The Bergman and Szego kernels and mapping problems in complex analysis

In chapter 2 it is proved that the uniform extendibility of the Bergman kernel is equivalent to a global regularity property of the Bergman projection. This result improves on the work of So-Chin Chen and simplifies the proof of Chen\u27s original result. Applications are given to Bergman kernel density and finite order vanishing theorems which arise in mapping problems between equidimensional domains. The Szego kernel of a smoothly bounded domain in the plane is known to be the solution of a certain Fredholm equation of the second kind known as the Kerzman-Stein equation. It follows that the boundary values of the Szego kernel are easy to compute. Then the Riemann map may be computed via a classical formula relating this map to the Szego kernel. In chapter 3, it is shown how to extend the Kerzman-Stein method to simply connected domains with piecewise smooth boundaries. In particular, this gives a new method for computing the inverse of the Schwarz-Christoffel map. The Kerzman-Stein integral equation is valid for smoothly bounded, multiply connected planar domains allowing the boundary values of the Szego kernel to be computed. The interior values of this kernel can be evaluated via the Cauchy integral formula. Unfortunately, the Cauchy integral is singular in nature for evaluation points near the boundary of the domain. In chapter 4, it is explained how to alleviate the singular nature of the Cauchy integral formula for domains with circular boundary components. This result is then used to give a method for computing the Ahlfors map of domains with circular boundary components onto the unit disk