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<p>We provide new characterizations of the <i>BMO</i>-Sobolev space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3215_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\(I_{\alpha }(BMO)\)</EquationSource> </InlineEquation> for the range <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3215_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(0&lt; \alpha &lt;2\)</EquationSource> </InlineEquation>. When <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3215_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(0&lt; \alpha &lt;1\)</EquationSource> </InlineEquation>, our characterizations are in terms of square functions measuring multiscale approximation of constants, and when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3215_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(1 \le \alpha &lt;2\)</EquationSource> </InlineEquation> our characterizations are in terms of square functions measuring multiscale approximation by linear functions.</p>

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Square functions controlling smoothness and higher-order quantitative rectifiability

  • 期刊论文

  • 发表日期：

    2025年8月18日

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  • John Hoffman

摘要

We provide new characterizations of the BMO-Sobolev space \(I_{\alpha }(BMO)\) for the range \(0< \alpha <2\) . When \(0< \alpha <1\) , our characterizations are in terms of square functions measuring multiscale approximation of constants, and when \(1 \le \alpha <2\) our characterizations are in terms of square functions measuring multiscale approximation by linear functions.