The harmonic Bergman kernel and the Friedrichs operator

Abstract

The harmonic Bergman kernel Q_Ω for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator F_Ω ║F_Ω║<1 in operator norm. Suppose that Ω is the image of a univalent analytic function ø in the unit disk with ø' (z)=1+ψ (z) where ψ(0)=0. We show that if the function ψ belongs to a space D_s (D), s>0, of Dirichlet type, then provided that ║ψ║∞<1 the series for Q_Ω also converges pointwise in $\bar \Omega \times \bar \Omega \backslash \Delta (\partial \Omega )$ , and the rate of convergence can be estimated. The proof uses the eigenfunctions of the Friedrichs operator as well as a formula due to Lenard on projections in Hilbert spaces. As an application, we show that for every s>0 there exists a constant C_s>0 such that if ║ψ║_Ds(D)≤C_s, then the biharmonic Green function for Ω=ø (D) is positive.