# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.332976

Arkiv for Matematik, 40, (1), 89-104, 2000.9
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║_{$D$}_{s}($D$)≤$C$_{$s$}, then the biharmonic Green function for Ω=ø ($D$) is positive.
@inproceedings{stefan2000the,
title={The harmonic Bergman kernel and the Friedrichs operator},
author={Stefan Jakobsson},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203603834732785},
booktitle={Arkiv for Matematik},
volume={40},
number={1},
pages={89-104},
year={2000},
}

Stefan Jakobsson. The harmonic Bergman kernel and the Friedrichs operator. 2000. Vol. 40. In Arkiv for Matematik. pp.89-104. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203603834732785.