Multiplier transformations on compact Lie groups and algebras

Abstract:

Let G be a semisimple compact Lie group and $Tf = \sum \phi (m){d_{m \chi m}} \ast f$ a bi-invariant operator on ${L^2}(G)$, where ${\chi _m}$ and ${d_m}$ are the characters and dimensions of the irreducible representations of G, which are indexed by a lattice of points m in the Lie algebra $\mathfrak {G}$ in a natural way. If $\Phi$ is a bounded ad-invariant function on $\mathfrak {G}$ and \begin{equation}\tag {$\ast $} \phi {\text {(}}m{\text {) = }}\Phi {\text {(}}m{\text { + }}\beta {\text {)}}\quad {\text {or}}\end{equation} \begin{equation}\tag {$\ast \ast $} \phi {\text {(}}m{\text {) = }}\int _G {\Phi (m + \beta - {\text {ad}}\;g\beta )dg} \end{equation} $\beta$ being half the sum of the positive roots, then various properties of T are related to properties of the Fourier multiplier transformation on $\mathfrak {G}$ with multiplier $\Phi$. These properties include boundedness on ${L^1}$, uniform boundedness on ${L^p}$ of a family of operators, and, in the special case $G = {\text {SO}}(3)$, boundedness in ${L^p}$ for ad-invariant functions with $1 \leq p < 3/2$.