The harmonic Bergman kernel and the Friedrichs operator
(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)
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
- 0004-2080
- DOI
- 10.1007/BF02384504
- language
- English
- LU publication?
- yes
- id
- 747d529f-57d9-4770-b398-fbfb3e34262a (old id 338729)
- date added to LUP
- 2016-04-01 16:45:46
- date last changed
- 2022-01-28 21:55:57
@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 <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.}}, 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}}, doi = {{10.1007/BF02384504}}, volume = {{40}}, year = {{2002}}, }