Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants

In this paper, we investigate the optimal asymptotic lower bound for the stability of the Sobolev inequality on the Heisenberg group. We first establish the optimal local stability of the Sobolev inequality on the CR sphere by means of bispherical harmonics and a refined orthogonality technique (see Lemma 3.1). The absence of both the Pólya–Szegö inequality and the Riesz rearrangement inequality on the Heisenberg group makes it impossible to apply any rearrangement flow method—either differential or integral—to deduce the global optimal stability of the Sobolev inequality on the CR sphere from its corresponding local stability. To overcome this difficulty, we develop a new approach based on the CR Yamabe flow, which enables us to pass from local to global stability and thereby establish the optimal stability of the Sobolev inequality on the Heisenberg group, with dimension-dependent constants (see Theorem 1.1). As an application, we also obtain the optimal stability of the Hardy–Littlewood–Sobolev (HLS) inequality for a special conformal index, again with dimension-dependent constants (see Theorem 1.2). Our approach is free of any rearrangement argument and can be applied to study the optimal stability problem for the fractional Sobolev or HLS inequalities on the Heisenberg group, once the corresponding continuous flow is established.