The harmonic Bergman kernel and the Friedrichs operator
(2002) In Arkiv för Matematik 40(1). p.89104 Abstract
 The harmonic Bergman kernel Q(Omega) for a simply connected planar domain Q can be expanded in terms of powers of the Friedrichs operator FOmega 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 Ds(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... (More)
 The harmonic Bergman kernel Q(Omega) for a simply connected planar domain Q can be expanded in terms of powers of the Friedrichs operator FOmega 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 Ds(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 Cs>0 such that if parallel topsiparallel to(Ds(D))less than or equal to Cs, then the bilharmonic Green function for Omega=phi(D) is positive. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/338729
 author
 Jakobsson, Stefan ^{LU}
 organization
 publishing date
 2002
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Arkiv för Matematik
 volume
 40
 issue
 1
 pages
 89  104
 publisher
 Springer
 external identifiers

 wos:000175359900006
 scopus:0036552274
 ISSN
 00042080
 DOI
 10.1007/BF02384504
 language
 English
 LU publication?
 yes
 id
 747d529f57d94770b398fbfb3e34262a (old id 338729)
 date added to LUP
 20160401 16:45:46
 date last changed
 20220128 21:55:57
@article{747d529f57d94770b398fbfb3e34262a, abstract = {{The harmonic Bergman kernel Q(Omega) for a simply connected planar domain Q can be expanded in terms of powers of the Friedrichs operator FOmega 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 Ds(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 Cs>0 such that if parallel topsiparallel to(Ds(D))less than or equal to Cs, then the bilharmonic Green function for Omega=phi(D) is positive.}}, author = {{Jakobsson, Stefan}}, issn = {{00042080}}, language = {{eng}}, number = {{1}}, pages = {{89104}}, publisher = {{Springer}}, series = {{Arkiv för Matematik}}, title = {{The harmonic Bergman kernel and the Friedrichs operator}}, url = {{http://dx.doi.org/10.1007/BF02384504}}, doi = {{10.1007/BF02384504}}, volume = {{40}}, year = {{2002}}, }