Clarkson-McCarthy inequality on a locally compact group
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| Format: | Preprint |
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2025
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| _version_ | 1866917937369055232 |
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| author | Kečkić, Dragoljub J. Lazović, Zlatko |
| author_facet | Kečkić, Dragoljub J. Lazović, Zlatko |
| contents | Let $G$ be a locally compact group, $μ$ its Haar measure, $\hat G$ its Pontryagin dual and $ν$ the dual measure. For any $A_θ\in L^1(G;\mathcal C_p)\cap L^2(G;\mathcal C_p)$, ($\mathcal C_p$ is Schatten ideal), and $1<p\le2$ we prove $$\int_{\hat G}\left\|\int_GA_θ\overline{ξ(θ)}\,\mathrm dμ(θ)\right\|_p^q\,\mathrm dν(ξ)\le
\left(\int_G\|A_θ\|_p^p\,\mathrm dμ(θ)\right)^{q/p}, $$ where $q=p/(p-1)$. This appears to be a generalization of some earlier obtained inequalities, including Clarkson-McCarthy inequalities (in the case $G=\mathbf Z_2$), and Hausdorff-Young inequality. Some corollaries are also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Clarkson-McCarthy inequality on a locally compact group Kečkić, Dragoljub J. Lazović, Zlatko Functional Analysis 47A30, 47B10, 43A25 Let $G$ be a locally compact group, $μ$ its Haar measure, $\hat G$ its Pontryagin dual and $ν$ the dual measure. For any $A_θ\in L^1(G;\mathcal C_p)\cap L^2(G;\mathcal C_p)$, ($\mathcal C_p$ is Schatten ideal), and $1<p\le2$ we prove $$\int_{\hat G}\left\|\int_GA_θ\overline{ξ(θ)}\,\mathrm dμ(θ)\right\|_p^q\,\mathrm dν(ξ)\le \left(\int_G\|A_θ\|_p^p\,\mathrm dμ(θ)\right)^{q/p}, $$ where $q=p/(p-1)$. This appears to be a generalization of some earlier obtained inequalities, including Clarkson-McCarthy inequalities (in the case $G=\mathbf Z_2$), and Hausdorff-Young inequality. Some corollaries are also given. |
| title | Clarkson-McCarthy inequality on a locally compact group |
| topic | Functional Analysis 47A30, 47B10, 43A25 |
| url | https://arxiv.org/abs/2502.19188 |