$L^p$- Heisenberg--Pauli--Weyl uncertainty inequalities on certain two-step nilpotent Lie groups

This article presents the $L^p$-Heisenberg--Pauli--Weyl uncertainty inequality for the group Fourier transform on a class of two-step nilpotent Lie groups, specifically the Métivier groups. This inequality quantitatively demonstrates that on Métivier groups, a nonzero function and its group Fourier transform cannot both be sharply localized. The proof primarily relies on utilizing the dilation structure inherent to two-step nilpotent Lie groups and estimating the Schatten class norms of the group Fourier transform. The inequality we establish is new, even in the simplest case of Heisenberg groups. Our result significantly sharpens all previously known $L^p$-Heisenberg--Pauli--Weyl uncertainty inequalities for $1 \leq p < 2$ on Métivier groups.