Clarkson-McCarthy inequality on a locally compact group

TL;DR

This paper generalizes Clarkson-McCarthy and Hausdorff-Young inequalities to the setting of locally compact groups, involving Schatten class-valued functions and their Fourier transforms.

Contribution

It extends classical inequalities to a broader context of locally compact groups and Schatten class operators, providing new bounds and corollaries.

Findings

Established a generalized Clarkson-McCarthy inequality for locally compact groups.

Connected the inequality to Hausdorff-Young and classical Fourier analysis results.

Provided corollaries that extend the applicability of these inequalities.

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 $1<p\le2$ we prove $$\int_{\hat G}\left\|\int_GA_\theta\overline{\xi(\theta)}\,\mathrm d\mu(\theta)\right\|_p^q\,\mathrm d\nu(\xi)\le \left(\int_G\|A_\theta\|_p^p\,\mathrm d\mu(\theta)\right)^{q/p}, $$ 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.