The Mathematical Intelligencer

, Volume 24, Issue 2, pp 4–12 | Cite as

Extension phenomena in multidimensional complex analysis: correction of the historical record

Article
First Online:

Conclusion

Even though Bochner should not be credited with the proof of any version of theCR extension theorem, his 1943 paper remains a landmark in the history of the Hartogs extension phenomenon. His vision to enlarge his horizon from holomorphic functions to certain harmonic functions set the stage for further generalizations by himself (for example [Bochner 1954]) as well as for Ehrenpreis’s investigations on related problems for solutions of more general elliptic partial-differential operators.

In closing, it should be pointed out that Bochner’s 1943 paper, in an ironic twist, includes an important result for which Bochner did not receive any credit until recently [Range 1986, p. 188]. Bochner proved the solution of ∂ on polydiscs (for (0, l)-forms in the real-analytic case, which was the case of interest to him), via the Cauchy transform with parameters in dimension one, and by induction on the number of differentialsdzj appearing in the given form (Theorem 11,op. cit., p. 665). This result, with essentially the same proof, 10 years later became widely known as the Dolbeault-Grothendieck Lemma. But this is another story….

Keywords

Holomorphic Function Mathematical Intelligencer Holomorphic Extension Cauchy Kernel Cauchy Integral Formula 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Andreotti and Hill 1972]
    Andreotti, A., and Hill, CD.: E. E. Levi convexity and the Hans Lewy problem, Part I: Reduction to vanishing theorems.Ann. Scuola Norm. Sup. Pisa 2b6 (1972), 325–363.MathSciNetGoogle Scholar
  2. [Behnke and Thullen 1934]
    Behnke, H., and Thullen, P.: Theorie der Funktionen mehrerer komplexer Veränderlichen. Springer-Verlag, Berlin 1934, 2nd. ed. 1970CrossRefMATHGoogle Scholar
  3. [Betsch and Hofmann 1998]
    Betsch, G., and Hofmann, K.H.: Hellmuth Kneser: Persönlichkeit, Werk und Wirkung. Preprint Tech. Univ. Darmstadt, 1998.Google Scholar
  4. [Bochner 1943]
    Bochner, S.: Analytic and meromorphic continuation by means of Green’s formula.Ann. of Math. (2)44 (1943), 652–673.CrossRefMATHMathSciNetGoogle Scholar
  5. [Bochner 1954]
    Bochner, S.: Green’s formula and analytic continuation. In:Contributions to the Theory of Partial Differential Equations, ed. L. Bers et al.,Ann. of Math. Studies 33, Princeton Univ. Press, 1954.Google Scholar
  6. [Boggess 1991]
    Boggess, A.:CR Manifolds and the Tangential Cauchy-Riemann Complex. CRC Press, Boca Raton, FL, 1991.MATHGoogle Scholar
  7. [Čirka 1975]
    Čirka, E. M.: Analytic representation of CR-functions. Math.USSR Sbornik 27 (1975), 526–553.CrossRefGoogle Scholar
  8. [Ehrenpreis 1961]
    Ehrenpreis, L.: A new proof and an extension of Hartogs’ theorem.Bull. Amer. Math. Soc. 67 (1961), 507–509.CrossRefMATHMathSciNetGoogle Scholar
  9. [Fichera 1957]
    Fichera, G.: Caratterizazione della traccia, sulla frontiera di un campo, di una funzione analitica di più variabili complesse.Rend. Accad. Naz. Lincei VIII,23 (1957), 706–715.MathSciNetGoogle Scholar
  10. [Fichera 1986]
    Fichera, G.: Unification of global and local existence theorems for holomorphic functions of several complex variables.AW Acc. Lincei Mem. fis., S. VIII,18 (1986), 61–83.MATHMathSciNetGoogle Scholar
  11. [Fueter 1939]
    Fueter, R.: Über einen Hartogs’schen Satz.Comm. Math. Helv. 12 (1939), 75–80.CrossRefMathSciNetGoogle Scholar
  12. [Fueter 1941/42]
    Fueter, R.: Über einen Hartogs’schen Satz in der Theorie der analytischen Funktionen von n komplexen Variablen.Comm. Math. Helv. 14 (1941/42), 394–400.CrossRefMathSciNetGoogle Scholar
  13. [Hartogs 1906]
    Hartogs, F.: Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen. Sitzungsber. Köngl. Bayer. Akad. Wissen36 (1906), 223–242.Google Scholar
  14. [Harvey and Lawson 1975]
    Harvey, R., and Lawson, B.: On boundaries of complex analytic varieties.I. Ann. of Math. (2)102 (1975), 223–290.CrossRefMATHMathSciNetGoogle Scholar
  15. [Hörmander 1966]
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton 1966, 3rd. ed. North-Holland Publ. Co., Amsterdam-New York (1990).MATHGoogle Scholar
  16. [Jacobowitz 1995]
    Jacobowitz, H.: Real hypersurfaces and complex analysis.Notices Amer. Math. Soc. 42 (1995), 1480–1488.MATHMathSciNetGoogle Scholar
  17. [Kneser 1936]
    Kneser, H.: Die Randwerte einer analytischen Funktion zweier Veränderlichen.Monats, f. Math. u. Phys. 43 (1936), 364–380.CrossRefMathSciNetGoogle Scholar
  18. [Kohn and Rossi 1965]
    Kohn, J. J., and Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifold.Ann. of Math. (2)81 (1965), 451–472.CrossRefMATHMathSciNetGoogle Scholar
  19. [Lewy 1956]
    Lewy, H.: On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables.Ann. of Math. (2)64 (1956), 514–522.CrossRefMATHMathSciNetGoogle Scholar
  20. [Lewy 1957]
    Lewy, H.: An example of a smooth linear partial differential equation without solution.Ann. of Math. (2)66 (1957), 155–158.CrossRefMathSciNetGoogle Scholar
  21. [Martinelli 1938]
    Martinelli, E.: Alcuni teoremi integrali per le funzioni analitiche di più variabili complesse.Mem. della R. Accad. d’ltalia 9 (1938), 269–283.Google Scholar
  22. [Martinelli 1942/43]
    Martinelli, E.: Sopra una dimostrazione di R. Fueter per un teorema di Hartogs.Comm. Math. Helv. 15 (1942/43), 340–349.CrossRefMathSciNetGoogle Scholar
  23. [Martinelli 1961]
    Martinelli, E.: Sulla determinazione di una funzione analitica di più variabili complesse in un campo, assegnatane la traccia sulla frontiera.Ann. Matem. Pura e Appl. 55 (1961), 191–202.CrossRefMATHMathSciNetGoogle Scholar
  24. [Polking and Wells 1975]
    Polking, J., and Wells, R. O.: Hyperfunction boundary values and a generalized Bochner-Hartogs theorem.Proc. Symp. Pure Math. 30, 187–194, Amer. Math. Soc., Providence, Rl, 1977.CrossRefMathSciNetGoogle Scholar
  25. [Range 1986]
    Range, R. M.: Holomorphic Functions and Integral Representations in Several Complex Variables. Springer-Verlag, New York 1986, 2nd. corrected printing (1998).CrossRefMATHGoogle Scholar
  26. [Serre 1953]
    Serre, J. P.: Quelques problèmes globaux rélatifs aux variétés de Stein. Coll. Plus. Var., Bruxelles, 1953, 57–68.Google Scholar
  27. [Severi 1931]
    Severi, F.: Risoluzione générale del problema di Dirichlet per le funzioni biarmoniche.Rend. Reale Accad. Lincei 23 (1931), 795–804.Google Scholar
  28. [Struppa 1988]
    Struppa, D.: The first eighty years of Hartogs’ Theorem.Seminari di Geometria 1987–88, Univ. Bologna, Italy 1988, 127–209.Google Scholar
  29. [Trépreau 1986]
    Trépreau, J. M.: Sur le prolongement holomorphe des fonctiones CR définies sur une hypersurface réelle de classe C2 dans ℂn.Invent. Math. 83 (1986), 583–592.CrossRefMATHMathSciNetGoogle Scholar
  30. [Weinstock 1976]
    Weinstock, B.: Continuous boundary values of holomorphic functions on Kähler domains,Can. J. Math. 28 (1976), 513–522.CrossRefMATHMathSciNetGoogle Scholar
  31. [Wells 1968]
    Wells, R. O., Jr.: Holomorphic hulls and holomorphic convexity of differentiable submanifolds.Trans. Amer. Math. Soc. 132 (1968), 245–262.CrossRefMATHMathSciNetGoogle Scholar
  32. [Wells 1974]
    Wells, R. O., Jr.: Function theory on differentiable submanifolds. In:Contributions to Analysis, 407–441, ed. L. Ahlfors et al., Academic Press, New York (1974).Google Scholar
  33. [Wirtinger 1926]
    Wirtinger, W.: Zur formalen Theorie der Funktionen von mehreren komplexen Veränderlichen.Math. Ann. 97 (1926), 357–375.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2002

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsState University of New York at AlbanyUSA

Personalised recommendations