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Keywords:
superbiharmonic function; biharmonic Green function; weighted Bergman space
superbiharmonic function; biharmonic Green function; weighted Bergman space
Summary:
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 $0\le w(z)\le C(1-|z|)$ 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.
References:
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[2] A. Abkar: Norm approximation by polynomials in some weighted Bergman spaces. J. Func. Anal. 191 (2002), 224–240. DOI 10.1006/jfan.2001.3851 | MR 1911185 | Zbl 1059.30049
[3] H. Hedenmalm: A computation of Green function for the weighted biharmonic operators $\Delta \vert z\vert ^{-2\alpha }\Delta $ with $\alpha >-1$. Duke Math. J. 75 (1994), 51–78. DOI 10.1215/S0012-7094-94-07502-9 | MR 1284815
[4] K. Hoffman: Banach Spaces of Analytic Functions. Dover Publications, Inc. New York, 1988. MR 1102893 | Zbl 0734.46033
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