Transformation formula for the reduced Bergman kernel and its application (Q2678410)

The aim of this paper is to prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded planar domains. As a special case, the transformation formula is obtained for the reduced Bergman kernels under proper holomorphic maps. The authors also establish a transformation formula for kernels of a more general class of spaces, namely weighted reduced Bergman kernels under proper holomorphic maps. Finally, an application of this transformation formula is given. Any proper holomorphic map from a bounded planar domain to the unit disc is rational if the reduced Bergman kernel of the domain is rational.