The Szegő kernel and proper holomorphic mappings to a half plane
The author uses the Szegő kernel \(S(z,w)\) and the Garabedian kernel \(L(z,w)\) to describe the proper holomorphic mappings from a bounded, \(n\)-connected planar domain (whose \(n\) boundary components are smooth, real-analytic Jordan curves) onto the right-hand half-plane. One of the main results states that if \(b_1\), \(\ldots\), \(b_n\) are arbitrary points in the boundary of the domain (one point from each boundary curve), then there exist another boundary point \(a\) and positive real numbers \(c_1\), \(\ldots\), \(c_n\) such that \[ \sum_{j=1}^n c_j \frac{S(z,b_j) L(b_j,a)} {L(z,a)} \] is an \(n\)-to-\(1\) proper holomorphic mapping to the right-hand half-plane having poles at the points \(b_{1}\), \(\ldots\), \(b_{n}\). Moreover, these particular mappings generate the additive semigroup of all proper holomorphic mappings to the right-hand half-plane. An application is an easy construction of a primitive pair for the space of meromorphic functions on the double of a finitely connected planar domain.
scientific article; zbMATH DE number 5965034
scientific article