Geodesic
On the integral representation of superbiharmonic functions
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 877-883 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 31A10, 31A30, 35C15
Keywords: superbiharmonic function; biharmonic Green function; weighted Bergman space 
@article{CMJ_2007_57_3_a7,
     author = {Abkar, Ali},
     title = {On the integral representation of superbiharmonic functions},
     journal = {Czechoslovak Mathematical Journal},
     pages = {877--883},
     year = {2007},
     volume = {57},
     number = {3},
     mrnumber = {2356287},
     zbl = {1174.31302},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a7/}
}
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Abkar, Ali. On the integral representation of superbiharmonic functions. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 877-883. http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a7/