Szegő kernel transformation law for proper holomorphic mappings (Q457939)

Let \(\Omega \subset \mathbb{C}\) be a bounded region with \(C^\infty\) smooth boundary. The Szegő projection \(\mathcal{S}\) is the orthogonal projection of \(L^2(\partial\Omega)\) onto the subspace \(H^2(\partial\Omega)\) of functions that extend holomorphically to \(\Omega\). It acts by integration against an Hermitian kernel \(S_\Omega(\cdot,\cdot)\) which is called the Szegő kernel, according to NEWLINE\[NEWLINE \mathcal{S}f(z) = \int_{\partial\Omega} S_\Omega(z,w)f(w)\,ds_w\,, \quad z \in \Omega\,. NEWLINE\]NEWLINE The integration is carried out with respect to the arc length measure \(ds\).NEWLINENEWLINEThe simplest transformation law for the Szegő kernel expresses the relationship between kernels for biholomorphically equivalent regions. \textit{M. Jeong} [Complex Variables, Theory Appl. 23, No. 3--4, 157--162 (1993; Zbl 0795.30010)] extended these results to proper holomorphic mappings provided that the target is simply connected. In this paper the author establishes a tranformation law for proper holomorphic mappings between doubly connected regions.