Dimension gap under conformal mappings
Let \(f : \Delta \rightarrow \Omega\) be a conformal map of the unit disc \(\Delta\) onto a simply connected domain \(\Omega\) and let \(p > 1\). The main result of this paper is the following theorem:NEWLINENEWLINEIf \(\int_\Delta |f'(z)|^2 \log^p (1-|z|)^{-1} dA < \infty\), then \(\psi\)-Hausdorff dimension of \(\partial \Omega\) is zero, i.e., \(H^{\psi}(\partial \Omega) = 0\), for the gauge function \(\psi(t) = t^2 (-\log t)^s\) and \(s < 2p\).NEWLINENEWLINEThis result is extended to multiply connected domains satisfying \(k_{\Omega}(z_0, \cdot) \in L^p(\Omega)\), where \(k_\Omega\) is the quasihyperbolic metric in \(\Omega\).NEWLINENEWLINEThese results are essentially sharp as is shown by constructing an example. The authors make an important point that one can have \(H^\psi (\partial \Omega) = \infty\) for all \(s>0\) with \(\psi(t) = t^2 (-\log t)^s\) if one only assumes \(|f^\prime| \in L^2(\Delta, |\log (1-|z|)| dA(z))\). Therefore, there is a dimension gap as \(p \to 1\).
scientific article; zbMATH DE number 6059034