On the Heisenberg group, the Sobolev inequality is stable with the asymptotically optimal lower bound β0/Q times the squared distance to extremals.
Loiudice, Improved Sobolev inequalities on the Heisenberg group , Nonlinear Anal
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Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants
On the Heisenberg group, the Sobolev inequality is stable with the asymptotically optimal lower bound β0/Q times the squared distance to extremals.