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

We prove in this paper a generalization of Heisenberg inequality 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 we obtain an optimal analogue of the Heisenberg inequality. A local uncertainty inequality for the Gabor transform is also provided, in the same context. This allows us to prove a couple of global uncertainty inequalities. The representation theory and Plancherel formula are fundamental tools in the proof of our results.