Advertisement

Extending Holomorphic Functions from Subvarieties

  • Kenzō Adachi
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 9)

Abstract

Let D be a pseudoconvex domain in ℂ n and V a subvariety of D. The purpose of this paper is to survey the extension problem of holomorphic functions from V to D in various function spaces. First, we study extension theorems in bounded strictly pseudoconvex domains. Next, we discuss extension theorems in bounded weakly pseudoconvex domains with support function and analytic polyhedra. The main tools to construct extension functions are integral formulas in subvarieties obtained by Hatziafratis[27] and Berndtsson[15]. Further, we state the L 2 extension theorem of Ohsawa-Takegoshi[36] in bounded pseudoconvex domains. Finally, we give some counterexamples for extension problems.

Keywords

Holomorphic Function Integral Formula Convex Domain Pseudoconvex Domain Stein Manifold 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Adachi, Continuation of A O functions from submanifolds to strictly pseudoconvex domains,J. Math. Soc. Japan. 32 (1980), 331–341.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    K. Adachi, Extending bounded holomorphic functions from certain subvarieties of a weakly pseudoconvex domain,Pacific J. Math. 110 (1984) 9–19.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    K. Adachi, Continuation of bounded holomorphic functions from certain subvarieties to weakly pseudoconvex domains,Pacific J. Math. 130 (1987), 1–8.MathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Adachi, Continuation of holomorphic functions from subvarieties to pseudoconvex do mains,Kobe J. Math. 11 (1994), 33–47.MathSciNetzbMATHGoogle Scholar
  5. 5.
    K.Adachi, Extending real analytic functions from boundaries of subvarieties to convex doamins,Proceedings of the Second Korean-Japanese Colloquium on Finite or Infinite Dimensional Complex Analysis (1994), 15–21.Google Scholar
  6. 6.
    M. Andersson and H. R. Cho, LP and HP extensions of holomorphic functions from subvarieties of analytic polyhedra, Pacific J. Math. 189 (1999), 201–210.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    K. Adachi and H. R. Cho, HP and LP extensions of holomorphic functions from subvrieties to certain convex domains, Math. J. Toyama Univ. 20 (1997), 1–13.MathSciNetzbMATHGoogle Scholar
  8. 8.
    K. Adachi and H. R. Cho, Lipschitz and BMO extensions of holomorphic functions from subvarieties to a convex domain, Complex Variables 34 (1997), 465–473.MathSciNetCrossRefGoogle Scholar
  9. 9.
    K. Adachi and H. Kajimoto, On the extension of Lipschitz functions from boundaries of subvarieties to strongly pseudoconvex domains, Pacific J. Math. 158 (1993), 201–222.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    P. Ahern and R. Schneider, Holomorphic Lipschitz functions in pseudoconvex domains, Amer. J. Math. 101 (1979), 543–565.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    H. Alexander, Extending bounded holomorphic functions from certain subvarieties of a polydisc, Pacific J. Math. 29 (1969), 485–490.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    E. Amax, Extension de fonctions holomorphes et courants, Bull. Sc. Math. 107 (1983), 25–48.Google Scholar
  13. 13.
    E. Amax, Cohomologie complexe et applications,J. London Math. Soc. 29 (1984), 127–140.MathSciNetCrossRefGoogle Scholar
  14. 14.
    F. Beatrous, LP estimates for extensions of holomorphic functions, Michigan Math. J. 32 (1985), 361–380.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    B. Berndtsson, A formula for interpolation and division in C“, Math. Ann. 263 (1983), 399–418.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    B. Berndtsson, The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly- Fet-terman, Ann. Inst. Fourier 46 (1996), 1083–1094.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    J. Boo, The HP corona theorem in analytic polyhedra, Ark. Mat. 35 (1997), 225–251.MathSciNetzbMATHGoogle Scholar
  18. 18.
    S. C. Chen, Real analytic boundary regularity of the Cauchy transform on convex domains, Proc. Amer. Math. Soc. 108 (1990), 423–432.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    S. J. Chen, General integral representation of the holomorphic functions on the analytic subvariety, Publ.RIMS, Kyoto Univ. 29 (1993), 511–533.zbMATHCrossRefGoogle Scholar
  20. 20.
    Z. Chen, Local real analytic boundary regularity of an integral solution operator of the 8 equation on convex domains, Pacific J. Math. 156 (1992), 97–105.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    H. R. Cho, A counterexample to the LP extension of holomorphic functions from subvarieties to pseudoconvex domains, Complex Variables 35 (1998), 89–91.CrossRefGoogle Scholar
  22. 22.
    A. Cumenge, Extension dans des classes de Hardy de fonctions holomorphes et estimation de type “measures de Carleson” pour l’equation 9, Ann. Inst. Fourier 33 (1983), 59–97.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    M. Elgueta, Extensions to strictly pseudoconvex domains of functions holomorphic in a submanifold in general position and Ccc up to the boundary, Ill. J. Math. 24 (1980), 1–17.MathSciNetzbMATHGoogle Scholar
  24. 24.
    J. E. Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math. 98 (1976), 529–569.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    J. E. Fornaess and N. Sibony, On LP estimates for 8,Proc. Symp. Pure Math. 52 (1991), 129–163.MathSciNetGoogle Scholar
  26. 26.
    H. Hamada and M. Tsuji, Counterexample of a bounded domain for Ohsawa’s problem, Complex Variables 28 (1996), 285–287.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    T. E. Hatziafratis, Integral representation formulas on analytic varieties, Pacific J.Math. 123 (1986), 71–91.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    G. M. Henkin, Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications, Math. USSR Sbornik 7 (1969), 597–616.CrossRefGoogle Scholar
  29. 29.
    G. M. Henkin, Continuation of bounded holomorphic functions from submanifods in general position to strictly pseudoconvex domains, Izv. Akad. Nauk SSSR. 36 (1972), 540–567.MathSciNetzbMATHGoogle Scholar
  30. 30.
    G. M. Henkin and J. Leiterer, Global integral formulas for solving the 8-equation on Stein man- ifolds, Ann. Polon. Math. XXXIX (1981), 93–116.Google Scholar
  31. 31.
    G. M. Henkin and J. Leiterer, Theory of functions on complex manifolds, Birkhäuser, 1984.Google Scholar
  32. 32.
    G.M. Henkin and P.L. Polyakov, Prolongement des fonctions holomorphes bornées d’une sous-variété du polydisque, C. R. Acad. Sc. Paris, Série I 298 (1984), 221–224.MathSciNetzbMATHGoogle Scholar
  33. 33.
    P. Jakobczak, On the regularity of extension to strictly pseudoconvex domains of functions holomorphic in a submanifold in general position, Ann. Polon. Math. 42 (1983), 115–124.MathSciNetzbMATHGoogle Scholar
  34. 34.
    E. Mazzilli, Extension des fonctions holomorphes, C. R. Acad. Sci. Paris, Série I321, Série I 321 (1995), 831–836.MathSciNetzbMATHGoogle Scholar
  35. 35.
    E. Mazzilli, Extension des fonctions holomorphes dans les pseudo-ellipsoides, Math. Z. 227 (1998), 607–622.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    T. Ohsawa and K. Takegoshi, On the exension of L 2 holomorphic functions, Math. Z. 195 (1987), 197–204.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    P.L. Polyakov, Extension of bounded holomorphic functions from an analytic curve in general position in a polydisc, Functional. Anal. Prilozen 17 (1983), 237–239.MathSciNetzbMATHGoogle Scholar
  38. 38.
    E. Ramirez, Ein Divisions problem und Randintegraldarstellungen in der Komplexen Analysis, Math. Ann. 184 (1970), 172–185.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    N. Sibony, Prolongement des fonctions holomorphes bornées et Métrique de Carathéodory, Inv. Math. 29 (1975), 205–230.MathSciNetzbMATHGoogle Scholar
  40. 40.
    Y.T. Siu, The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi,Proc. 3rd Int. RIMSJ., Geometric Complex Analysis (1996), World Sci., 577–592.Google Scholar
  41. 41.
    E.L. Stout, An integral formula for holomorphic functions on strictly pseudoconvex hypersurfaces, Duke Math. J. 42 (1975), 347–356.MathSciNetzbMATHGoogle Scholar
  42. 42.
    M. Tsuji, Counterexample of an unbounded domain for Ohsawa’s problem, Complex Variables 27 (1995), 335–338.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    M. Tsuji, Counterexample in case of a family of hyperplanes for Ohsawa’s problem, Pro-ceedings of the third international colloquium on finite or infinite dimensional complex analysis (1995), 253–260.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Kenzō Adachi
    • 1
  1. 1.Department of Mathematics, Faculty of EducationNagasaki UniversityNagasakiJapan

Personalised recommendations