Hardy–Littlewood–Paley inequalities and Fourier multipliers on SU(2)
Volume 234 / 2016
Studia Mathematica 234 (2016), 1-29
MSC: Primary 43A85, 43A15; Secondary 35S05.
DOI: 10.4064/sm8106-4-2016
Published online: 17 June 2016
Abstract
We prove noncommutative versions of Hardy–Littlewood and Paley inequalities relating a function and its Fourier coefficients on the group ${\rm SU(2)}$. We use it to obtain lower bounds for the $L^p$-$L^q$ norms of Fourier multipliers on ${\rm SU(2)}$ for $1 \lt p\leq 2\leq q \lt \infty $. 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 ${\rm SU(2)}$.
