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/K and vector $$i{\mathfrak {p}},$$ where Lie algebra of U has Cartan decomposition $$\mathfrak {k\oplus }i{\mathfrak {p}}$$ . This generalizes classic deLeeuw relating $$ L^{p}(\mathbb {T)}$$ $$L^{p}(\mathbb {R)}$$