Transferring spherical multipliers on compact symmetric spaces

We prove a two-sided transference theorem between $L^p$ spherical multipliers on the compact symmetric space $U\mathrm{/}K$ and $L^p$ multipliers on the vector space $i\mathfrak{p},$ where the Lie algebra of $U$ has Cartan decomposition $\mathfrak{k}\oplus i\mathfrak{p}$. This generalizes the classic theorem transference theorem of deLeeuw relating multipliers on $% L^{p}($\mathbb{T})$$and$L^{p}($\mathbb{R})$$.