Geodesic
The Green function for the weighted biharmonic operator $\Delta(1-|z|^2)^{-\alpha}\Delta$, and factorization of analytic functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 23, Tome 222 (1995), pp. 203-221

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An explicit integral formula is obtained for the Green function of the weighted biharmonic operator $\Delta(1-|z|^2)^{-\alpha}\Delta$ in the unit disc of the complex plane for the case $\alpha\in(-1,0)$. The formula shows the positivity of the Green function. This is a basis for a theorem on factorization of analytic functions in the weighted Bergman spaces with the weights $w(z)=(1-|z|^2)^\alpha$ as product of a nonvanishing function and a function of special form responsible for the zeros. Bibliography: 16 titles. 
@article{ZNSL_1995_222_a7,
     author = {S. M. Shimorin},
     title = {The {Green} function for the weighted biharmonic operator $\Delta(1-|z|^2)^{-\alpha}\Delta$, and factorization of analytic functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {203--221},
     publisher = {mathdoc},
     volume = {222},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1995_222_a7/}
}
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S. M. Shimorin. The Green function for the weighted biharmonic operator $\Delta(1-|z|^2)^{-\alpha}\Delta$, and factorization of analytic functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 23, Tome 222 (1995), pp. 203-221. http://geodesic.mathdoc.fr/item/ZNSL_1995_222_a7/