## Abstract

Let $G$ be a compact Abelian group, let $\mu $ be the corresponding Haar measure, and let $\stackrel{\u02c6}{G}$ 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 $${\sum}_{k\in \stackrel{\u02c6}{G}}\Vert {\int}_{G}\overline{k\left(\theta \right)}{A}_{\theta}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\theta {\Vert}_{p}^{p}\le {\int}_{G}\Vert {A}_{\theta}{\Vert}_{p}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\theta ,\phantom{\rule{1em}{0ex}}p\ge 2,\phantom{\rule{0ex}{0ex}}{\sum}_{k\in \stackrel{\u02c6}{G}}\Vert {\int}_{G}\overline{k\left(\theta \right)}{A}_{\theta}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\theta {\Vert}_{p}^{p}\le ({\int}_{G}\Vert {A}_{\theta}{\Vert}_{p}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\theta {)}^{p/q},\phantom{\rule{1em}{0ex}}p\ge 2,\phantom{\rule{0ex}{0ex}}{\sum}_{k\in \stackrel{\u02c6}{G}}\Vert {\int}_{G}\overline{k\left(\theta \right)}{A}_{\theta}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\theta {\Vert}_{p}^{q}\le ({\int}_{G}\Vert {A}_{\theta}{\Vert}_{p}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\theta {)}^{q/p},\phantom{\rule{1em}{0ex}}p\le 2.$$ 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.

## Citation

Dragoljub J. Kečkić. "Continuous generalization of Clarkson–McCarthy inequalities." Banach J. Math. Anal. 13 (1) 26 - 46, January 2019. https://doi.org/10.1215/17358787-2018-0014

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