Advertisement

Acta Mathematica

, Volume 142, Issue 1, pp 79–122 | Cite as

Subellipticity of the\(\bar \partial\)-Neumann problem on pseudo-convex domains: Sufficient conditionsproblem on pseudo-convex domains: Sufficient conditions

  • J. J. Kohn
Article

Keywords

Holomorphic Function NEUMANN Problem Pseudodifferential Operator Regular Point Pseudoconvex Domain 
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].
    Bedford, E. & Fornaess, J. E., A construction of peak functions on weakly pseudo-convex domains. Preprint.Google Scholar
  2. [2].
    Bloom, T. & Graham, I., A geometric characterization of points of typem on real hypersurfaces.J. Differential Geometry, to appear.Google Scholar
  3. [3].
    Boutet De Monvel, L. &Sjöstrand, J., Sur la singularité des noyaux de Bergman et de Szegö.Soc. Math. de France Astérisque, 34–35 (1976), 123–164.Google Scholar
  4. [4].
    Cartan, H., Variétés analytiques reélles et variétés analytiques complexes.Bull. Soc. Math. France, 85 (1957), 77–99.MATHMathSciNetGoogle Scholar
  5. [5].
    Catlin, D.,Boundary behaviour of holomorphic functions on weakly pseudo-convex domains. Thesis, Princeton Univ. 1978.Google Scholar
  6. [6].(a)
    D'Angelo, J., Finite type conditions for real hypersurfaces.J. Differential Geometry, to appear.Google Scholar
  7. [6].(b)
    D'Angelo, J., A note on the Bergman kernel.Duke Math. J., to appear.Google Scholar
  8. [7].
    Derridj, M., Sur la régularité des solutions du problème de Neumann pour\(\bar \partial\) dans quelques domains faiblement pseudo-convexes.J. Differential Geometry, to appear.Google Scholar
  9. [8].
    Derridj, M. &Tartakoff, D., On the global real-analyticity of solutions of the\(\bar \partial\)-Neumann problem.Comm. Partial Differential Equations, 1 (1976), 401–435.MATHMathSciNetGoogle Scholar
  10. [9].
    Diederich, K. &Fornaess, J. E., Pseudoconvex domains with real-analytic boundary.Ann. of Math., 107 (1978), 371–384.MATHMathSciNetCrossRefGoogle Scholar
  11. [10].
    Egorov, Yu. V., Subellipticity of the\(\bar \partial\)-Neumann problem.Dokl. Akad. Nauk. SSSR, 235, No. 5 (1977), 1009–1012.MATHMathSciNetGoogle Scholar
  12. [11].
    Ephraim, R.,C and analytic equivalence of singularities.Rice Univ. Studies, Complex Analysis, Vol. 59, No. 1 (1973), 11–31.MATHMathSciNetGoogle Scholar
  13. [12].
    Fefferman, C., The Bergman kernel and biholomorphic mappings.Invent. Math., 26 (1974), 1–65.MATHMathSciNetCrossRefGoogle Scholar
  14. [13].
    Folland, G. B. & Kohn, J. J.,The Neumann problem for the Cauchy-Riemann complex. Ann. of Math. Studies, No. 75, P.U. Press, 1972.Google Scholar
  15. [14].
    Greiner, P. C., On subelliptic estimates of the\(\bar \partial\)-Neumann problem inC 2.J. Differential Geometry, 9 (1974), 239–250.MATHMathSciNetGoogle Scholar
  16. [15].(a)
    Greiner, P. C. & Stein, E. M., On the solvability of some differential operators of type □b, preprint.Google Scholar
  17. [15].(b)
    Greiner, P. C. & Stein, E. M.,Estimates for the \(\bar \partial\) problem., Math. Notes No. 19, Princeton University Press 1977.Google Scholar
  18. [16].
    Henkin, G. M. &Čirka, E. M.,Boundary properties of holomorphic functions of several complex variables. Problems of Math. Vol. 4, Moscow 1975, 13–142.MATHGoogle Scholar
  19. [17].(a)
    Hörmander, L.,L 2 estimates and existence theorems for the\(\bar \partial\) operator.Acta Math., 113 (1965), 89–152.MATHMathSciNetCrossRefGoogle Scholar
  20. [17].(b)
    —, Hypoelliptic second order differential equations.Acta Math., 119 (1967), 147–171.MATHMathSciNetCrossRefGoogle Scholar
  21. [17].(c)
    —, Pseudo-differential operators and non-elliptic boundary problems.Ann. of Math., 83 (1966), 129–209.MATHMathSciNetCrossRefGoogle Scholar
  22. [18].
    Kashiwara, M., Analyse micro-locale du noyau de Bergman.Sem. Goulaouic-Schwartz 1976–1977. Exposé No VIII.Google Scholar
  23. [19].(a)
    Kerzman, N., The Bergman-kernel function: differentiability at the boundary.Math. Ann., 195 (1972), 149–158.MathSciNetCrossRefGoogle Scholar
  24. [19].(b)
    —, Hölder andL p estimates for solutions of\(\bar \partial u = f\) in strongly pseudo-convex domains.Comm. Pure Appl. Math., 24 (1971), 301–379.MATHMathSciNetGoogle Scholar
  25. [20].(a)
    Kohn, J. J., Lectures on degenerate elliptic problems.Proc. CIME Conf. on Pseudo-differential Operators, Bressanone (1977), to appear.Google Scholar
  26. [20].(b)
    —, Sufficient conditions for subellipticity on weakly pseudo-convex domains.Proc. Nat. Acad. Sci. U.S.A., 74 (1977), 2214–2216.MATHMathSciNetCrossRefGoogle Scholar
  27. [20].(c)
    —, Methods of partial differential equations in complex analysis.Proc. of Symp. in Pure Math., vol. 30, part 1 (1977), 215–237.MATHMathSciNetGoogle Scholar
  28. [20].(d)
    —, Global regularity for\(\bar \partial\) on weakly pseudo-convex manifolds.Trans. Amer. Math. Soc., 181 (1973), 273–292.MATHMathSciNetCrossRefGoogle Scholar
  29. [20].(e)
    —, Boundary behaviour of\(\bar \partial\) on weakly pseudo-convex manifolds of dimension two.J. Differential Geometry, 6 (1972), 523–542.MATHMathSciNetGoogle Scholar
  30. [21].(a)
    Kohn, J. J. &Nirenberg, L., A pseudo-convex domain not admitting a holomorphic support function.Math. Ann., 201 (1973), 265–268.MATHMathSciNetCrossRefGoogle Scholar
  31. [21].(b)
    —, Non-coercive boundary value problems.Comm. Pure Appl. Math., 18 (1965), 443–492.MATHMathSciNetGoogle Scholar
  32. [22].(a)
    Krantz, S. G., Characterizations of various domains of holomorphy via\(\bar \partial\) estimates and applications to a problem of Kohn, preprint.Google Scholar
  33. [22].(b)
    —, Optimal Lipschitz andL p regularity for the equation,\(\bar \partial u = f\) on strongly pseudo-convex domains.Math. Ann., 219 (1976), 223–260.MathSciNetCrossRefGoogle Scholar
  34. [23].
    Lieb, I., Ein Approximationssatz auf streng pseudo-konvexen GebietenMath. Ann., 184 (1969), 55–60.MathSciNetCrossRefGoogle Scholar
  35. [24].
    Lojasiewicz, S.,Ensembles semi-analytiques. Lecture note (1965) at I.H.E.S.; Reproduit no A66-765, Ecole polytechnique, Paris.Google Scholar
  36. [25].
    Narasimhan, R.,Introduction to the theory of analytic spaces Lectures notes in Math. No. 25, Springer Verlag 1966.Google Scholar
  37. [26].
    Range, R. M., On Hölder estimates for\(\bar \partial u = f\) on weakly pseudoconvex domains, preprint.Google Scholar
  38. [27].
    Rothschild, L. P. &Stein, E. M., Hypoelliptic differential operators and nilpotent groups.Acta Math., 137 (1976), 247–320.MathSciNetCrossRefGoogle Scholar
  39. [28].
    Tartakoff, D., The analytic hypoellipticity of □b and related operators on non-degenerate C-R manifolds, preprint.Google Scholar
  40. [29].
    Treves, F., Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and application to the\(\bar \partial\)-Neumann problem, preprint.Google Scholar
  41. [30].
    Spencer, D. C., Overdetermined systems of linear partial differential equations.Bull. Amer. Math. Soc., 75 (1969), 176–239.CrossRefMathSciNetGoogle Scholar
  42. [31].(a)
    Sweeney, W. J., TheD-Neumann problem.Acta Math., 120 (1968), 223–277.MATHMathSciNetCrossRefGoogle Scholar
  43. [31].(b)
    —, A condition for subellipticity in Spencer's Neumann problem.J. Differential Equations, 21 (1976), 316–362.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Almqvist & Wiksell 1979

Authors and Affiliations

  • J. J. Kohn
    • 1
  1. 1.Princeton UniversityPrincetonUSA

Personalised recommendations