Hardy’s uncertainty principle for Gabor transform on compact extensions of \(\mathbb {R}^n\)

We prove in this paper a generalization of Hardy’s theorem for Gabor transform in the setup of the semidirect product \(\mathbb {R}^n\rtimes K\) ${\mathbb{R}}^n⋊K$ , where K is a compact subgroup of automorphisms of \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . We also solve the sharpness problem and thus obtain a complete analogue of Hardy’s theorem for Gabor transform. The representation theory and Plancherel formula are fundamental tools in the proof of our results.