One sided conformal collars and the reflection principle

TL;DR

The paper demonstrates that for Jordan curves with a certain type of conformal collar, all other collars from the same side share the same regularity properties, extending results beyond analytic curves.

Contribution

It establishes that the regularity of one-sided conformal collars is invariant under the reflection principle for general Jordan curves, not limited to analytic cases.

Findings

Conformal collar regularity is preserved across all collars from the same side.

The reflection principle applies to non-analytic Jordan curves.

Regularity class A^p is invariant for conformal collars from the same side.

Abstract

If a Jordan curve {\sigma} has a one-sided conformal collar with "good" properties, then, using the Reflection principle, we show that any other conformal collar of {\sigma} from the same side has the same "good" properties. A particular use of this fact concerns analytic Jordan curves, but in general the Jordan arcs we consider do not have to be analytic. We show that if an one-sided conformal collar bounded by {\sigma} is of class A^p, then any other collar bounded by {\sigma} and from the same side of {\sigma} is of class A^p.