We obtain the best known quantitative estimates for the L^p-Poincaré and log-Sobolev inequalities on domains in various sub-Riemannian manifolds, including ideal Carnot groups and in particular ideal generalized H-type Carnot groups and the Heisenberg groups, corank 1 Carnot groups, the Grushin plane, and various H-type foliations, Sasakian and 3-Sasakian manifolds. Moreover, this constitutes the first time that a quantitative estimate independent of the dimension is established on these spaces. For instance, the Li-Yau / Zhong-Yang spectral-gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of 4.

中文翻译：

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准曲率维条件在亚黎曼流形中的应用

我们获得了各种亚黎曼流形中域上L ^p -Poincaré 和 log-Sobolev 不等式的最著名的定量估计，包括理想的卡诺群，特别是理想的广义 H 型卡诺群和海森堡群，corank 1 Carnot 群、Grushin 平面和各种 H 型叶理、Sasakian 和3 -Sasakian流形。此外，这是第一次在这些空间上建立独立于维度的定量估计。例如，Li-Yau/Zhong-Yang 谱隙估计适用于任意维数的所有海森堡群，最高为4。