Boundary regularity of domains satisfying a wedge condition

Abstract

Certain geometrical properties of Jordan domains arise in the study of boundary behavior of Riemann maps. The property investigated here is the following wedge condition: there is a fixed wedge such that for each boundary point of the domain the wedge can be moved rigidly to have its vertex at the boundary point and its interior in the domain. Geometrical consequences of this property include that the boundary is rectifiable and has bounded arc length-interior distance ratio. It follows from work of Pommerenke that the domain is a Smirnov domain and that the derivative of a conformal mapping of the unit disk onto the domain satisfies the Muckenhoupt A condition.