Logarithmic capacity and Bergman functions
Let \(D\) be a bounded domain in \(\mathbb{C}\) and let \(L^2_h(D)\) be square integrable holomorphic functions on \(D\). \(L^2_h(D)\) is a Hilbert space with the scalar product induced from \(L^2 (D)\). For a bounded domain \(D\subset\mathbb{C}\), \(n=0,1,\dots\) and \(z\in D\) let \(K^{(n)}_D(z)\) be the square of the norm of the linear mapping \[ L^2_h(D) \ni f \mapsto f^{(n)}(z)\in \mathbb{C}. \] In other words, \[ K^{(n)}_D(z):=\sup\left\{\frac{| f^{(n)}(z)| ^2}{\| f\| ^2_D}:f\in L^2_h(D),\;f\not\equiv 0\right\} , \] where \(\| \cdot\| _D\) is the \(L^2\) norm on \(D\). The authors also introduce the following potential theoretic concept: \[ \gamma^{(n)}_D(z):=\int\limits_0^{1/4} \frac{d\delta}{\delta^{2n+3}(-\log{\text{ cap}}(\bar\triangle(z,\delta)\setminus D))},\;n\in \mathbb{N}_0, \;z\in \mathbb{C}. \] where cap\((E)\) is the logarithmic capacity of a set \(E\). The main result of the paper is the following theorem. Theorem. Fix \(n\in \mathbb{N}_0\) and \(d>1\). Then there is a \(C>0\) such that (1) for any bounded domain \(D \subset \mathbb{C}\) with \(\text{diam} \, D<d\) \[ C\gamma^{(n)}_D(z) \leq K^{(n)}_D(z), \;z\in D; \] (2) for any bounded domain \(D \subset \mathbb{C}\) with \(1/d < \text{diam}\, D<d\) \[ K^{(n)}_D(z) \leq C \max\{1, \gamma^{(n)}_D(z)(\text{log}(\gamma^{(n)}_D(z))^2\}, \;z\in D. \]
scientific article; zbMATH DE number 1993487