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A Loewner-type lemma for weighted biharmonic operators. (English) Zbl 0871.31003

The author gives a simpler proof of the recent result of Hedenmalm that the Green function for the weighted biharmonic operator \(\Delta |z|^{2 \alpha} \Delta\), \(\alpha> -1\), on the unit disc \({\mathbf D}\) with Dirichlet boundary conditions is positive. The main ingredient, which in the special case of the unweighted biharmonic operator \(\Delta^2\) is due to Loewner and which is of independent interest, is a lemma characterizing, for a positive \(C^2\) weight function \(w\), the second-order linear differential operators which take any function \(u\) satisfying \(\Delta w^{-1} \Delta u=0\) into a harmonic function. Another application of this lemma concerning positivity of the Poisson kernels for the biharmonic operator \(\Delta^2\) is also given.
Reviewer: M.Engliš (Praha)

MSC:

31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J40 Boundary value problems for higher-order elliptic equations
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