$L^p$- 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.

💡 Research Summary

The paper establishes a new family of $L^{p}$-Heisenberg–Pauli–Weyl (HPW) uncertainty inequalities for the group Fourier transform on a class of two‑step nilpotent Lie groups known as Métivier groups. Classical HPW inequalities assert that a function and its Fourier transform cannot both be sharply localized; in Euclidean space this is expressed by an inequality involving weighted $L^{2}$ norms of $f$ and its Fourier transform $\widehat f$. Extending such results to non‑commutative groups is non‑trivial because the Fourier transform becomes operator‑valued.

The authors first review the Euclidean $L^{p}$‑HPW inequalities (Theorem 1.2) and the existing literature on stratified and two‑step nilpotent groups, highlighting the limitations of previous works: most results either treat only the $p=2$ case, or they rely on geometric arguments that cannot handle higher powers of the sub‑Laplacian. In particular, the works of Ciatti–Cowling–Ricci and Dall’Ara–Trevizan provide $L^{p}$‑type inequalities but either exclude the endpoint $p=1$ or are restricted to first‑order sub‑Laplacians.

Métivier groups are defined by the non‑degeneracy of the skew‑symmetric bilinear form $B_{\lambda}(V,V’)=\lambda(