On Multipliers on Compact Lie Groups

In this note we announce $L^p$ multiplier theorems for invariant and noninvariant operators on compact Lie groups in the spirit of the well-known Hörmander–Mikhlin theorem on $\mathbb R^n$ and its versions on the torus $\mathbb T^n$. Applications to mapping properties of pseudo-differential operators on $L^p$-spaces and to a priori estimates for nonhypoelliptic operators are given.