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<p>Weighted <i>k</i>-holomorphic functions constitute a new class of functions, which is a further generalization of weighted monogenic functions. First, we use the parametric equation of the sphere given under non-Euclidean distance to find the relationship between <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1698_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(d\sigma _{x}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mi>x</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> under non-Euclidean distance and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1698_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(d\mu _{r}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <msub> <mi>μ</mi> <mi>r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> under Euclidean distance. Then, based on the characteristics of the high-order kernel function of the weighted <i>k</i>-holomorphic functions, the Borel-Pompeiu formula and a Cauchy integral formula of weighted <i>k</i>-holomorphic functions are given. Finally, on the basis of Cauchy integral formula, the Cauchy integral formula outside a ball and the boundary properties of Cauchy type integral operators of weighted <i>k</i>-holomorphic functions are studied.</p>

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The Borel-Pompeiu Formula of Weighted k-Holomorphic Functions and its Related Properties

  • 期刊论文

  • 发表日期：

    2025年5月17日

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  • Xin Jiang,
  • Liping Wang,
  • Ying Li,
  • Chenyi He

摘要

Weighted k-holomorphic functions constitute a new class of functions, which is a further generalization of weighted monogenic functions. First, we use the parametric equation of the sphere given under non-Euclidean distance to find the relationship between \(d\sigma _{x}\) d σ x under non-Euclidean distance and \(d\mu _{r}\) d μ r under Euclidean distance. Then, based on the characteristics of the high-order kernel function of the weighted k-holomorphic functions, the Borel-Pompeiu formula and a Cauchy integral formula of weighted k-holomorphic functions are given. Finally, on the basis of Cauchy integral formula, the Cauchy integral formula outside a ball and the boundary properties of Cauchy type integral operators of weighted k-holomorphic functions are studied.