A maximum principle a la Hadamard for biharmonic operators with applications to the Bergman spaces

Let D be the open unit disk and Sigma area measure, normalized so that D ha s mass 1. Suppose the weight omega : D --> [0; +infinity[ has log omega sub harmonic, and furthermore, that it has the reproducing property: h(0) = integral(D)h(z)omega(z)d Sigma(z) for all bounded harmonic functions h on D. Let Gamma(omega) be the Green fu nction for the weighted biharmonic operator Delta omega(-1) Delta on D with vanishing Dirichlet boundary data. We prove that 0 less than or equal to G amma(omega) holds on D x D. This result has interesting applications to the operator theory of the Bergman spaces. (C) Academie des Sciences/Elsevier, Paris.