Arkiv för Matematik
- Ark. Mat.
- Volume 40, Number 1 (2002), 89-104.
The harmonic Bergman kernel and the Friedrichs operator
The harmonic Bergman kernel QΩ for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator FΩ ║FΩ║<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 Ds (D), s>0, of Dirichlet type, then provided that ║ψ║∞<1 the series for QΩ 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 Cs>0 such that if ║ψ║Ds(D)≤Cs, then the biharmonic Green function for Ω=ø (D) is positive.
Ark. Mat., Volume 40, Number 1 (2002), 89-104.
Received: 18 September 2000
First available in Project Euclid: 31 January 2017
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2002 © Institut Mittag-Leffler
Jakobsson, Stefan. The harmonic Bergman kernel and the Friedrichs operator. Ark. Mat. 40 (2002), no. 1, 89--104. doi:10.1007/BF02384504. https://projecteuclid.org/euclid.afm/1485898755