The harmonic Bergman kernel and the Friedrichs operator

Stefan Jakobsson Centre for Mathematical Sciences, Lund University

TBD mathscidoc: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.
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  title={The harmonic Bergman kernel and the Friedrichs operator},
  author={Stefan Jakobsson},
  booktitle={Arkiv for Matematik},
Stefan Jakobsson. The harmonic Bergman kernel and the Friedrichs operator. 2000. Vol. 40. In Arkiv for Matematik. pp.89-104.
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