Bergman Kernel and its Transformations

The Bergman kernel is a very important object to study in many different areas of mathematics, namely, complex analysis, partial differential equations, differential geometry and operator theory to mention a few. Bergman kernel is an important tool for studying biholomorphic (conformal) mappings and their boundary behaviour. We discuss Hilbert spaces of complex-valued functions defined on some X such that the point evaluations are continuous for each point of X. The Bergman kernel transforms nicely under biholomorphic mappings. We discuss this transformation property in detail and include two proofs of the transformation rule using the Hilbert space structure of the Bergman space. The relationship between the Bergman kernel function of a simply connected domain and the Riemann mapping function of the domain is also mentioned. The relation between the Bergman projection and the Bergman kernel is also discussed.