Hardy-Littlewood inequalities and Fourier multipliers on SU(2)

TL;DR

This paper extends Hardy-Littlewood inequalities to the noncommutative setting of SU(2), providing bounds for Fourier multipliers' norms and advancing harmonic analysis on noncommutative groups.

Contribution

It introduces a noncommutative Hardy-Littlewood inequality for SU(2) and derives new bounds for Fourier multipliers' $L^p-L^q$ norms in this setting.

Findings

Established noncommutative Hardy-Littlewood inequalities for SU(2)

Derived lower bounds for Fourier multiplier norms on SU(2)

Provided upper bounds analogous to classical results on the torus

Abstract

In this paper we prove a noncommutative version of Hardy-Littlewood inequalities relating a function and its Fourier coefficients on the group $SU(2)$. As a consequence, we use it to obtain lower bounds for the $L^p-L^q$ norms of Fourier multipliers on the group $SU(2)$, for $1 < p \leq 2 \leq q < 1$. In addition, we give upper bounds of a similar form, analogous to the known results on the torus, but now in the noncommutative setting of $SU(2)$.