Skip to main content
Springer Nature Link
Log in

On the Dimension of the Bergman Space for Some Unbounded Domains

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of D’Angelo.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adachi, M., Brinkschulte, J.: A global estimate for the Diederich-Fornæss index of weakly pseudoconvex domains. Nagoya Math. J. 220, 67–80 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahn, T., Gaussier, H., Kim, K.-T.: Positivity and Completeness of Invariant Metrics. J. Geom. Anal. 26(2), 1173–1185 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carleson, L.: Selected Problems on Exceptional Sets. Van Nostrand Mathematical Studies, No. 13. D. Van Nostrand Co., Inc., Princeton (1967)

    Google Scholar 

  4. Chen, B.-Y.: An essay on Bergman completeness. Ark. Mat. 51(2), 269–291 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, B.-Y., Zhang, J.-H.: The Bergman metric on a Stein manifold with a bounded plurisubharmonic function. Trans. Am. Math. Soc. 354(8), 2997–3009 (electronic) (2002)

  6. Cho, S.: A lower bound on the Kobayashi metric near a point of finite type in \({ C}^n\). J. Geom. Anal. 2(4), 317–325 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. D’Angelo, J.P.: Real hypersurfaces, orders of contact, and applications. Ann. Math. (2) 115(3), 615–637 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Demailly, J.-P.: Analytic methods in algebraic geometry. Surveys of Modern Mathematics, 1. International Press/Higher Education Press, Somerville, MA/Beijing (2012)

    Google Scholar 

  9. Diederich, K., Fornæss, J.E.: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39(2), 129–141 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dinew, Ż.: The Ohsawa-Takegoshi extension theorem on some unbounded sets. Nagoya Math. J. 188, 19–30 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Globevnik, J.: On Fatou-Bieberbach domains. Math. Z. 229(1), 91–106 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harvey, F.R., Lawson Jr., H.B.: Geometric plurisubharmonicity and convexity: an introduction. Adv. Math. 230(4–6), 2428–2456 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Harvey, F.R., Lawson Jr., H.B.: \(p\)-convexity, \(p\)-plurisubharmonicity and the Levi problem. Indiana Univ. Math. J. 62(1), 149–169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Harz, T., Shcherbina, N., Tomassini, G.: On defining functions for unbounded pseudoconvex domains. arXiv:1405.2250, ver. 3 (2014)

  15. Herbort, G.: Über das Randverhalten der Bergmanschen Kernfunktion und Metrik für eine spezielle Klasse schwach pseudokonvexer Gebiete des \({ C}^{n}\). Math. Z. 184(2), 193–202 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hörmander, L.: \(L^{2}\) estimates and existence theorems for the \(\bar{\partial }\) operator. Acta Math. 113, 89–152 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jucha, P.: A remark on the dimension of the Bergman space of some Hartogs domains. J. Geom. Anal. 22(1), 23–37 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kobayashi, S.: Geometry of bounded domains. Trans. Am. Math. Soc. 92, 267–290 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  19. Krantz, S.G., Peloso, M.M., Stoppato, C.: Bergman kernel and projection on the unbounded Diederich–Fornæss worm domain. Annali della Scuola Normale Superiore. doi:10.2422/2036-2145.201503_012 (2014)

  20. Ligocka, E.: On the Forelli-Rudin construction and weighted Bergman projections. Studia Math. 94(3), 257–272 (1989)

    MathSciNet  MATH  Google Scholar 

  21. Pflug, P., Zwonek, W.: Bergman completeness of unbounded Hartogs domains. Nagoya Math. J. 180, 121–133 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Riemenschneider, O.: Über den Flächeninhalt analytischer Mengen und die Erzeugung \(k\)-pseudokonvexer Gebiete. Invent. Math. 2, 307–331 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  23. Słodkowski, Z., Tomassini, G.: Minimal kernels of weakly complete spaces. J. Funct. Anal. 210(1), 125–147 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wiegerinck, J.J.O.O.: Domains with finite-dimensional Bergman space. Math. Z. 187(4), 559–562 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Special thanks to the Institute of Mathematics, particularly Professor Ines Kath, at the University of Greifswald for their hospitality and support of the first author during the academic year 2015–2016. The second author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean government (MSIP) (No. 2011-0030044).

Author information

Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, Stillwater, USA

    A.-K. Gallagher

  2. The Center for Geometry and Its Applications, Pohang University of Science and Technology, Pohang, Republic of Korea

    T. Harz

  3. Department of Mathematics, University of Wuppertal, Wuppertal, Germany

    G. Herbort

Authors
  1. A.-K. Gallagher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Harz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Herbort
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A.-K. Gallagher.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gallagher, AK., Harz, T. & Herbort, G. On the Dimension of the Bergman Space for Some Unbounded Domains. J Geom Anal 27, 1435–1444 (2017). https://doi.org/10.1007/s12220-016-9725-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-016-9725-8

Keywords

Mathematics Subject Classification

Search

Navigation