Localization properties and boundary behavior of the Bergman kernel

In this work, we study localization of the Bergman kernel and the behavior of the Bergman kernel at different types of boundary points. The goal of the former is to show that the behavior of the Bergman kernel on a finitely connected domain Ω near a smooth boundary point P can be localized. That is, given a simply connected subdomain W of Ω that shares a piece of boundary with Ω, where P lies on that shared piece, then we show that the behavior of the Bergman kernel on W near P is the same as the behavior of the Bergman kernel on Ω near P. We also provide a localization result for the Bergman kernel near corner boundary points. One useful tool for this part of the thesis is the striking relationship between the Bergman kernel and the Green's function. We then turn our attention to studying the behavior of the Bergman kernel at three types of boundary points: smooth points, corner points, and cusp points. We calculate Bergman kernels for each type of domain. With these formulas, we show that the Bergman kernel blows-up at each type of boundary point, and we provide estimates for the rate of blow-up along the diagonal and off the diagonal. We are able to distinguish domains based on the rate of blow-up of the Bergman kernel.