Smoothness of a conformal mapping on a subset of the boundary

Abstract:

A conformal mapping $f$ of the unit disk onto a Jordan domain $G$ is considered. The boundary of $G$ has the following structure. Another Jordan domain $H$ is fixed whose boundary has Hölder smoothness $a>1$, and a countable family of open arcs dense in the boundary is specified. $G$ is obtained by replacement of each of these distinguished arcs with a Hölder arc of smoothness $b$, $1<b<a$, having the same end-points. Thus, $G$ has Hölder smoothness $b$. It is shown that if the lengths of the distinguished arcs decay sufficiently fast (depending on $a$ and $b$), the function $f$ still has Hölder smoothness $a$ on a set of positive measure on the unit circle. The numbers $a$ and $b$ are assumed to be nonintegers.