Clarkson-McCarthy inequality on a locally compact group

Let $G$ be a locally compact group, $\mu$ its Haar measure, $\widehat{G}$ 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 $${\int }_{\widehat{G}}{\parallel {\int }_G{A}_{\theta }\overline{\xi (\theta )}\mathrm{d}\mu (\theta )\parallel }_p^q\mathrm{d}\nu (\xi )\le {({\int }_G\mathrm{\parallel }{A}_{\theta }{\mathrm{\parallel }}_p^p\mathrm{d}\mu (\theta ))}^{q\mathrm{/}p},$$ where $q=p\mathrm{/}(p-1)$. 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.