Generalized quasidisks and the associated John domains (Q2870400)

A Jordan domain \(\Omega \subset \mathbb{R}^2\) is a \(K\)-quasidisk if it is the image of the unit disk \(D\) under a \(K\)-quasiconformal mapping \(f : \mathbb{R}^2 \rightarrow \mathbb{R}^2\). Equivalently, \(\Omega\) is a quasidisk if there is a \(K^2\)-quasiconformal mapping of \(\mathbb{R}^2\) which is conformal in \(D\) with \(f(D) = \Omega\). Using the latter approach the author studies quasidisks when quasiconformality is relaxed to the exponential integrability of the distortion \(K_f\) of \(f\). This means that \(\exp(\lambda K_f)\) is locally integrable for some \(\lambda > 0\). The article consists of five papers which form the authors thesis. Two of them, [\textit{C.-Y. Guo} et al., Publ. Mat., Barc. 58, No. 1, 193--212 (2014; Zbl 1286.30021)] and [\textit{C.-Y. Guo} and \textit{P. Koskela}, Cent. Eur. J. Math. 12, No. 2, 349--361 (2014; Zbl 1294.30037)], are about to appear soon. The author considers generalized local connectivity, a generalized three-point property for \(\partial \Omega\) and generalized John disks. All these involve a function, instead of a constant, which controls the behavior. Exponential integrability allows generalized quasidisks to have sharp cusps but, as it is shown, the relation between the sharpness of the cusp and the exponential integrability condition is quite subtle. The main part of the thesis is devoted to generalized John domains and John disks. Their relations to the other generalized concepts are studied in detail. Uniform continuity of ordinary quasiconformal mappings \(f\) from a domain \(\Omega'\) to a generalized John domain \(\Omega\) in terms of the internal metrics of \(\Omega\) and \(\Omega'\) is also studied. The articles include many examples and the main emphasis is on the sharpness of the results.