Clarkson-McCarthy inequality on a locally compact group
Dragoljub J. Kečkić, Zlatko Lazović
TL;DR
The paper generalizes Clarkson–McCarthy inequalities to locally compact abelian groups by leveraging a Parseval identity for Bochner spaces and complex interpolation. For $1\le p\le 2$ and $A_\theta\in L^1(G;C_p)\cap L^2(G;C_p)$, it defines $B_\xi=\int_G A_\theta\overline{\xi(\theta)} d\mu(\theta)$ and proves $\int_{\hat G} \|B_\xi\|_p^q d\nu(\xi) \le (\int_G \|A_\theta\|_p^p d\mu(\theta))^{q/p}$ with $q=p/(p-1)$. This unifies Clarkson–McCarthy and Hausdorff–Young inequalities in the setting of locally compact groups, and yields corollaries for specific groups (e.g., $\mathbb{Z}_2$, $\mathbb{R}^n$, $\mathbb{Q}_p^n$) as well as weighted Schatten classes. The approach combines a Bochner-Parseval identity with exact interpolation of Schatten classes, providing a flexible framework for noncommutative harmonic analysis and operator-norm inequalities on group contexts. The results enrich the toolkit for analyzing unitarily invariant norms under abstract Fourier transforms and pave the way for further applications in noncommutative harmonic analysis and quantum information theory.
Abstract
Let $G$ be a locally compact group, $μ$ its Haar measure, $\hat G$ its Pontryagin dual and $ν$ the dual measure. For any $A_θ\in L^1(G;\mathcal C_p)\cap L^2(G;\mathcal C_p)$, ($\mathcal C_p$ is Schatten ideal), and $1<p\le2$ we prove $$\int_{\hat G}\left\|\int_GA_θ\overline{ξ(θ)}\,\mathrm dμ(θ)\right\|_p^q\,\mathrm dν(ξ)\le \left(\int_G\|A_θ\|_p^p\,\mathrm dμ(θ)\right)^{q/p}, $$ 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.