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<p>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 <i>D</i> is a John domain if and only if, for every weak tangent <i>Y</i> of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\partial D\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>D</mi> </mrow> </math></EquationSource> </InlineEquation>, every connected component of the complement of <i>Y</i> that “originates” from <i>D</i> is a John domain, not necessarily with uniform constants. Our main theorem improves a result of Kinneberg (Trans. Amer. Math. Soc. <b>369</b>(9), 6511–6536, <CitationRef CitationID="CR10">2017</CitationRef>), 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.</p>

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Characterization of John Domains via Weak Tangents

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  • 发表日期：

    2026年2月21日

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  • Christina Karafyllia

摘要

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 \(\partial D\) ∂ 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 (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.