Boundary decomposition of the Bergman kernel (Q635276)

An interesting asymptotic formula for the Bergman kernel on a planar finitely connected domain \(\Omega\) with smooth boundary is obtained. \smallskip Let \(S_1,\dots,S_k\) denote the connected components of \(\partial \Omega\), where \(S_1\) bounds the unbounded connected component of \(\mathbb C\setminus \Omega\). Denote by \(\Omega_1\) the bounded domain having \(S_1\) as its boundary and by \(\Omega_j\), \(j> 1\), the unbounded domain that is bounded by \(S_j\). The author shows that the Bergman kernel function \(K_\Omega(z,w)\) of \(\Omega\) can be represented as \[ K_\Omega = \frac{1}{k}(K_{\Omega_1}+ \dots + K_{\Omega_k})+ \mathcal E \] with an error term \(\mathcal E\) that is smooth on \(\overline{\Omega\times \Omega}\). This result is carried over to the case of several complex variables, where it is proved in the following form: Let \(n>1\) and \(\Omega\) be a smoothly bounded domain in \(\mathbb C^n\). Denote the connected components of its boundary by \(S_1,\dots,S_k\), where \(S_1\) bounds the unbounded portion of \(\mathbb C^n\setminus \overline \Omega\). Let \(\Omega_1\) be the bounded region that is bounded by \(S_1\). Then \[ K_\Omega = K_{\Omega_1} + \mathcal E \] with an error term \(\mathcal E\) that, together with its derivatives, is bounded.