Sharpness of the Heisenberg-group Lieb inequality

Determine whether the stated Lieb inequality for the short-time Fourier transform on the Heisenberg group is sharp, including whether equality can be attained despite the absence of extremizers for the Heisenberg-group Young convolution inequality.

Background

The paper proves an upper bound for the Lp norm of the short-time Fourier transform on the Heisenberg group by combining the sharp Euclidean Hausdorff–Young inequality with the sharp Young convolution inequality on the Heisenberg group. In the Euclidean setting, Gaussian extremizers for both component inequalities imply sharpness of the resulting Lieb inequality.

For the Heisenberg group, the relevant Young convolution inequality has no extremizers. Consequently, the proof does not establish whether the resulting constant in the Lieb inequality is optimal or whether equality is attainable. This leaves the sharpness question unresolved.

References

In contrast, Young's convolution inequality on the Heisenberg group admits no extremizers (see ). Consequently, the above argument does not allow us to conclude that resultlieb is sharp.

resultlieb:

∫Hn∫Hn∣Vgf(Q,P)∣pdQ dP≤(2p)2n+1(∥f∥2 ∥g∥2)p.\int_{\mathbb{H}^n} \int_{\mathbb{H}^n}\left|V_gf(Q,P)\right|^p dQ\, dP \leq \left(\frac{2}{p}\right)^{2n+1} \left(\|f\|_2 \, \|g\|_2\right)^p.

— Uncertainty Principles for the Short-Time Fourier Transform on the Heisenberg Group  (2608.20952 - Dabra et al., 21 Aug 2026) in Remark following the Lieb inequality theorem, Section 5 ("Lieb's Uncertainty Principle")

Moreover, in the Euclidean setting, the proof for the case 1\leq p\leq2 relies on the converse of Young's inequality, which to the best of our knowledge is not yet known for the Heisenberg group.

— Uncertainty Principles for the Short-Time Fourier Transform on the Heisenberg Group  (2608.20952 - Dabra et al., 21 Aug 2026) in Remark following the Lieb inequality theorem, Section 5 ("Lieb's Uncertainty Principle")