Continuous generalization of Clarkson–McCarthy inequalities

Dragoljub J. Kečkić

doi:10.1215/17358787-2018-0014

January 2019 Continuous generalization of Clarkson–McCarthy inequalities

Dragoljub J. Kečkić

Banach J. Math. Anal. 13(1): 26-46 (January 2019). DOI: 10.1215/17358787-2018-0014

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.

Citation

Download 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