Clarkson-McCarthy inequality on a locally compact group

Abstract

Let $G$ be a locally compact group, $\mu$ its Haar measure, $\hat{G}$ its Pontryagin dual and $\nu$ the dual measure. For any $A_{\theta }\in L^1(G;{\mathcal{C}}_p)\cap L^2(G;{\mathcal{C}}_p)$, (${\mathcal{C}}_p$ is Schatten ideal), and $1<p\le 2$ we prove $${\int }_{\hat{G}}{\Vert {\int }_G{A}_{\theta }\overline{\xi (\theta )}\mathrm{d}\mu (\theta )\Vert }_p^q\mathrm{d}\nu (\xi )\le {\left({\int }_G\Vert A_{\theta }{\Vert }_p^p\mathrm{d}\mu (\theta )\right)}^{q/p},$$ where $q=p/(p-1)$. This appears to be a generalization of some earlier obtained inequalities, including Clarkson-McCarthy inequalities (in the case $G={\mathbf{Z}}_2$), and Hausdorff-Young inequality. Some corollaries are also given.

11 pages