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<p>In this paper, we investigate the invariance of John domains under quasisymmetric maps. We establish that if a homeomorphism is quasisymmetric relative to the boundary of the domain then it maps a length John domain to a diameter John domain. Also, we study the invariance of the class of distance John domains under quasisymmetric maps relative to the boundary. Moreover, we prove a necessary and sufficient condition for a diameter John domain to be length John and thereby prove that if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f:G\rightarrow G'\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> is (<i>M</i>,&#xa0;<i>C</i>)-CQH map, where <i>G</i> is a John domain, and the map extends to the boundary such that the extension is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\eta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-QS relative to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\delta G\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(G'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>G</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> is a John domain.</p>

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On Invariance of John Domains Under Quasisymmetric Mappings

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Vasudevarao Allu,
  • Alan P. Jose

摘要

In this paper, we investigate the invariance of John domains under quasisymmetric maps. We establish that if a homeomorphism is quasisymmetric relative to the boundary of the domain then it maps a length John domain to a diameter John domain. Also, we study the invariance of the class of distance John domains under quasisymmetric maps relative to the boundary. Moreover, we prove a necessary and sufficient condition for a diameter John domain to be length John and thereby prove that if \(f:G\rightarrow G'\) f : G → G ′ is (M, C)-CQH map, where G is a John domain, and the map extends to the boundary such that the extension is \(\eta \) η -QS relative to \(\delta G\) δ G then \(G'\) G ′ is a John domain.