Characterization of John domains via weak tangents

TL;DR

This paper characterizes simply connected John domains in the plane using weak tangents of their boundaries, providing a necessary and sufficient condition that enhances previous results and explores properties of these tangents.

Contribution

It establishes a new characterization of John domains via weak tangents, improving upon prior necessary conditions by providing a full equivalence.

Findings

A bounded simply connected domain is a John domain iff all weak tangents of its boundary satisfy a related John condition.

The paper extends understanding of boundary behavior in John domains through weak tangents.

Several properties of weak tangents of John domains are also established.

Abstract

We characterize simply connected John domains in the plane with the aid of weak tangents of the boundary. Specifically, we prove that a bounded simply connected domain $D$ is a John domain if and only if, for every weak tangent $Y$ of $\partial D$, every connected component of the complement of $Y$ that ``originates" from $D$ is a John domain, not necessarily with uniform constants. Our main theorem improves a result of Kinneberg (arXiv:1507.04698), who obtains a necessary condition for a John domain in terms of weak tangents but not a sufficient one. We also establish several properties of weak tangents of John domains.