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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> be a hypocycloid-type Jordan domain, and <i>f</i> be a conformal mapping from the unit disk onto <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Omega .\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We extend <i>f</i> to the whole plane as a homeomorphism with exponentially integrable distortion. The sharp integrability exponent is related to the polynomial degrees of cusps on the boundary of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Omega .\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Furthermore, we generalize these results to the case when <i>f</i> is quasiconformal.</p>

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Homeomorphic Extensions of Conformal Mappings onto Hypocycloid-Type Domains

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Haiqing Xu

摘要

Let \(\Omega \) Ω be a hypocycloid-type Jordan domain, and f be a conformal mapping from the unit disk onto \(\Omega .\) Ω . We extend f to the whole plane as a homeomorphism with exponentially integrable distortion. The sharp integrability exponent is related to the polynomial degrees of cusps on the boundary of \(\Omega .\) Ω . Furthermore, we generalize these results to the case when f is quasiconformal.