A multiplier theorem for $SU(n)$

Abstract:

Let $G = {\text {SU}}(n)$, let $\mathfrak {g}$ be its Lie algebra and let $m$ be a function on $\mathfrak {g}$, invariant under the adjoint action of $G$, which is continuous at the points of $\hat G$ (which can be imbedded in $\mathfrak {g}$). If $1 \leqslant p < 2[1 - {(n + 2)^{ - 1}}]$ and $m$ is a multiplier for the ${\operatorname {Ad} _G}$-invariant ${L^p}$ functions on $\mathfrak {g}$, then the restriction of a translate of $m$ to $\hat G$ is a multiplier for the central ${L^p}$ functions on $G$.