. 1, 44–59. Engl. transl.: Math. USSR, Sb. 45, 45-61 (1983), Zbl.497.32001.MathSciNetGoogle Scholar
  32. Drozhzhinov, Yu.N., Vladimirov, V.S., Zavialov, B.I. (1984): Tauberian type theorems for generalized functions. Tr. Math. Inst. Steklova 163, 42–48. Engl. transl.: Proc. Steklov Inst. Math. 163, 53-60 (1985), Zbl.568.46032.zbMATHGoogle Scholar
  33. Drozhzhinov, Yu.N., Zavialov, B.I. (1979): Tauberian theorems for generalized functions supported in cones. Mat. Sb., Nov. Ser. 108, No. 1, 78–90. Engl. transl.: Math. USSR, Sb. 36, 75-86 (1980), Zbl.405.46033.Google Scholar
  34. Drozhzhinov, Yu.N., Zavialov, B.I. (1982): On a multidimensional analog of Lindelöf’s theorem. Dokl. Akad. Nauk SSSR 262, No. 2, 269–270 (Russian).MathSciNetGoogle Scholar
  35. Drozhzhinov, Yu.N., Zavialov, B.I. (1985): Multidimensional Tauberian comparison theorems for generalized functions in cones. Mat. Sb., Nov. Ser. 126, No. 4, 515–542. Engl. transl.: Math. USSR, Sb. 54, 499-524 (1986), Zbl.585.46033.Google Scholar
  36. Dyson, F.J., (1958): Integral representations of causal commutators. Phys. Rev., II. Ser. 110, No. 6, 1460–1464, Zbl.85, 434.MathSciNetzbMATHGoogle Scholar
  37. Fefferman, C. (1970): Inequalities for strongly singular convolution operators. Acta Math. 124, No. 1–2, 9–36, Zbl. 188, 426.MathSciNetzbMATHCrossRefGoogle Scholar
  38. Freeman, M. (1977): Real submanifolds with degenerate Levi form. Proc. Symp. Pure Math. 30, part 1, 141–147, Zbl.354.53010.MathSciNetGoogle Scholar
  39. Freeman, M. (1977): Local biholomorphic straightening of real submanifolds. Ann. Math., II. Ser. 106, No. 2, 319–352, Zbl.372.32005.MathSciNetzbMATHCrossRefGoogle Scholar
  40. Fuks, B.A. (1963): Special Topics from the Theory of Analytic Functions of Several Complex Variables. Moscow: Fizmatgiz. 427 pp. English transl.: transl. Math. Monogr., Vol. 14, Providence (1965), Zbl. 146, 308.Google Scholar
  41. Furstenberg, H. (1963): A Poisson formula for semi-simple Lie groups. Ann. Math., II. Ser. 77, No. 2, 335–386, Zbl. 192, 127.MathSciNetzbMATHCrossRefGoogle Scholar
  42. Gindikin, S.G. (1964): Analysis on homogeneous domains. Usp. Mat. Nauk 19, No. 4, 3–92. Engl. transl.: Russ. Math. Surv. 19, No. 4, 1-89 (1964), Zbl.144, 81.MathSciNetzbMATHGoogle Scholar
  43. Hahn, K.T. (1972): Properties of holomorphic functions of bounded characteristic on star-shaped circular domains. J. Reine Angew. Math. 254, 33–40, Zbl.246.32002.MathSciNetzbMATHGoogle Scholar
  44. Heinzner, P., Sergeev, A.G. (1991): The extended matrix disk is a domain of holomorphy. Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 3, 647–657. Engl. transl.: Math. USSR, Izv. 38, 637-645 (1992).MathSciNetzbMATHGoogle Scholar
  45. Helgason, S. (1978): Differential Geometry, Lie Groups and Symmetric Spaces. New York: Academic Press, 628 pp., Zbl.451.53038.zbMATHGoogle Scholar
  46. Hill, C.D., Kazlow, M. (1977): Function theory on tube manifolds. Proc. Symp. Pure Math. 30, part 1, 153–156, Zbl.383.32002.MathSciNetGoogle Scholar
  47. Hörmander, L. (1971): Fourier integral operators, I. Acta Math. 127, 79–183, Zbl.212, 466.MathSciNetzbMATHCrossRefGoogle Scholar
  48. Hua, L.-K. (1958): Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Peking: Science Press. Engl. transl.: Providence: Am. Math. Soc. 1979.186 pp., Zbl.90, 96.Google Scholar
  49. Hua, L.-K., Look, K.H. (1983): Theory of harmonic functions in classical domains. Hua L.K., Selected Papers. Berlin, Heidelberg, New York: Springer-Verlag, 743–806, Zbl.518.01022.Google Scholar
  50. Johnson, K.D. (1978): Differential equations and the Bergman-Shilov boundary on the Siegel upper half-plane. Ark. Mat. 16, No. 1, 95–108, Zbl.395.22013.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Johnson, K.D. (1984a): Generalized Hua-operators and parabolic subgroups. The cases of SL(n, ℂ) and SL(n, ∝). Trans. Am. Math. Soc. 281, No. 1, 417–429, Zbl.531.22010.zbMATHGoogle Scholar
  52. Johnson, K.D. (1984b): Generalized Hua-operators and parabolic subgroups. Ann. Math., II. Ser. 120, No. 3, 477–496, Zbl.576.22016.zbMATHCrossRefGoogle Scholar
  53. Johnson, K.D., KoraĔyi, A. (1980): The Hua operators on bounded symmetric domains of tube type. Ann. Math., II. Ser. 111, No. 3, 589–608, Zbl.468.32007.zbMATHCrossRefGoogle Scholar
  54. Jöricke, B. (1982): The two constants theorem for functions of several complex variables. Math. Nachr. 107, 17–52 (Russian), Zbl.526.32003.MathSciNetzbMATHCrossRefGoogle Scholar
  55. Jöricke, B. (1983): Continuity of the Cauchy projection in Hölder norms for classical domains. Math. Nachr. 112, 227–244 (Russian), Zbl.579.32006.MathSciNetzbMATHCrossRefGoogle Scholar
  56. Jost, R., Lehmann, H. (1957): Integral Darstellung kausaler Kommutatoren. Nuovo Cimento, X. Ser. 5, No. 7, 1598–1610, Zbl.77, 424.MathSciNetzbMATHCrossRefGoogle Scholar
  57. Kataoka, K. (1981): On the theory of Radon transformations of hyperfunctions. J. Fac. Sci., Univ. Tokyo, Sect. 1A 28, No. 2, 331–413, Zbl.576.32008.MathSciNetzbMATHGoogle Scholar
  58. Khenkin, G.M., Henkin, G.M., Leiterer, J. (1984): Theory of Functions on Complex Manifolds. Berlin: Akademie-Verlag, 226 pp., Zbl.573.32001.Google Scholar
  59. Khenkin, G.M., Henkin, G.M., Sergeev, A.G. (1980): Uniform estimates of solutions of the ∂-equation in pseudoconvex polyhedra. Mat. Sb., Nov. Ser. 112, No. 4, 522–567. Engl. transl.: Math. USSR, Sb. 40, 469-507 (1981), Zbl.452.32012.MathSciNetGoogle Scholar
  60. Khenkin, G.M., Henkin, G.M., Tumanov, A.E. (1976): Interpolation submanifolds of pseudoconvex manifolds, Math. Program. Rel. Probi., Cent. Ehkon. Mat. Inst. Akad. Nauk SSSR, Mosk. 1974, 74–86. Engl. transl.: transl., II. Ser., Am. Math. Soc. 115, 59-69 (1980), Zbl.455.32009.Google Scholar
  61. Khenkin, G.M., Henkin, G.M., Tumanov, A.E. (1983): Local characterization of holomorphic automorphisms of Siegel domains. Funkts. Anal. Prilozh. 17, No. 4, 49–61. Engl. transl.: Funct. Anal. Appl. 17, 285-294 (1983), Zbl.572.32018.MathSciNetGoogle Scholar
  62. Khurumov, Yu.V. (1983): Lindelöf’s theorem in ℂn. Dokl. Akad. Nauk SSSR 273, No. 6, 1325–1328. Engl. transl.: Sov. Math. Dokl. 28, 806-809 (1983), Zbl.567.32002.MathSciNetGoogle Scholar
  63. Knapp, A.V., Williamson, R.E. (1971): Poisson integrals and semisimple groups. J. Anal. Math. 24, 53–76, Zbl.247.31002.MathSciNetzbMATHCrossRefGoogle Scholar
  64. Koecher, M. (1957): Positivitätsbereiche im ∝n. Am. J. Math. 79, No. 3, 575–596, Zbl.78, 12.MathSciNetzbMATHCrossRefGoogle Scholar
  65. Komatsu, H. (1972): A local version of Bochner’s tube theorem. J. Fac. Sci., Univ. Tokyo, Sect. IA 19, No. 2, 201–214, Zbl.239.32012.MathSciNetzbMATHGoogle Scholar
  66. KoraĔyi, A. (1965): The Poisson integral for generalized half-planes and bounded symmetric domains. Ann. Math., II. Ser. 82, No. 2, 332–350, Zbl.138, 66.CrossRefGoogle Scholar
  67. KoraĔyi, A. (1969): Boundary behavior of Poisson integrals on symmetric spaces. Trans. Am. Math. Soc. 140, 393–409, Zbl.179, 151.Google Scholar
  68. KoraĔyi, A. (1972): Harmonic functions on symmetric spaces. Symmetric Spaces. Pure Appl. Math. 8, 379–412, Zbl.291.43016.Google Scholar
  69. KoraĔyi, A. (1976): Poisson integrals and boundary components of symmetric spaces. Invent. Math. 34, No. 1, 19–35, Zbl.328.22017.MathSciNetCrossRefGoogle Scholar
  70. KoraĔyi, A. (1979): Compactifications of symmetric spaces and harmonic functions. Lect. Notes Math. 739, Berlin, Heidelberg, New York: Springer-Verlag, 341–366, Zbl.425.43014.Google Scholar
  71. KoraĔyi, A., Malliavin, P. (1975): Poisson formula and compound diffusion associated to an over-determined elliptic system on the Siegel half-plane of rank two. Acta Math. 134, No. 1–2, 185–209, Zbl.318.60066.MathSciNetCrossRefGoogle Scholar
  72. KoraĔyi, A., Pukanszky, L. (1963): Holomorphic functions with positive real part on polycylinders. Trans. Am. Math. Soc. 108, 449–456, Zbl. 136, 71.CrossRefGoogle Scholar
  73. KoraĔyi, A., Vagi, S. (1976): Isometries of H p spaces of bounded symmetric domains. Can. J. Math. 28, No. 2, 334–340, Zbl.344.32025.CrossRefGoogle Scholar
  74. KoraĔyi, A., Vagi, S. (1979): Rational inner functions on bounded symmetric domains. Trans. Am. Math. Soc. 254, 179–193, Zbl.439.32006.Google Scholar
  75. KoraĔyi, A., Wolf, J.A. (1965): Realisation of hermitian symmetric spaces as generalized half-planes. Ann. Math., II. Ser. 81, No. 2, 265–288, Zbl.137, 274.CrossRefGoogle Scholar
  76. Labonde, J.-M. (1985): Ensembles pics pour A(U n). C.R. Acad. Sci., Paris, Sér. I 301, No. 13, 671–673, Zbl.584.32031.MathSciNetzbMATHGoogle Scholar
  77. Lassalle, M. (1984a): Les équations de Hua d’un domaine borné symétrique du type tube. Invent. Math. 77, No. 1, 129–161, Zbl.582.32042.MathSciNetzbMATHCrossRefGoogle Scholar
  78. Lassalle, M. (1984b): Sur la valeur au bord du noyau de Poisson d’un domaine borné symétrique. Math. Ann. 268, No. 4, 417–423, Zbl.579.32052.MathSciNetzbMATHCrossRefGoogle Scholar
  79. Leray, J. (1959): Le calcul différentiel et intégral sur une variété analytique complexe (Problème de Cauchy, III). Bull. Soc. Math. Fr. 87, 81–180, Zbl.199, 412.MathSciNetzbMATHGoogle Scholar
  80. Lindahl, L.-A. (1972): Fatou’s theorem for symmetric spaces. Ark. Mat. 10, No. 1, 33–47, Zbl.246.22010.MathSciNetzbMATHCrossRefGoogle Scholar
  81. Löw, E. (1984): Inner functions and boundary values in H (Ω) and A(Ω) in smoothly bounded pseudoconvex domains. Math. Z. 185, No. 2, 191–210, Zbl.508.32005.MathSciNetCrossRefGoogle Scholar
  82. Lu Qui-keng (1965): On the Cauchy-Fantappiè formula. Acta Math. Sin. 16, No. 3, 344–363, Zbl. 173, 329.Google Scholar
  83. Manin, Yu.I. (1984): Gauge Fields and Complex Geometry. Moscow: Nauka. Engl. transl.: Berlin, Heidelberg, New York: Springer-Verlag, 1988, Zbl.576.53002.zbMATHGoogle Scholar
  84. Martineau, A. (1964): Distributions et valeurs au bord des fonctions holomorphes. Proc. Intern. Summer Course on the Theory of Distributions. Lisboa, 195–326.Google Scholar
  85. Martineau, A. (1970): Le “edge of the wedge theorem” en théorie des hyperfonctions de Sato. Proc. Int. Conf. Funct. Anal Rel. Topics, Tokyo 1969, 95–106, Zbl.193, 415.Google Scholar
  86. Matsushima, Y. (1972): On tube domains. In: Symmetric Spaces, Pure Appl. Math. 8, 255–270, Zbl.232.32001.MathSciNetGoogle Scholar
  87. Mitchell, J., Sampson, G. (1982): Singular integrals on bounded symmetric domains in ℂn. J. Math. Anal Appl. 90, No. 2, 371–380, Zbl.506.32017.MathSciNetzbMATHCrossRefGoogle Scholar
  88. Monopoles (1985): (Collection of papers translated into Russian). Ed.: Monastyrski, M.I., Sergeev, A.G.; Moscow: Mir.Google Scholar
  89. Morimoto, M. (1973): Edge of the wedge theorem and hyperfunction. Lect. Notes Math. 287. Berlin, Heidelberg, New York: Springer-Verlag, 41–81, Zbl.262.46043.Google Scholar
  90. Morimoto, M. (1980): Analytic functionals on the Lie sphere. Tokyo J. Math. 3, No. 1, 1–35, Zbl.454.46032.MathSciNetzbMATHCrossRefGoogle Scholar
  91. Murakami, S. (1972): On automorphisms of Siegel domains, Lect. Notes Math. 286. Berlin, Heidelberg, New York: Springer-Verlag, 95 pp., Zbl.245.32001.Google Scholar
  92. Nagel, A. (1976): Smooth zero sets and interpolation sets for some algebras of holomorphic functions on strictly pseudoconvex domains. Duke Math. J. 43, No. 2, 323–348, Zbl.343.32016.MathSciNetzbMATHCrossRefGoogle Scholar
  93. Penrose, R. (1980): The complex geometry of the natural world. Proc. Int. Congr. Math., Helsinki 1978, Vol. 1, 189–194, Zbl.425.53033.MathSciNetGoogle Scholar
  94. Penrose, R. (1968): The structure of space-time. Battelle Rencontres, 1967, Lect. Math. Phys., 121–235, Zbl. 174, 559.Google Scholar
  95. Pflug, P. (1974): über polynomiale Funktionen auf Holomorphiegebieten. Math. Z. 139, No. 2, 133–139, Zbl.278.32011.MathSciNetzbMATHCrossRefGoogle Scholar
  96. Pinchuk, S.I. (1992): CR transformations of real manifolds in ℂn. Indiana Univ. Math. J. 41, No. 1, 1–16.MathSciNetzbMATHCrossRefGoogle Scholar
  97. Piatetski-Shapiro, I.I. (1961): Geometry of Classical Domains, and Theory of Automorphic Functions. Moscow: GOSIZDAT. 191 pp. French transl.: Paris: Dunod 1966, Zbl.137, 275, Zbl.142, 51.Google Scholar
  98. Polyakov, P.L. (1985): Solution of the ∂-equation with estimates in tube domains. Usp. Mat. Nauk 40, No. 1, 213–214. Engl. transl.: Russ. Math. Surv. 40, No. 1, 235-236 (1985), Zbl.593.32013.MathSciNetGoogle Scholar
  99. Rigoli, M., Travaglini, G. (1983): A remark on mappings of bounded symmetric domains into balls. Lect. Notes Math. 992. Berlin, Heidelberg, New York: Springer-Verlag, 387–390, Zbl.552.32020.Google Scholar
  100. Rossi, H., Vergne, M. (1976): Equations de Cauchy-Riemann tangentielles associées à un domaine de Siegel. Ann. Sci. Ec. Norm. Supér., IV Sér. 9, No. 1, 31–80, Zbl.398.32018.MathSciNetzbMATHGoogle Scholar
  101. Rothaus, O.S. (1960): Domains of positivity. Abh. Math. Semin. Univ. Hamb. 24, 189–235, Zbl.96, 279.MathSciNetzbMATHCrossRefGoogle Scholar
  102. Rudin, W. (1969): Function Theory in Polydiscs. New York: Benjamin, 188 pp. Zbl. 177, 341.zbMATHGoogle Scholar
  103. Rudin, W. (1971a): Harmonic analysis in polydiscs. Actes Congr. Int. Math., Nice, 1970, t. 2, 489–493, Zbl.233.32002.Google Scholar
  104. Rudin, W. (1971b): Lectures on the Edge-of-the-Wedge theorem. Reg. Conf. Ser. Math. 6. Providence: Am. Math. Soc., 30 pp., Zbl.214, 90.Google Scholar
  105. Rudin, W. (1978): Peak-interpolation sets of class C 1. Pac. J. Math. 75, No. 1, 267–279, Zbl.383.32007.MathSciNetzbMATHGoogle Scholar
  106. Rudin, W. (1980): Function Theory in the Unit Ball of ℂn. New York, Berlin, Heidelberg: Springer-Verlag, 436 pp., Zbl.495.32001.CrossRefGoogle Scholar
  107. Saerens, R. (1984): Interpolation manifolds. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11, No. 2, 177–211, Zbl.579.32023.MathSciNetzbMATHGoogle Scholar
  108. Sato, M. (1959-1960): Theory of hyperfunctions, I, II. J. Fac. Sci. Univ. Tokyo, Sect. I 8, No. 1, 139–193, Zbl.87, 314. No. 2, 387-437, Zbl.97, 314.Google Scholar
  109. Sato, M., Kawai, T., Kashiwara, M. (1973): Microfunctions and pseudodifferential equations. Lect. Notes Math. 287. Berlin, Heidelberg, New York: Springer-Verlag, 265–529, Zbl.277.46039.Google Scholar
  110. Schapira, P. (1970): Théorie des Hyperfonctions. Lect. Notes Math. 126, Berlin, Heidelberg, New York: Springer-Verlag, 157 pp., Zbl. 192, 473.Google Scholar
  111. Schmid, W. (1969): Die Randwerte holomorpher Funktionen auf Hermiteschen symmetrischen Räumen. Invent. Math. 9, No. 1, 61–80, Zbl.219.32013.MathSciNetzbMATHCrossRefGoogle Scholar
  112. Sergeev, A.G. (1975): Multiplicative theory of hyperfunctions. Usp. Mat. Nauk 30, No. 1, 257–258 (Russian), Zbl.379, 46034.MathSciNetzbMATHGoogle Scholar
  113. Sergeev, A.G. (1978): Multidimensional factorization problem. Proc. All-Union Conf. on PDEs. Moscow, 440–441 (Russian).Google Scholar
  114. Sergeev, A.G. (1983): Complex geometry and integral representations in the future tube in ℂ3. Teor. Mat. Fiz. 54, No. 1, 99–110. Engl. transl.: Theor. Math. Phys. 54, 62-70 (1983), Zbl.529.32001.MathSciNetzbMATHCrossRefGoogle Scholar
  115. Sergeev, A.G. (1985): Estimates for the Bergman projector in the future tube. Multidim. Compl. Anal., Krasnojarsk, SOAN SSSR, 161–172 (Russian).Google Scholar
  116. Sergeev, A.G. (1986): Complex geometry and integral representations in the future tube. Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 6, 1241–1275, 1343-1344. Engl. transl.: Math. USSSR, Izv. 29, 597-628 (1987), Zbl.618.32001.MathSciNetzbMATHGoogle Scholar
  117. Sergeev, A.G. (1988): On the behavior of solutions of the ∂-equation on the boundary of the future tube. Dokl. Akad. Nauk SSSR 298, No. 2, 294–298. Engl. transl.: Sov. Math., Dok. 37, No. 1, 83-87 (1988), Zbl.691.32007.Google Scholar
  118. Sergeev, A.G. (1989): On complex analysis in the future tube. Compl. Anal. Appl. 87, Sofia, 450–459.MathSciNetGoogle Scholar
  119. Sergeev, A.G. (1991): On complex analysis in tube cones. Proc. Sympos. Pure Math. 52, Part 1, 173–190.MathSciNetGoogle Scholar
  120. Sergeev, A.G., Vladimirov, V.S. (1985): A compactification of Minkowski space and complex analysis in the future tube. Ann. Pol. Math. 46, No. 1, 439–454 (Russian), Zbl.602.32010.MathSciNetzbMATHGoogle Scholar
  121. Siciak, J. (1969): Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of ℂn. Ann. Pol. Math. 22, No. 2, 145–171, Zbl.185, 152.MathSciNetzbMATHGoogle Scholar
  122. Siegel, C.L. (1949): Analytic Functions of Several Complex Variables. Princeton: Inst. Adv. Stud., 200 pp., Zbl.36, 50.Google Scholar
  123. Stein, E.M. (1971): Some problems in harmonic analysis suggested by symmetric spaces and semisimple groups. Actes Congr. Int. Math., Nice, 1970, 1, 173–189, Zbl.252.43022.Google Scholar
  124. Stein, E.M. (1972): Boundary behaviour of holomorphic functions of several complex variables. Princeton: Princeton Univ. Press, 72 pp., Zbl.242.32005.Google Scholar
  125. Stein, E.M. (1983): Boundary behavior of harmonic functions on symmetric spaces: maximal estimates for Poisson integrals. Invent. Math. 74, No. 1, 63–83, Zbl.522.43007.MathSciNetzbMATHCrossRefGoogle Scholar
  126. Stein, E.M., Weiss, G. (1971): Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton Univ. Press, 297 pp., Zbl.232.42007.zbMATHGoogle Scholar
  127. Stein, E.M., Weiss, G., Weiss, M. (1964): H p classes of holomorphic functions in tube domains. Proc. Natl. Acad. Sci. USA 52 No. 4, 1035–1039, Zbl. 126, 94.MathSciNetzbMATHCrossRefGoogle Scholar
  128. Stein, E.M., Weiss, N.J. (1969): On the convergence of Poisson integrals. Trans. Am. Math. Soc. 140, 34–54, Zbl. 182.108.MathSciNetCrossRefGoogle Scholar
  129. Stoll, M. (1974): Integral formulae for pluriharmonic functions on bounded symmetric domains. Duke Math. J. 41, No. 2, 393–404, Zbl.287.32020.MathSciNetzbMATHCrossRefGoogle Scholar
  130. Stoll, M. (1976a): Harmonic majorants for plurisubharmonic functions on bounded symmetric domains with applications to the spaces H ϕ and N*. J. Reine Angew. Math. 282, 80–87, Zbl.318.32014.MathSciNetzbMATHGoogle Scholar
  131. Stoll, M. (1976b): The space N* of holomorphic functions on bounded symmetric domains. Ann. Pol. Math. 32, No. 1, 95–110, Zbl.284.32013.MathSciNetzbMATHGoogle Scholar
  132. Stoll, M. (1985): Mean growth and Fourier coefficients of some classes of holomorphic functions on bounded symmetric domains. Ann. Pol. Math. 45, No. 2, 161–183, Zbl.579.32022.MathSciNetzbMATHGoogle Scholar
  133. Stout, E.L. (1981): Interpolation manifolds. In: Recent Developments in Several Complex Variables. Ann. Math. Stud. 100, 373–391, Zbl.486.32010.MathSciNetGoogle Scholar
  134. Tillmann, H.G. (1961a): Distributionen als Randverteilungen analytischer Funktionen. Math. Z. 76, No. 1, 5–21, Zbl.97, 96.MathSciNetzbMATHCrossRefGoogle Scholar
  135. Tillmann, H.G. (1961b): Darstellung der Schwartzschen Distributionen durch analytische Funktionen. Math. Z. 77, No. 2, 106–124, Zbl.99, 97.MathSciNetzbMATHCrossRefGoogle Scholar
  136. Twistors and Gauge Fields (1983): (Collection of papers translated into Russian). Ed.: Zharinov, V.V.; Moscow: Mir, 364 pp.Google Scholar
  137. Uhlmann, A. (1963): The closure of Minkowski space. Acta Phys. Pol. 24, No. 2, 295–296, Zbl. 115, 423.MathSciNetzbMATHGoogle Scholar
  138. Upmeier, H. (1984): Toeplitz C*-algebras on bounded symmetric domains. Ann. Math., II. Ser. 119, No. 3, 549–576, Zbl.549.46031.MathSciNetzbMATHCrossRefGoogle Scholar
  139. Vinberg, E.B. (1963): The theory of convex homogeneous cones. Tr. Mosk. Mat. O.-va 12, 303–358. Engl. transl: Trans. Mosc. Math. Soc. 1963, 340-403 (1965), Zbl.138, 433.MathSciNetGoogle Scholar
  140. Vladimirov, V.S. (1960): On constructing the envelope of holomorphy for domains of special type. Dokl. Akad. Nauk SSSR 134, No. 2, 251–254. Engl. transl.: Sov. Math., Dokl. 1, 1039-1042 (1960), Zbl.118, 303.Google Scholar
  141. Vladimirov, V.S. (1961): On constructing the envelope of holomorphy for domains of special type and their applications. Tr. Mat. Inst. Steklova 6; 101–144. Engl. transl.: Am. Math. Soc., transl, II. Ser. 48, 107-150 (1965), Zbl.118, 303.Google Scholar
  142. Vladimirov, V.S. (1964): Methods of the Theory of Functions of Several Complex Variables. Moscow: Nauka, 410 pp. French transl: Les fonctions de plusieurs variables complexes et leur application. Paris: Dunod 1967, 338 pp., Zbl.125, 319.Google Scholar
  143. Vladimirov, V.S. (1965): The problem of linear conjugation of holomorphic functions of several complex variables. Izv. Akad. Nauk SSSR, Ser. Mat. 29, No. 4, 807–834. Engl transl: Am. Math. Soc., transl, II. Ser. 71, 203-232 (1968), Zbl.166, 337.MathSciNetzbMATHGoogle Scholar
  144. Vladimirov, V.S. (1969a): A generalization of the Cauchy-Bochner integral representation. Izv. Akad. Nauk SSSR, Ser. Mat. 33, No. 1, 90–108. Engl transl: Math. USSR, Izv. 3, 87-104 (1969), Zbl.183, 87.MathSciNetzbMATHGoogle Scholar
  145. Vladimirov, V.S. (1969b): Holomorphic functions with nonnegative imaginary part in a tube domain over a cone. Mat. Sb., Nov. Ser. 79, No. 1, 128–152. Engl transl: Math. USSR, Sb. 8, 125-146 (1969), Zbl.183, 87.Google Scholar
  146. Vladimirov, V.S. (1969c): Bogolubov’s “edge-of-the-wedge” theorem, its development and applications. Problems of Theoretical Physics. Moscow: Nauka, 61–67 (Russian).Google Scholar
  147. Vladimirov, V.S. (1969d): Linear passive systems. Theor. Mat. Fiz. 1, No. 1, 67–94. Engl transl: Theor. Math. Phys.Google Scholar
  148. Vladimirov, V.S. (1971): Analytic functions of several complex variables and axiomatic quantum field theory. Actes Congr. Int. Math. Nice, 1970, t. 3. Paris: Gauthier-Villars, 21–26.Google Scholar
  149. Vladimirov, V.S. (1972): Multidimensional linear passive systems. Meh. Splosn. Sredy rodstv. Probl Anal, 121–134 (Russian), Zbl.263.93019.Google Scholar
  150. Vladimirov, V.S. (1974, 1974, 1977): Holomorphic functions with positive imaginary part in the future tube. Mat. Sb. 93, No. 1, 3–17; II, 94, No. 4, 499-515: IV, 104, No. 3, 341-370. Engl transl: Math. USSR, Sb. 22 1-16; II, 23, 467-482; III, 27, 263-268; IV, 33, 301-325 (1975–1977); Zbl.291.32003; Zbl.313.32001; Zbl.319.32004; Zbl.383.32001.Google Scholar
  151. Vladimirov, V.S. (1976): Multidimensional generalization of a Tauberian theorem of Hardy-Littlewood. Izv. Akad. Nauk SSSR, Ser. Mat. 40, No. 5, 1084–1101. Engl transl: Math. USSR, Izv. 10, 1031-1048 (1978), Zbl.359.40001.MathSciNetzbMATHGoogle Scholar
  152. Vladimirov, V.S. (1978a): Holomorphic functions with nonnegative imaginary part in tube domains over cones. Dokl Akad. Nauk SSSR 239, No. 1, 26–29. Engl transl: Sov. Math., Dokl 19, 254-258 (1978), Zbl.448.32003.MathSciNetGoogle Scholar
  153. Vladimirov, V.S. (1978b): Growth estimates for boundary values of positive pluriharmonic functions in a tube domain over a proper cone. Complex Analysis and its Applications. Moscow: Nauka, 137–148 (Russian), Zbl.447.31006.Google Scholar
  154. Vladimirov, V.S. (1979): Generalized Functions in Mathematical Physics, 2nd. ed. Moscow: Nauka, 319 pp. Engl. transl: Moscow: Mir, 362 pp, Zbl.515.46034.Google Scholar
  155. Vladimirov, V.S. (1982): Several complex variables in mathematical physics. Lect. Notes Math. 919. Berlin, Heidelberg, New York: Springer-Verlag, 358–386, Zbl.493.32014.Google Scholar
  156. Vladimirov, V.S. (1983): Functions of several complex variables in mathematical physics. In: Problems of Mathematics and Mechanics. Novosibirsk: Nauka, 15–32. Engl transl: transl, II. Ser., Am. Math. Soc. 136, 19-33 (1987), Zbl.625.32001.Google Scholar
  157. Vladimirov, V.S. (1984): Blaschke products in the generalized unit disc and complete orthonormal systems in the future tube. Tr. Mat. Inst. Steklova 166, 44–51. Engl transl: Proc. Steklov Inst. Math. 166, 45-52 (1986), Zbl.574.32007.zbMATHGoogle Scholar
  158. Vladimirov, V.S., Zharinov, V.V. (1970): On a representation of Jost-Lehmann-Dyson type. Teor. Mat. Fiz. 3, 305–319. Engl. transl: Theor. Math. Phys. 3, No. 3, 525-536 (1970), Zbl.201, 582.zbMATHCrossRefGoogle Scholar
  159. Weiss, N.J. (1972): Fatou’s theorem for symmetric spaces. In: Symmetric Spaces, Pure Appl. Math. 8, 413–441, Zbl.242.43011.Google Scholar
  160. Wolf, J.A. (1972): Fine structure of Hermitian symmetric spaces. In: Symmetric Spaces, Pure Appl. Math. 8, 271–357, Zbl.257.32014.Google Scholar
  161. Yang, P. (1984): Geometry of tube domains. Proc. Symp. Pure Math. 41, 277–283, Zbl.579.32050.Google Scholar
  162. Yang, P.C. (1982): Automorphisms of tube domains. Am. J. Math. 104, No. 5, 1005–1024, Zbl.514.32018.zbMATHCrossRefGoogle Scholar
  163. Zakharyuta, V.P. (1976): Separately analytic functions, generalization of the Hartogs theorem and envelopes of holomorphy. Mat. Sb., Nov. Ser. 101, No. 1, 57–76. Engl. transl.: Math. USSR, Sb. 30, 51-67 (1978), Zbl.357.32002.Google Scholar
  164. Zharinov, V.V. (1980): On an exact squence of modules and Bogolubov’s “edge-of-the-wedge” theorem. Dokl. Akad. Nauk SSSR 251, No. 1, 19–22. Engl. transl.: Sov. Math., Dokl. 21, 357-360 (1980), Zbl.478.46046.MathSciNetGoogle Scholar
  165. Zharinov, V.V. (1983): Distributive lattices and their applications in complex analysis. Tr. Mat. Inst. Steklova 162, 3–80. Engl. transl.: Proc. Steklov Inst. Math. 162, Providence, 79 pp. (1985), Zbl.574.32017.MathSciNetGoogle Scholar
  166. Zygmund, A. (1958): Trigonometric Series, Vol. 1, 2. 2nd. ed. Cambridge: Cambridge University Press, Zbl.85, 56, Zbl.1 1, 17.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • A. G. Sergeev
  • V. S. Vladimirov

There are no affiliations available

Personalised recommendations