Lp Fourier multipliers on compact Lie groups

In this paper we prove Fourier multiplier theorems for invariant and also non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on and its variants on tori T-n . We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.