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<p>In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^1(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^{1,\infty }(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>∞</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The methods we use are entirely elementary, and rely only on pigeonholing and stationary phase considerations.</p>

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An approach to endpoint problems in oscillatory singular integrals

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月23日

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  • Alex Iosevich,
  • Ben Krause,
  • Hamed Mousavi

摘要

In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from \(L^1(\mathbb {R})\) L 1 ( R ) to \(L^{1,\infty }(\mathbb {R})\) L 1 , ∞ ( R ) . The methods we use are entirely elementary, and rely only on pigeonholing and stationary phase considerations.