Continuous generalization of Clarkson-McCarthy inequalities

TL;DR

This paper extends Clarkson-McCarthy inequalities to continuous settings over compact abelian groups, providing new bounds for Schatten class operators with applications to finite groups and related inequalities.

Contribution

It introduces a continuous generalization of Clarkson-McCarthy inequalities for operators over compact abelian groups, unifying and extending previous finite group results.

Findings

Derived new inequalities for Schatten class operators over compact abelian groups.

Unified finite and infinite group cases under a common framework.

Provided related inequalities and special cases for finite groups.

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, $$\sum_{k\in\hat G}\left\|\int_G\overline{k(\theta)}A_\theta\,\mathrm{d}\theta\right\|_p^p\le\int_G\|A_\theta\|_p^p\,\mathrm{d}\theta,\qquad p\ge2$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(\theta)}A_\theta\,\mathrm{d}\theta\right\|_p^p\le\left(\int_G\|A_\theta\|_p^q\,\mathrm{d}\theta\right)^{p/q},\qquad p\ge2.$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(\theta)}A_\theta\,\mathrm{d}\theta\right\|_p^q\le\left(\int_G\|A_\theta\|_p^p\,\mathrm{d}\theta\right)^{q/p},\qquad p\le2.$$ 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.