The Faber operator and its boundedness
Let \(G\) be a domain bounded by a rectifiable Jordan curve \(\Gamma\), and let \(A(G) := \mathcal O(G) \cap\mathcal C(\overline G)\) be the Banach space of functions holomorphic in \(G\) and continuous on \(\overline G\). Let \[ \psi(w) = bw +b_o+ {{b_1}\over w} + \dots ,\quad |w|>1, \] be the normalized exterior mapping which maps \(\{w:|w|>1\}\) onto the exterior of \(\Gamma \). Let \(\mathbb D\) be the unit disk. The Faber operator \(T:A(\mathbb D) \mapsto \mathcal O(G)\) is defined by the formula \[ \left (Tf\right)(z) := {{1}\over {2\pi i}}\int_{\partial \mathbb D} f(w) {{\psi'(w)}\over {\psi(w)-z}} dw, \quad z \in G, \quad f\in A(\mathbb D). \] Put \(\|T\|:= \sup \{\|Tf\|_{G}/\|f\|_{\mathbb D}; f \in A(\mathbb D)\}\). If \(T\) is bounded, i.e. \(\|T\|<\infty \), then \(T(A(D)) \subset A(G)\). The author shows that \(\|T\|<\infty\), if \(\Gamma\) is piecewise Dini-smooth, and gives an example of a quasicircle \(\Gamma\) for which \(T\) is unbounded.
scientific article; zbMATH DE number 1398585
scientific article