Remainder terms for several inequalities on some groups of Heisenberg-type

We give estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group. By considering the variations of associated functionals, we give a stability for two dual inequalities: The fractional Sobolev (FS) and Hardy-Littlewood-Sobolev (HLS) inequalities, in terms of distance to the submanifold of extremizers. Then we compare their remainder terms to improve the inequalities in another way. We also compare, in the limit case, the remainder terms of Beckner-Onofri (BO) inequality and its dual logarithmic Hardy-Littlewood-Sobolev (Log-HLS) inequality. Besides, we also list without proof some results for other groups of Iwasawa-type. Our results generalize earlier works on Euclidean spaces of Chen et al. (2013) and Dolbeault and Jankowiak (2014) onto some groups of Heisenberg-type. We worked for "almost" all fractions especially for comparing results, and the stability of HLS is also absolutely new, even for Euclidean case.