Skip to main content
Springer Nature Link
Log in

On the integral representation of superbiharmonic functions

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

We consider a nonnegative superbiharmonic function w satisfying some growth condition near the boundary of the unit disk in the complex plane. We shall find an integral representation formula for w in terms of the biharmonic Green function and a multiple of the Poisson kernel. This generalizes a Riesz-type formula already found by the author for superbihamonic functions w satisfying the condition 0 ⩽ w(z) ⩽ C(1-|z|) in the unit disk. As an application we shall see that the polynomials are dense in weighted Bergman spaces whose weights are superbiharmonic and satisfy the stated growth condition near the boundary.

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. A. Abkar: Application of a Riesz-type formula to weighted Bergman spaces. Proc. Amer. Math. Soc. 131 (2003), 155–164.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Abkar: Norm approximation by polynomials in some weighted Bergman spaces. J. Func. Anal. 191 (2002), 224–240.

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Hedenmalm: A computation of Green function for the weighted biharmonic operators Δ| z|−2α Δ with α > − 1. Duke Math. J. 75 (1994), 51–78.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Hoffman: Banach Spaces of Analytic Functions. Dover Publications, Inc. New York, 1988.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, Imam Khomeini International University, Qazvin, 34194, Iran

    Ali Abkar

  2. Mathematics Department, Institute for Studies in Theoretical Physics and Mathematics, Tehran, 19395-5746, Iran

    Ali Abkar

Authors
  1. Ali Abkar
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by IPM under the grant number 83310011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abkar, A. On the integral representation of superbiharmonic functions. Czech Math J 57, 877–883 (2007). https://doi.org/10.1007/s10587-007-0082-4

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-007-0082-4

Keywords

Search

Navigation