Szegő kernel transformation law for proper holomorphic mappings

Abstract

Let $\Om_1, \Om_2$ be smoothly bounded doubly connected regions in the complex plane. We establish a transformation law for the Szeg\H{o} kernel under proper holomorphic mappings. This extends known results concerning biholomorphic mappings between multiply connected regions as well as proper holomorphic mappings from multiply connected regions to simply connected regions.