Kac型紧量子群上的Hardy-Littlewood不等式
Hardy-Littlewood inequalities on compact quantum groups of Kac type
摘要：
$\mathbb{T}$上的Hardy-Littlewood不等式比较了函数的$L^p$ -范数与其傅立叶系数的加权$\ell^p$ -范数。最近研究了紧凑的齐性空间的方法，我们研究了紧凑的量子群的框架中的自然类似物。特别地，在约化群$C^*$ -代数和自由量子群的情况下，我们利用量子群的内在信息，如增长率和快速衰减性质，建立了显式的$L^p-\ell^p$不等式.此外，我们还证明了一类不等式的尖锐性，包括紧李群的$C(G)$，多项式增长离散群的$C_r^*(G)$和自由量子群的$O_N^+$，$S_N^+$ .
英文摘要：
The Hardy-Littlewood inequality on $\mathbb{T}$ compares the $L^p$-norm of a function with a weighted $\ell^p$-norm of its Fourier coefficients. The approach has recently been studied for compact homogeneous spaces and we study a natural analogue in the framework of compact quantum groups. Especially, in the case of the reduced group $C^*$-algebras and free quantum groups, we establish explicit $L^p-\ell^p$ inequalities through inherent information of underlying quantum group, such as growth rate and rapid decay property. Moreover, we show sharpness of the inequalities in a large class, including $C(G)$ with compact Lie group, $C_r^*(G)$ with polynomially growing discrete group and free quantum groups $O_N^+$, $S_N^+$.
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