The harmonic Bergman kernel and the Friedrichs operator

The harmonic Bergman kernel Q(Omega) for a simply connected planar domain Q can be expanded in terms of powers of the Friedrichs operator F-Omega if parallel toF(Omega)parallel to <1 in operator norm. Suppose that &UOmega; is the image of a univalent analytic function φ in the unit disk with φ '(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 parallel topsiparallel to(infinity) < 1 the series for Q(&UOmega;) also converges pointwise in <(Omega)over bar>x (&UOmega;) over barDelta(partial derivativeOmega), 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 parallel topsiparallel to(Ds(D))less than or equal to C-s, then the bilharmonic Green function for Omega=phi(D) is positive.