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.
Similar content being viewed by others
References
A. Abkar: Application of a Riesz-type formula to weighted Bergman spaces. Proc. Amer. Math. Soc. 131 (2003), 155–164.
A. Abkar: Norm approximation by polynomials in some weighted Bergman spaces. J. Func. Anal. 191 (2002), 224–240.
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.
K. Hoffman: Banach Spaces of Analytic Functions. Dover Publications, Inc. New York, 1988.
Author information
Authors and Affiliations
Additional information
Research supported in part by IPM under the grant number 83310011.
Rights 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
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10587-007-0082-4