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The harmonic Bergman kernel and the Friedrichs operator

Jakobsson, Stefan LU (2002) In Arkiv för matematik 40(1). p.89-104
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 F-Omega 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 D-s(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 F-Omega 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 D-s(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 C-s>0 such that if parallel topsiparallel to(Ds(D))less than or equal to C-s, then the bilharmonic Green function for Omega=phi(D) is positive. (Less)
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author
organization
publishing date
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
0004-2080
DOI
10.1007/BF02384504
language
English
LU publication?
yes
id
747d529f-57d9-4770-b398-fbfb3e34262a (old id 338729)
date added to LUP
2007-10-24 12:52:01
date last changed
2017-01-01 07:14:28
@article{747d529f-57d9-4770-b398-fbfb3e34262a,
  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 F-Omega if parallel toF(Omega)parallel to &lt;1 in operator norm. Suppose that &amp;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 D-s(D), s&gt;0, of Dirichlet type, then provided that parallel topsiparallel to(infinity) &lt; 1 the series for Q(&amp;UOmega;) also converges pointwise in &lt;(Omega)over bar&gt;x (&amp;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&gt;0 there exists a constant C-s&gt;0 such that if parallel topsiparallel to(Ds(D))less than or equal to C-s, then the bilharmonic Green function for Omega=phi(D) is positive.},
  author       = {Jakobsson, Stefan},
  issn         = {0004-2080},
  language     = {eng},
  number       = {1},
  pages        = {89--104},
  publisher    = {Springer},
  series       = {Arkiv för matematik},
  title        = {The harmonic Bergman kernel and the Friedrichs operator},
  url          = {http://dx.doi.org/10.1007/BF02384504},
  volume       = {40},
  year         = {2002},
}