Extremal disks and composition operators on convex domains in \(\mathbb{C}^ n\)
Let \(\Omega\) be a bounded domain in \(\mathbb{C}^ n\), strongly convex with smooth boundary. Let \(k_ \Omega\) be the Kobayashi distance on \(\Omega\); let \(\varphi \in H(\Delta, \Omega)\) be an associated extremal disk \((k_ \Omega (\varphi (\lambda), \varphi (\mu)) = k_ \Delta (\lambda, \mu)\), \(\Delta\) is the unit disk in \(\mathbb{C})\) and let \(p = p_ \varphi \in H (\Omega, \Omega)\) be an associated retraction. The author proves the following proposition generalizing a theorem of Rudin: \[ | f \circ \varphi^{-1} \circ p |_{H^ (\Omega)} \asymp | f |_{H^ p_{n-2}} (\Delta),\;\forall f \in H^ p_{n-2} (\Delta). \]
scientific article; zbMATH DE number 757785
scientific article