Decomposition and removability properties of John domains
A domain \(D\) is a \(c\)--John domain if for all points \(z_1, z_2 \in D\) there is a rectifiable curve \(\alpha \subset D\) joining them and \(\min_{j = 1,2} l(\alpha[z_j,z]) \leq c\, dist(z, \partial D)\) for each \(z \in \alpha\), [see \textit{O. Martio, J. Sarvas}, Ann. Acad. Sci. Fenn., Ser. A I 4, 383-401 (1979; Zbl 0406.30013)] for another definition. The domain \(D\) is said to have the John decomposition property if for some constant \(c\) each pair of points \(z_1, z_2 \in D\) is contained in a simply connected \(c\)--John domain which is included in \(D\). It is shown that \(D\) is a John domain iff it has the John decomposition property. In [\textit{O. Martio}, Rev. Roum. Math. Pures Appl. 33, No.1-2, 107-112 (1988; Zbl 0652.30012)] bilipschitz balls are used instead of John domains to prove a similar result. The authors also show that a removal of a finite number of points does not destroy the John property.
scientific article; zbMATH DE number 5354748