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

Abstract

Listen

Translate article

Let D be the open unit disk and ∑ area measure, normalized so that D has mass 1. Suppose the weight ω : D → [ 0 , + ∞ [ has log ω subharmonic, and furthermore, that it has the reproducing property: h ( 0 ) = ∫ D h ( z ) ω ( z ) d Σ ( z ) for all bounded harmonic functions h on D . Let Γ in gw be the Green function for the weighted biharmonic operator Δω 1 on D with vanishing Dirichlet boundary data. We prove that 0 ≤ Γ ω holds on D × D . This result has interesting applications to the operator theory of the Bergman spaces.