A generalizaton of Hardy’s uncertainty principle on compact extensions of $$\mathbb R ^n$$

Abstract

Let \(K\) be a compact subgroup of automorphisms of \(\mathbb R ^n\). We prove in this paper a generalization of Hardy’s uncertainty principle on the semi-direct product \(K\ltimes \mathbb R ^n\), and we solve the sharpness problem. As a consequence, a complete analogue of classical Hardy’s theorem is obtained. The representation theory and the Plancherel formula play an important role in the proofs.