## Banach Journal of Mathematical Analysis

### Continuous generalization of Clarkson–McCarthy inequalities

Dragoljub J. Kečkić

#### Abstract

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 $\begin{eqnarray*}\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{p}&\le&\int_{G}\Vert A_{\theta}\Vert _{p}^{p}\,\mathrm{d}\theta,\quad p\ge2,\\\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{p}&\le&(\int_{G}\Vert A_{\theta}\Vert _{p}^{q}\,\mathrm{d}\theta )^{p/q},\quad p\ge2,\\\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{q}&\le&(\int_{G}\Vert A_{\theta}\Vert _{p}^{p}\,\mathrm{d}\theta )^{q/p},\quad p\le2.\end{eqnarray*}$ 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.

#### Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 26-46.

Dates
Accepted: 19 April 2018
First available in Project Euclid: 28 September 2018

https://projecteuclid.org/euclid.bjma/1538121808

Digital Object Identifier
doi:10.1215/17358787-2018-0014

Mathematical Reviews number (MathSciNet)
MR3894063

Zentralblatt MATH identifier
07002030

#### Citation

Kečkić, Dragoljub J. Continuous generalization of Clarkson–McCarthy inequalities. Banach J. Math. Anal. 13 (2019), no. 1, 26--46. doi:10.1215/17358787-2018-0014. https://projecteuclid.org/euclid.bjma/1538121808

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