Quasilines and conformal mappings
Let \(\Omega\) be a simply connected plane domain and f a conformal mapping of \(\Omega\) onto the unit disk. A notable theorem of Hayman and Wu (1982) asserts the existence of an absolute constant C such that \[ (1)\quad \int_{L\cap \Omega}| f'(x)| dx\leq C \] for every line or circle L. The paper under review contains a number of related results. For instance, the authors use symmetrization and hyperbolic convexity considerations to show that \(C=4\pi^ 2\) is permissible in (1). They also offer a conjecture for the sharp value of C. Hayman and Wu had obtained \(C\leq 10^{35}.\) The authors also consider the \(L^ p\) case of (1), and show that there exists \(\epsilon >0\) such that \(f'\in L^{1+\epsilon}(L\cap \Omega)\) for every L. It had been conjectured that \(f'\in L^ p(L\cap \Omega)\) for every \(p\in (1,2)\), but this has been disproved by the reviewer [J. Anal. Math. 53, 253-268 (1989)]. Other results assert that if D is a quasidisk contained in \(\Omega\) then f(D) is also a quasidisk, and if L is a K- quasicircle then \(f(L_ i)\) is a ``quasiarc'' for each component \(L_ i\) of \(L\cap \Omega\). Moreover, \[ (2)\quad \sum_{i}(diam f(L_ i))^ K<\infty. \] The authors call this the quasiconformal analogue of the Hayman-Wu theorem. They leave open the question of whether the series in (2) converges when K is replaced by 1. In the past few months two further results inspired by the Hayman-Wu theorem have been proved. \textit{C. Bishop} and \textit{P. Jones} [Ann. Math., to appear] have characterized plane curves L with the ``Hayman-Wu property'' as precisely those which are ``Ahlfors-regular'', thereby settling a conjecture discussed in the paper under review. Also, J.-M. Wu has proved a version of (1) when L is a hyperplane in \({\mathbb{R}}^ n\), \(\Omega\) satisfies a suitable thickness condition at the boundary, and \(f'\) is replaced by the gradient of the Green's function of \(\Omega\).
scientific article; zbMATH DE number 4109082
scientific article