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<p>In this paper, we introduce Dunkl-Besov spaces using the Dunkl-type generalized translation operator and derive the embedding relations of these spaces by analyzing the changes of different indices and examining their endpoint values. In particular, we establish completeness, density, the min-max property and Sobolev-type inequalities for Dunkl-Besov spaces. In addition, we prove the boundedness of Hardy-Littlewood maximal operators on Dunkl-Besov spaces. Finally, we investigate the capacity theory related to Dunkl-Besov spaces, examine the measure-theoretic properties of this capacity, and obtain several equivalent definitions. As applications, we characterize the continuous embedding of Dunkl-Besov spaces into the spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2156_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{p}(\mu )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of the Dunkl Besov capacity or the associated variational functional of a nonnegative Radon measure <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2156_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>.</p>

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Besov spaces via generalized translation on the Dunkl setting and its capacity

  • 期刊论文

  • 发表日期：

    2025年8月20日

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  • M. Shi,
  • N. Zhao,
  • Yu Liu

摘要

In this paper, we introduce Dunkl-Besov spaces using the Dunkl-type generalized translation operator and derive the embedding relations of these spaces by analyzing the changes of different indices and examining their endpoint values. In particular, we establish completeness, density, the min-max property and Sobolev-type inequalities for Dunkl-Besov spaces. In addition, we prove the boundedness of Hardy-Littlewood maximal operators on Dunkl-Besov spaces. Finally, we investigate the capacity theory related to Dunkl-Besov spaces, examine the measure-theoretic properties of this capacity, and obtain several equivalent definitions. As applications, we characterize the continuous embedding of Dunkl-Besov spaces into the spaces \(L^{p}(\mu )\) L p ( μ ) in terms of the Dunkl Besov capacity or the associated variational functional of a nonnegative Radon measure \(\mu \) μ .