Continuous generalization of Clarkson-McCarthy inequalities

Abstract

Let $G$ be a compact abelian group, let $\mu$ be the corresponding Haar measure, and let $\hat G$ be the Pontryagin dual of $G$. Further, let $C_p$ denote the Schatten class of operators on some separable infinite dimensional Hilbert space, and let $L^p(G;C_p)$ denote the corresponding Bochner space. If $G\ni\theta\mapsto A_\theta$ is the mapping belonging to $L^p(G;C_p)$ then, $$\sum_{k\in\hat G}\left\|\int_G\overline{k(\theta)}A_\theta\,\mathrm{d}\theta\right\|_p^p\le\int_G\|A_\theta\|_p^p\,\mathrm{d}\theta,\qquad p\ge2$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(\theta)}A_\theta\,\mathrm{d}\theta\right\|_p^p\le\left(\int_G\|A_\theta\|_p^q\,\mathrm{d}\theta\right)^{p/q},\qquad p\ge2.$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(\theta)}A_\theta\,\mathrm{d}\theta\right\|_p^q\le\left(\int_G\|A_\theta\|_p^p\,\mathrm{d}\theta\right)^{q/p},\qquad p\le2.$$ If $G$ is a finite group, the previous comprises several earlier obtained generalizations of Clarkson-McCarthy inequalities (e.g. $G=\mathbf{Z}_n$ or $G=\mathbf{Z}_2^n$), as well as the original inequalities, for $G=\mathbf{Z}_2$. Other related inequalities are also obtained.