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<p>In this paper, we apply a direct method instead of a limit approach, for proving the Levin–Cochran–Lee inequalities. First, we state and prove Levin–Cochran–Lee type inequalities on a homogeneous group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2024_1143_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {G}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">G</mi> </math></EquationSource> </InlineEquation> with parameters <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2024_1143_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">\(0&lt;p\le q&lt;\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>p</mi> <mo>≤</mo> <mi>q</mi> <mo>&lt;</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, for the case <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2024_1143_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(p=q\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>, we prove the sharp inequalities with power weights and derive some other new inequalities.</p>

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Levin–Cochran–Lee inequalities and best constants on homogeneous groups

  • 期刊论文

  • 发表日期：

    2025年1月13日

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  • Michael Ruzhansky,
  • Markos Fisseha Yimer

摘要

In this paper, we apply a direct method instead of a limit approach, for proving the Levin–Cochran–Lee inequalities. First, we state and prove Levin–Cochran–Lee type inequalities on a homogeneous group \(\mathbb {G}\) G with parameters \(0<p\le q<\infty \) 0 < p ≤ q < ∞ . Furthermore, for the case \(p=q\) p = q , we prove the sharp inequalities with power weights and derive some other new inequalities.