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<p>In this paper, I prove that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13348_2025_496_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Vert \chi _{[a,b]}\Vert _{\mathcal {M}_p} &lt;\Vert H\Vert _{L^p \rightarrow L^p}\)</EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13348_2025_496_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\ne 2, p\in (1,\infty )\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13348_2025_496_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="136" /> </InlineMediaObject> <EquationSource Format="TEX">\(-\infty&lt;a&lt;b&lt;\infty \)</EquationSource> </InlineEquation>. This contradicts a previous conclusion in [<CitationRef CitationID="CR1">1</CitationRef>].</p>

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The segment multiplier operator has a smaller norm than the Hilbert transform for \(p\ne 2\)

  • 期刊论文

  • 发表日期：

    2025年10月22日

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  • Di Wu

摘要

In this paper, I prove that \(\Vert \chi _{[a,b]}\Vert _{\mathcal {M}_p} <\Vert H\Vert _{L^p \rightarrow L^p}\) for \(p\ne 2, p\in (1,\infty )\) and \(-\infty<a<b<\infty \) . This contradicts a previous conclusion in [1].