Clarkson-McCarthy不等式的连续推广
Continuous generalization of Clarkson-McCarthy inequalities
摘要：
设$G$是紧交换群，$μ$是相应的Haar测度，$\hat G$是$G$的Pontryagin对偶.进一步地，令$C_p$表示可分无穷维Hilbert空间上的Schatten算子类，令$L^p(G;C_p)$表示相应的Bochner空间。如果$G\niθ\mapsto A_θ$是属于$L^p(G;C_p)$的映射，则$$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^p\le\int_G\|A_θ\|_p^p\,\mathrm{d}θ,\qquad p\ge2$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^p\le\left(\int_G\|A_θ\|_p^q\,\mathrm{d}θ\right)^{p/q},\qquad p\ge2.$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^q\le\left(\int_G\|A_θ\|_p^p\,\mathrm{d}θ\right)^{q/p},\qquad p\le2.$$如果$G$是有限群，则前一个包括几个早期获得的Clarkson-McCarthy不等式的推广（例如$G=\mathbf{Z}_n$或$G=\mathbf{Z}_2^n$），以及$G=\mathbf{Z}_2$的原始不等式。还得到了其它相关的不等式。
英文摘要：
Let $G$ be a compact abelian group, let $μ$ 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 $G\niθ\mapsto A_θ$ is the mapping belonging to $L^p(G;C_p)$ then, $$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^p\le\int_G\|A_θ\|_p^p\,\mathrm{d}θ,\qquad p\ge2$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^p\le\left(\int_G\|A_θ\|_p^q\,\mathrm{d}θ\right)^{p/q},\qquad p\ge2.$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^q\le\left(\int_G\|A_θ\|_p^p\,\mathrm{d}θ\right)^{q/p},\qquad p\le2.$$ 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.
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