Generality of the Heisenberg Hardy method beyond ℍ^N

Investigate the extent to which the change-of-variables-based proof of the Heisenberg Hardy inequality (used to obtain lower bounds for the Dirichlet sub-Laplacian without boundary regularity assumptions) can be generalized to broader sub-Riemannian settings beyond the Heisenberg group.

Background

The authors prove a Hardy inequality on the Heisenberg group via a simple change-of-variables argument that avoids boundary regularity assumptions, contrasting with approaches relying on sub-Riemannian Santaló formulas.

They explicitly state uncertainty about the scope of this method’s applicability to more general structures considered in prior work, highlighting a methodological open question.