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April 2002 The harmonic Bergman kernel and the Friedrichs operator
Stefan Jakobsson
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Ark. Mat. 40(1): 89-104 (April 2002). DOI: 10.1007/BF02384504


The harmonic Bergman kernel QΩ for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator FΩFΩ║<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.


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Stefan Jakobsson. "The harmonic Bergman kernel and the Friedrichs operator." Ark. Mat. 40 (1) 89 - 104, April 2002.


Received: 18 September 2000; Published: April 2002
First available in Project Euclid: 31 January 2017

zbMATH: 1075.47505
MathSciNet: MR1948888
Digital Object Identifier: 10.1007/BF02384504

Rights: 2002 © Institut Mittag-Leffler


Vol.40 • No. 1 • April 2002
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