Weighted Bergman kernels and biharmonic Green functions

Sammanfattning: 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.