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

Abstract

We give estimates of the remainder terms for several conformally-invariant Sobolevtype 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 λ = 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.

Sharp Hardy-Littlewood-Sobolev inequalities on a class of H-type groups

“…Another main result, obtained in [LZ15], is about the improvements of the sharp inequalities obtained in the first main theorem. We derive a stability for the extremizers of the HLS and fractional Sobolev inequalities, extending [CFW13].…”

On sharp inequalities on nilpotent Lie groups

“…I 型群上另一个主要结果, 是对定理 2.1 中尖锐不等式的优化 [28] . 我们推导 HLS 不等式和分数 阶 Sobolev 不等式的稳定性, 推广了文献 [29] 的结果, 并类似于文献 [7,30] 对对偶余项进行比较.…”

“…他们的结果验证了关于一般的 HLS 问题的猜 想, 也启发了 Frank 和 Lieb 运用反变分不等式方法来研究 Heisenberg 群上的 HLS 问题. 文献 [28] 改 进了这些不等式, 证明了极值函数的稳定性并比较了 HLS 不等式与分数阶 Sobolev 不等式的对偶余 项, 从而推广了 Euclid 空间上相应的结果 (参见文献 [7,29,30]). 在 Euclid 空间和球面上, 还可以得到 一个次临界版本的尖锐不等式 [31] .…”

Stability of Trace Theorems on the Sphere

“…The inequality (14) appears to have been first proved by Liu-Zhang via a direct derivation ([36], Theorem 2.2), however the approach in [16] permits some simplifications compared to this. For instance, a lack of smoothness of the functional given by the left-hand side of ( 14) occurs since q > 2, and this causes a failure of the second-order Taylor expansion used in the derivation of (2) (necessitating a result of Christ from [24]; see Section 4.2 of [36]).…”

“…where G is as above, G * := D q T g * ∈ M (T * ), and η := (min{q, 2} − 1) −1 . The second inequality here follows from (33), and the third follows from (36). By raising (37) to an appropriate power and keeping track of the constants, one may obtain an explicit dependence of c 2 on c 1 and the other quantities as claimed.…”