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, $$\sum _{k\in \hat{G}}{\Vert {\int }_G\overline{k(\theta )}{A}_{\theta }\mathrm{d}\theta \Vert }_p^p\le {\int }_G\Vert A_{\theta }{\Vert }_p^p\mathrm{d}\theta ,p\ge 2$$ $$\sum _{k\in \hat{G}}{\Vert {\int }_G\overline{k(\theta )}{A}_{\theta }\mathrm{d}\theta \Vert }_p^p\le {\left({\int }_G\Vert A_{\theta }{\Vert }_p^q\mathrm{d}\theta \right)}^{p/q},p\ge 2.$$ $$\sum _{k\in \hat{G}}{\Vert {\int }_G\overline{k(\theta )}{A}_{\theta }\mathrm{d}\theta \Vert }_p^q\le {\left({\int }_G\Vert A_{\theta }{\Vert }_p^p\mathrm{d}\theta \right)}^{q/p},p\le 2.$$ 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.