Hardy-Littlewood inequalities on compact quantum groups of Kac type

Abstract

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 $C^{\ast }$-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, $C_r^{\ast }(G)$ with polynomially growing discrete group and free quantum groups $O_N^{+}$, $S_N^{+}$.

22 pages

References in corpus (2)

• Lacunary Fourier series for compact quantum groups
• Hypercontractivity of heat semigroups on free quantum groups

Cited by in corpus (5)

• Smooth dense subalgebras and Fourier multipliers on compact quantum groups
• Property RD and hypercontractivity for orthogonal free quantum groups
• $L^p-L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces
• Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications
• Entropic uncertainty relations under localizations on discrete quantum groups