Sharp L^2-Sobolev constant for Métivier groups

Determine the sharp constant in the L^2-Sobolev inequality for the sublaplacian on Métivier groups, enabling extension of Pleijel-type results to this broader class of step-two stratified Lie groups.

Background

The method used in the paper relies critically on having the sharp L2-Sobolev constant (recently obtained by Yang) for H-type groups, along with control of the Weyl constant via spectral analysis. Extending the program to Métivier groups requires analogous sharp constants, which are not currently available.

The authors point out that the lack of a known sharp constant and the more complicated spectral decomposition of the sublaplacian on Métivier groups present obstacles to generalizing Pleijel’s theorem beyond H-type groups.

References

Another possible direction may be extension to the larger class of Metivier groups where the sharp L2-Sobolev constant is not known and the spectral decomposition of the sublaplacian is more complicated, see for instance \S2.

— A note on the Pleijel theorem for $H$-type groups  (2510.19381 - Qiu, 22 Oct 2025) in Section 2 (Proof of main result), closing paragraph

We restrict our focus to the present setting because the precise energy quantization relies fundamentally on the explicit classification of critical extremals (the Jerison--Lee bubbles) and the sharp Sobolev constant, which to the best of our knowledge remain unknown in those more general frameworks.

— Profile decomposition and multiple positive solutions for the perturbed CR Yamabe equation on the Heisenberg group  (2608.16352 - Basak et al., 17 Aug 2026) in Remark following Theorem 1.2, Section 1; see also Remark 3.6