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<p>This paper investigates the boundedness properties of the Cauchy singular integral operator on chord-arc curves in the complex plane. By completing and expanding the approach stemming from prior work by Melnikov and Verdera, we bridge between the uniform local estimates obtained by the respective authors and the global estimates that imply the boundedness of the truncated Cauchy operators on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> spaces. In addition, we provide proofs for end-point results, such as uniform boundedness of the truncated Cauchy operators from the Hardy space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>, and from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> into the John-Nirenberg space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textrm{BMO}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>BMO</mtext> </math></EquationSource> </InlineEquation>. The results discussed here also accommodate the principal value Cauchy singular integral operator and the maximal Cauchy operator, considered directly on the chord-arc curve.</p>

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ON THE BOUNDEDNESS OF THE CAUCHY OPERATOR ON CHORD-ARC CURVES

  • 期刊论文

  • 发表日期：

    2026年2月28日

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  • Dorina Mitrea,
  • Maricela Ramirez

摘要

This paper investigates the boundedness properties of the Cauchy singular integral operator on chord-arc curves in the complex plane. By completing and expanding the approach stemming from prior work by Melnikov and Verdera, we bridge between the uniform local estimates obtained by the respective authors and the global estimates that imply the boundedness of the truncated Cauchy operators on \(L^p\) L p spaces. In addition, we provide proofs for end-point results, such as uniform boundedness of the truncated Cauchy operators from the Hardy space \(H^1\) H 1 into \(L^1\) L 1 , and from \(L^\infty \) L ∞ into the John-Nirenberg space \(\textrm{BMO}\) BMO . The results discussed here also accommodate the principal value Cauchy singular integral operator and the maximal Cauchy operator, considered directly on the chord-arc curve.