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Abstract <p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(D\)</EquationSource> <!--RusMath2570053Kytmanov-m1--> </InlineEquation> be a bounded domain in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathbb{C}}^{n}}\)</EquationSource> <!--RusMath2570053Kytmanov-m2--> </InlineEquation> (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(n &gt; 1\)</EquationSource> <!--RusMath2570053Kytmanov-m3--> </InlineEquation>) with a real analytic connected boundary <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\partial D = \Gamma \)</EquationSource> <!--RusMath2570053Kytmanov-m4--> </InlineEquation>. The Bochner–Martinelli integral (integral operator) <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(M(f)\)</EquationSource> <!--RusMath2570053Kytmanov-m5--> </InlineEquation> is considered for real analytic functions <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource> <!--RusMath2570053Kytmanov-m6--> </InlineEquation> on <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <!--RusMath2570053Kytmanov-m7--> </InlineEquation>. It is shown that the integral <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(M(f)\)</EquationSource> <!--RusMath2570053Kytmanov-m8--> </InlineEquation> is real analytic up to <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <!--RusMath2570053Kytmanov-m9--> </InlineEquation>. Iterations of the Bochner–Martinelli integral <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq10.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\({{M}^{k}}(f)\)</EquationSource> <!--RusMath2570053Kytmanov-m10--> </InlineEquation> are considered. It is proved that they converge to a function holomorphic in <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq11.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\overline D \)</EquationSource> <!--RusMath2570053Kytmanov-m11--> </InlineEquation> as <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(k \to \infty \)</EquationSource> <!--RusMath2570053Kytmanov-m12--> </InlineEquation>. The Bochner–Martinelli transform <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(M(T)(z)\)</EquationSource> <!--RusMath2570053Kytmanov-m13--> </InlineEquation> is defined for analytical functionals <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq14.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\)</EquationSource> <!--RusMath2570053Kytmanov-m14--> </InlineEquation>. It is proved that the iterations of <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq15.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\({{M}^{k}}(T)(z)\)</EquationSource> <!--RusMath2570053Kytmanov-m15--> </InlineEquation> converge weakly to a <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq16.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(CR\)</EquationSource> <!--RusMath2570053Kytmanov-m16--> </InlineEquation>-functional as <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10510_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(k \to \infty \)</EquationSource> <!--RusMath2570053Kytmanov-m17--> </InlineEquation>.</p>

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The Bochner–Martinelli Integral Operator for Real Analytic Functions

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • A. M. Kytmanov,
  • S. G. Myslivets

摘要

Abstract

Let \(D\) be a bounded domain in \({{\mathbb{C}}^{n}}\) ( \(n > 1\) ) with a real analytic connected boundary \(\partial D = \Gamma \) . The Bochner–Martinelli integral (integral operator) \(M(f)\) is considered for real analytic functions \(f\) on \(\Gamma \) . It is shown that the integral \(M(f)\) is real analytic up to \(\Gamma \) . Iterations of the Bochner–Martinelli integral \({{M}^{k}}(f)\) are considered. It is proved that they converge to a function holomorphic in \(\overline D \) as \(k \to \infty \) . The Bochner–Martinelli transform \(M(T)(z)\) is defined for analytical functionals \(T\) . It is proved that the iterations of \({{M}^{k}}(T)(z)\) converge weakly to a \(CR\) -functional as \(k \to \infty \) .