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Weighted Bergman kernels and biharmonic Green functions

Jakobsson, Stefan LU (2000) In Doctorial Theses in Mathematical Sciences 2000:5.
Abstract
The main theme of this thesis is the connection between weighted biharmonic Green functions and weighted Bergman kernels. In the first paper, which is a joint work with H. Hedenmalm and S. Shimorin, we prove that weighted biharmonic Green functions are positive for weights which satisfy a mean-value condition and whose logarithms are subharmonic. To achieve this, we use a variational formula due to J. Hadamard, weighted Hele-Shaw flow, as well as a new structural formula for the analytic Bergman kernel. The result has applications to the factorization theory in weighted Bergman spaces. In the subsequent papers, we continue to investigate Bergman kernels and Green functions. In the second paper, we analyze the singularity of the weighted... (More)
The main theme of this thesis is the connection between weighted biharmonic Green functions and weighted Bergman kernels. In the first paper, which is a joint work with H. Hedenmalm and S. Shimorin, we prove that weighted biharmonic Green functions are positive for weights which satisfy a mean-value condition and whose logarithms are subharmonic. To achieve this, we use a variational formula due to J. Hadamard, weighted Hele-Shaw flow, as well as a new structural formula for the analytic Bergman kernel. The result has applications to the factorization theory in weighted Bergman spaces. In the subsequent papers, we continue to investigate Bergman kernels and Green functions. In the second paper, we analyze the singularity of the weighted analytic and harmonic Bergman kernels for a general smooth weight in a domain with smooth boundary. In the third paper, we apply the theory of semigroups to one of the arguments in the first paper. It turns out that if the harmonic Bergman kernel for a starshaped domain satisfies a certain inequality, then the biharmonic Green function for the domain is positive. In the fourth paper, we show how the Friedrichs operator can be used to expand the harmonic Bergman kernel in terms of the analytic counterpart for a simply connected domain. Under certain conditions on the conformal map from the unit disk to the domain, we obtain a pointwise estimate of the harmonic Bergman kernel. (Less)
Please use this url to cite or link to this publication:
author
opponent
  • Professor Lindqvist, Peter, NTNU, Trondheim
organization
publishing date
type
Thesis
publication status
published
subject
keywords
singularity resolution of Bergman kernels, Hadamard's variational formula, weighted Bergman kernel, weighted biharmonic Green function, Friedrichs operator, Plant biochemistry, Växtbiokemi
in
Doctorial Theses in Mathematical Sciences
volume
2000:5
pages
130 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Centre for Mathematical Sciences Sölvegatan 18, room MH:A
defense date
2000-10-13 10:15
external identifiers
  • other:ISRN: LUNFMA-1014-2000
ISSN
1404-0034
ISBN
91-7874-085-4
language
English
LU publication?
yes
id
deee2490-a3db-4feb-a99f-1371b7171360 (old id 40796)
date added to LUP
2007-08-01 12:02:05
date last changed
2016-09-19 08:44:58
@phdthesis{deee2490-a3db-4feb-a99f-1371b7171360,
  abstract     = {The main theme of this thesis is the connection between weighted biharmonic Green functions and weighted Bergman kernels. In the first paper, which is a joint work with H. Hedenmalm and S. Shimorin, we prove that weighted biharmonic Green functions are positive for weights which satisfy a mean-value condition and whose logarithms are subharmonic. To achieve this, we use a variational formula due to J. Hadamard, weighted Hele-Shaw flow, as well as a new structural formula for the analytic Bergman kernel. The result has applications to the factorization theory in weighted Bergman spaces. In the subsequent papers, we continue to investigate Bergman kernels and Green functions. In the second paper, we analyze the singularity of the weighted analytic and harmonic Bergman kernels for a general smooth weight in a domain with smooth boundary. In the third paper, we apply the theory of semigroups to one of the arguments in the first paper. It turns out that if the harmonic Bergman kernel for a starshaped domain satisfies a certain inequality, then the biharmonic Green function for the domain is positive. In the fourth paper, we show how the Friedrichs operator can be used to expand the harmonic Bergman kernel in terms of the analytic counterpart for a simply connected domain. Under certain conditions on the conformal map from the unit disk to the domain, we obtain a pointwise estimate of the harmonic Bergman kernel.},
  author       = {Jakobsson, Stefan},
  isbn         = {91-7874-085-4},
  issn         = {1404-0034},
  keyword      = {singularity resolution of Bergman kernels,Hadamard's variational formula,weighted Bergman kernel,weighted biharmonic Green function,Friedrichs operator,Plant biochemistry,Växtbiokemi},
  language     = {eng},
  pages        = {130},
  publisher    = {Centre for Mathematical Sciences, Lund University},
  school       = {Lund University},
  series       = {Doctorial Theses in Mathematical Sciences},
  title        = {Weighted Bergman kernels and biharmonic Green functions},
  volume       = {2000:5},
  year         = {2000},
}