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Continuous generalization of Clarkson-McCarthy inequalities. (English) Zbl 07002030

Summary: Let \(G\) be a compact Abelian group, let \(\mu\) be the corresponding Haar measure, and let \(\hat{G}\) be the Pontryagin dual of \(G\). Furthermore, let \(\mathcal{C}_{p}\) denote the Schatten class of operators on some separable infinite-dimensional Hilbert space, and let \(L^{p}(G;\mathcal{C}_{p})\) denote the corresponding Bochner space. If \(G\ni\theta\mapsto A_{\theta}\) is the mapping belonging to \(L^{p}(G;\mathcal{C}_{p})\), then \[ \begin{aligned}\sum_{k\in\hat{G}}\| \int_{G}\overline{k(\theta)}A_{\theta}\,\text{d}\theta\|_{p}^{p}&\leq\int_{G}\| A_{\theta}\|_{p}^{p}\,\text{d}\theta,\quad p\geq2,\\ \sum_{k\in\hat{G}}\| \int_{G}\overline{k(\theta)}A_{\theta}\,\text{d}\theta\|_{p}^{p}&\leq(\int_{G}\| A_{\theta}\|_{p}^{q}\,\text{d}\theta )^{p/q},\quad p\geq2,\\ \sum_{k\in\hat{G}}\| \int_{G}\overline{k(\theta)}A_{\theta}\,\text{d}\theta\|_{p}^{q}&\leq(\int_{G}\| A_{\theta}\|_{p}^{p}\,\text{d}\theta )^{q/p},\quad p\leq2.\end{aligned} \] If \(G\) is a finite group, then the previous equations comprise several generalizations of Clarkson-McCarthy inequalities obtained earlier (e.g., \(G=\mathbf{Z}_{n}\) or \(G=\mathbf{Z}_{2}^{n}\)), as well as the original inequalities, for \(G=\mathbf{Z}_{2}\). We also obtain other related inequalities.

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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References:

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