Remainder Terms for Several Inequalities on Some Groups of Heisenberg-type

TL;DR

This paper investigates the stability and remainder terms of conformally-invariant Sobolev inequalities on Heisenberg groups, extending Euclidean results and comparing inequalities in various limit cases.

Contribution

It introduces new estimates for remainder terms and stability results for Sobolev and Hardy-Littlewood-Sobolev inequalities on Heisenberg-type groups, generalizing prior Euclidean findings.

Findings

Stability of fractional Sobolev and Hardy-Littlewood-Sobolev inequalities on Heisenberg groups.

Comparison and improvement of remainder terms for these inequalities.

Extension of Euclidean space results to Heisenberg-type groups.

Abstract

We give some estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group, in analogy with the Euclidean case. By considering the variation of associated functionals, we give a stability of two dual forms: the fractional Sobolev (Folland-Stein) and Hardy-Littlewood-Sobolev inequality, 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 s = Q (or $\lambda$ = 0), the remainder terms of Beckner-Onofri inequality and its dual Logarithmic Hardy-Littlewood-Sobolev inequality. Besides, we also list without proof some results for the other two cases of groups of Iwasawa-type. Our results generalize earlier works on Euclidean spaces by Chen, Frank, Weth [CFW13] and Dolbeault, Jankowiakin [DJ14] onto some groups of Heisenberg-type.