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<p>The purpose of this paper is to establish local regularity of the solution operator to the Kohn–Laplace equation, called the complex Green operator, on abstract CR manifolds of hypersurface type. For a cut-off function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2024_10143_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>, we introduce the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2024_10143_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-superlogarithmic property, a potential theoretical condition on CR manifolds. We prove that if the given datum is smooth on an open set containing the support of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2024_10143_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> then the solution is smooth on the interior of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2024_10143_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{x\in M:\sigma (x)=1\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>M</mi> <mo>:</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we also study the smoothness of the integral kernel of the complex Green operator.</p>

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The Kohn–Laplace Equation on Abstract CR Manifolds: Local Regularity

  • 期刊论文

  • 发表日期：

    2025年1月29日

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  • Tran Vu Khanh

摘要

The purpose of this paper is to establish local regularity of the solution operator to the Kohn–Laplace equation, called the complex Green operator, on abstract CR manifolds of hypersurface type. For a cut-off function \(\sigma \) σ , we introduce the \(\sigma \) σ -superlogarithmic property, a potential theoretical condition on CR manifolds. We prove that if the given datum is smooth on an open set containing the support of \(\sigma \) σ then the solution is smooth on the interior of \(\{x\in M:\sigma (x)=1\}\) { x ∈ M : σ ( x ) = 1 } . Furthermore, we also study the smoothness of the integral kernel of the complex Green operator.