Weighted Kernel Functions on Planar Domains

We study the variation of weighted Szegő and Garabedian kernels on planar domains as a function of the weight. A Ramadanov type theorem is shown to hold as the weights vary. As a consequence, we derive properties of the zeros of the weighted Szegő and Garabedian kernel for weights close to the constant function $1$ on the boundary. We further study the weighted Ahlfors map and strengthen results concerning its boundary behaviour. Explicit examples of the weighted kernels are presented for certain classes of weights. We highlight an interesting property of the weighted Szegő and Garabedian kernels, implicit in Nehari's work, and explore several of its consequences. Finally, we discuss the weighted Carathéodory metric, and describe relations of the weighted Szegő and Garabedian kernel with certain classical kernel functions.