A Loewner-type lemma for weighted biharmonic operators (Q1816519)

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.