In the present note we give
a simpler proof of the recent result of Hedenmalm that the Green function for the
weighted biharmonic operator Δ|z|2αΔ, 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 Δ2 is due to Loewner and which is of an
independent interest, is a lemma characterizing, for a positive C2 weight function w,
the second-order linear differential operators which take any function u satisfying
Δw−1Δu = 0 into a harmonic function. Another application of this lemma
concerning positivity of the Poisson kernels for the biharmonic operator Δ2 is also
given.
