Weighted Bergman kernels and biharmonic Green functions
(2000) In Doctoral 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 meanvalue condition and whose logarithms are subharmonic. To achieve this, we use a variational formula due to J. Hadamard, weighted HeleShaw 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 meanvalue condition and whose logarithms are subharmonic. To achieve this, we use a variational formula due to J. Hadamard, weighted HeleShaw 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:
https://lup.lub.lu.se/record/40796
 author
 Jakobsson, Stefan ^{LU}
 supervisor
 opponent

 Professor Lindqvist, Peter, NTNU, Trondheim
 organization
 publishing date
 2000
 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
 Doctoral 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
 20001013 10:15:00
 external identifiers

 other:ISRN: LUNFMA10142000
 ISSN
 14040034
 ISBN
 9178740854
 language
 English
 LU publication?
 yes
 id
 deee2490a3db4feba99f1371b7171360 (old id 40796)
 date added to LUP
 20160401 16:50:25
 date last changed
 20190521 13:48:37
@phdthesis{deee2490a3db4feba99f1371b7171360, 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 meanvalue condition and whose logarithms are subharmonic. To achieve this, we use a variational formula due to J. Hadamard, weighted HeleShaw 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 = {{9178740854}}, issn = {{14040034}}, keywords = {{singularity resolution of Bergman kernels; Hadamard's variational formula; weighted Bergman kernel; weighted biharmonic Green function; Friedrichs operator; Plant biochemistry; Växtbiokemi}}, language = {{eng}}, publisher = {{Centre for Mathematical Sciences, Lund University}}, school = {{Lund University}}, series = {{Doctoral Theses in Mathematical Sciences}}, title = {{Weighted Bergman kernels and biharmonic Green functions}}, volume = {{2000:5}}, year = {{2000}}, }