Mathematische Annalen

, Volume 347, Issue 1, pp 1–13 | Cite as

On the growth of the Bergman kernel near an infinite-type point

Article

Abstract

We study diagonal estimates for the Bergman kernels of certain model domains in \({\mathbb{C}^{2}}\) near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range—roughly speaking—from being “mildly infinite-type” to very flat at the infinite-type points.

Mathematics Subject Classification (2000)

Primary 32A25 32A36 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boas H.P., Straube E.J., Yu J.: Boundary limits of the Bergman kernel and metric. Mich. Math. J. 42, 449–461 (1995)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Diederich K., Herbort G., Ohsawa T.: The Bergman kernel on uniformly extendable pseudoconvex domains. Math. Ann. 273, 471–478 (1986)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Haslinger F.: Bergman and Hardy spaces on model domains. Ill. J. Math. 42, 458–469 (1998)MATHMathSciNetGoogle Scholar
  4. 4.
    Kim K.T., Lee S.: Asymptotic behavior of the Bergman kernel and associated invariants in certain infinite-type pseudoconvex domains. Forum Math. 14, 775–795 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kohn J.J.: Boundary behavior of \({\overline{\partial} }\) on weakly pseudoconvex manifolds of dimension two. J. Diff. Geom. 6, 523–542 (1972)MATHMathSciNetGoogle Scholar
  6. 6.
    Krantz S.G., Yu J.: On the Bergman invariant and curvatures of the Bergman metric. Ill. J. Math. 40, 226–244 (1996)MATHMathSciNetGoogle Scholar
  7. 7.
    McNeal J.D.: Boundary behavior of the Bergman kernel function in \({\mathbb{C}^{2}}\) . Duke Math. J. 58, 499–512 (1989)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Nagel A., Rosay J.-P., Stein E.M., Wainger S.: Estimates for the Bergman and Szegö kernels in certain weakly pseudoconvex domains. Bull. Am. Math. Soc. (N.S.) 18, 55–59 (1988)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Nagel A., Rosay J.-P., Stein E.M., Wainger S.: Estimates for the Bergman and Szegö kernels in \({\mathbb{C}^2}\) . Ann. Math. 129(2), 113–149 (1989)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Ohsawa T.: Boundary behavior of the Bergman kernel function on pseudoconvex domains. Publ. Res. Inst. Math. Sci. 20, 897–902 (1984)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations