Characterization of John Domains via Weak Tangents

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 $$\partial D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> , 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 (Trans. Amer. Math. Soc. 369 (9), 6511–6536, 2017), 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.