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Abstract <p> Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3009_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(D\subset \mathbb{R}^d\)</EquationSource> </InlineEquation> be a bounded Lipschitz domain, let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3009_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega\)</EquationSource> </InlineEquation> be a modulus of continuity of a high order of smoothness, and let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3009_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\)</EquationSource> </InlineEquation> be a Calderón–Zygmund convolution operator with even kernel. Using the recent T(P) boundedness criterion found by the author with E. Doubtsov, we prove that the operator <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3009_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\)</EquationSource> </InlineEquation> is bounded in the Zygmund space <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3009_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{C}_{\omega}(D)\)</EquationSource> </InlineEquation> if the smoothness of the boundary of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3009_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(D\)</EquationSource> </InlineEquation> is by one higher than that of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3009_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{C}_{\omega}(D)\)</EquationSource> </InlineEquation>. The proof is based on estimates for the potentials with Calderón–Zygmund kernels of the characteristic function of a domain with a polynomial boundary. </p>

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Regularity of Calderón–Zygmund Operators in Domains

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • A. V. Vasin

摘要

Abstract

Let \(D\subset \mathbb{R}^d\) be a bounded Lipschitz domain, let \(\omega\) be a modulus of continuity of a high order of smoothness, and let \(T\) be a Calderón–Zygmund convolution operator with even kernel. Using the recent T(P) boundedness criterion found by the author with E. Doubtsov, we prove that the operator \(T\) is bounded in the Zygmund space \(\mathcal{C}_{\omega}(D)\) if the smoothness of the boundary of \(D\) is by one higher than that of \(\mathcal{C}_{\omega}(D)\) . The proof is based on estimates for the potentials with Calderón–Zygmund kernels of the characteristic function of a domain with a polynomial boundary.