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<p>We obtain sharp rotation bounds for homeomorphisms <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f:\mathbb {C}\rightarrow \mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> whose distortion is in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^p_{loc}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mrow> <mi mathvariant="italic">loc</mi> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and whose inverse have controlled modulus of continuity. The motivation to study this class of maps comes from so-called Yudovich solutions to planar Euler equations. Furthermore, we present examples proving sharpness in a strong sense, thereby settling the borderline case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(p=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in [<CitationRef CitationID="CR5">5</CitationRef>, Theorem 3].</p>

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Pointwise rotation for homeomorphisms with integrable distortion and controlled compression

  • 期刊论文

  • 发表日期：

    2026年6月26日

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  • Lauri Hitruhin,
  • Banhirup Sengupta

摘要

We obtain sharp rotation bounds for homeomorphisms \(f:\mathbb {C}\rightarrow \mathbb {C}\) f : C → C whose distortion is in \(L^p_{loc}\) L loc p , \(p\ge 1\) p ≥ 1 , and whose inverse have controlled modulus of continuity. The motivation to study this class of maps comes from so-called Yudovich solutions to planar Euler equations. Furthermore, we present examples proving sharpness in a strong sense, thereby settling the borderline case \(p=1\) p = 1 in [5, Theorem 3].