Just analysis: the Poisson-Szegő-Bergman kernel (Q2091074)

In the paper, new connections are derived between the Szegő kernel, the Poisson kernel, the Dirichlet-to-Neumann map, and the Bergman kernel in planar domains. Assume that \(\Omega\) is a bounded simply connected domain in the plane with \(C^{\infty}\) smooth boundary. Let \(a\in\Omega\), and let \(k(z,w)\) be a complex antiderivative in the \(z\) variable of the Bergman kernel \(K(z,w)\) for \(\Omega\) that vanishes at \(a\). Let \(T(w)\), with \(|T(w)|=1\), point in the direction of the standard orientation of the boundary at \(w\) in the boundary of \(\Omega\). The following theorem relates the Poisson kernel to the antiderivative of the Bergman kernel. Theorem 2.1. The Poisson kernel \(P(z,w)\) associated to a bounded simply connected domain in the plane with \(C^{\infty}\) smooth boundary \(\Omega\) is given by \[ P(z,w)=P(a,w)+\mathrm{Re}[ik(z,w)\overline{T(w)}], \] where \(a\) is a fixed point in \(\Omega\) and \(k(z,w)\) is a holomorphic antiderivative in the \(z\) variable of the Bergman kernel \(K(z,w)\) associated to \(\Omega\) that vanishes at \(z=a\). The next theorem shows relations between the Bergman kernel and the Dirichlet-to-Neumann map which is the mapping that takes a smooth function on \(\partial\Omega\) to the normal derivative of its harmonic extension to \(\Omega\). Theorem 2.2. Suppose \(\Omega\) is a bounded finitely connected domain with \(C^{\infty}\) smooth boundary. The Dirichlet-to-Neumann map applied to a \(C^{\infty}\) smooth real valued function \(\varphi\) on the boundary is given by the function on the boundary \(\mathrm{Re}[h(z)T(z)]\) where the boundary values of \(h\) are obtained from the holomorphic function \[ h(z)=\int_{w\in\partial\Omega}K(z,w)\varphi(w)d\overline w \] on \(\Omega\) by letting \(z\) tend to the boundary. Also, the author discusses Poisson, Szegő, Bergman kernel connections in the multiply connected case.