Differential structure on the dual of a compact Lie group

Abstract In this paper we define difference operators and homogeneous Sobolev-type spaces on the dual of a compact Lie group. As an application and to show that this defines a relevant differential structure, we state and prove multiplier theorems of Hormander, Mihlin and Marcinkiewicz types together with the sharpness in the Sobolev exponent for the result of Hormander type.