<p>Let <i>G</i> be a locally compact group, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> its Haar measure, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\hat{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation> its Pontryagin dual and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> the dual measure. For any <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="193" /> </InlineMediaObject> <EquationSource Format="TEX">$A_\theta \in L^1(G;\mathcal {C}_p)\cap L^2(G;\mathcal {C}_p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>θ</mi> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>;</mo> <msub> <mi mathvariant="script">C</mi> <mi>p</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>;</mo> <msub> <mi mathvariant="script">C</mi> <mi>p</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, (<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">C</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> is Schatten ideal), and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p\le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> we prove <Equation ID="Equ19"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_Equ19.gif" Format="GIF" Height="51" Rendition="HTML" Resolution="72" Type="Linedraw" Width="379" /> </MediaObject> <EquationSource Format="TEX">$\int _{\hat{G}}\left\| \int _GA_\theta \overline{\xi (\theta )}\textrm{d}\mu (\theta )\right\| _p^q\textrm{d}\nu (\xi )\le \left( \int _G\Vert A_\theta \Vert _p^p\textrm{d}\mu (\theta )\right) ^{q/p}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mo>∫</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="false">^</mo> </mover> </msub> <msubsup> <mfenced close="∥" open="∥"> <msub> <mo>∫</mo> <mi>G</mi> </msub> <msub> <mi>A</mi> <mi>θ</mi> </msub> <mover> <mrow> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mtext>d</mtext> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mi>p</mi> <mi>q</mi> </msubsup> <mtext>d</mtext> <mi>ν</mi> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msup> <mfenced close=")" open="("> <msub> <mo>∫</mo> <mi>G</mi> </msub> <msubsup> <mrow> <mo stretchy="false">‖</mo> <msub> <mi>A</mi> <mi>θ</mi> </msub> <mo stretchy="false">‖</mo> </mrow> <mi>p</mi> <mi>p</mi> </msubsup> <mtext>d</mtext> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mrow> <mi>q</mi> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$q=p/(p-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mi>p</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This appears to be a generalization of some earlier obtained inequalities, including Clarkson-McCarthy inequalities (in the case <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1854_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$G=\textbf{Z}_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mi mathvariant="bold">Z</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>) and Hausdorff-Young inequality. Some corollaries are also given.</p>

Clarkson-McCarthy Inequality on a Locally Compact Group

期刊论文
发表日期：

2025年11月11日

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Dragoljub J. Kečkić,
Zlatko Lazović

摘要

Let G be a locally compact group, $\mu $ $μ$ its Haar measure, $\hat{G}$ $\hat{G}$ its Pontryagin dual and $\nu $ $ν$ the dual measure. For any $A_\theta \in L^1(G;\mathcal {C}_p)\cap L^2(G;\mathcal {C}_p)$ $A_{θ} \in L^{1} (G; C_{p}) \cap L^{2} (G; C_{p})$ , ( $\mathcal {C}_p$ $C_{p}$ is Schatten ideal), and $1<p\le 2$ $1 < p \leq 2$ we prove $\int _{\hat{G}}\left\| \int _GA_\theta \overline{\xi (\theta )}\textrm{d}\mu (\theta )\right\| _p^q\textrm{d}\nu (\xi )\le \left( \int _G\Vert A_\theta \Vert _p^p\textrm{d}\mu (\theta )\right) ^{q/p}, $ $\int_{\hat{G}} {∥\int_{G}, A_{θ}, \bar{ξ (θ)}, d, μ, (θ)∥}_{p}^{q} d ν (ξ) \leq {(\int_{G}, {‖ A_{θ} ‖}_{p}^{p}, d, μ, (θ))}^{q / p},$ where $q=p/(p-1)$ $q = p / (p - 1)$ . This appears to be a generalization of some earlier obtained inequalities, including Clarkson-McCarthy inequalities (in the case $G=\textbf{Z}_2$ $G = Z_{2}$ ) and Hausdorff-Young inequality. Some corollaries are also given.