A Loewner-type lemma for weighted biharmonic operators

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In the present note we give a simpler proof of the recent result of Hedenmalm that the Green function for the weighted biharmonic operator ∆|z|2α∆, α > −1, on the unit disc D with the Dirichlet boundary conditions is positive. The main ingredient, which in the special case of the unweighted biharmonic operator ∆ is due to Loewner and which is of an independent interest, is a lemma characterizing, for a positive C weight function w, the second-order linear differential operators which take any function u satisfying ∆w−1∆u = 0 into a harmonic function. Another application of this lemma concerning positivity of the Poisson kernels for the biharmonic operator ∆ is also given.