$q$-Poincaré 不等式在具有 filiform 型李代数的 Carnot 群上

Universal Differentiability Sets in Carnot Groups of Arbitrarily High Step

We show that every model filiform group $\mathbb{E}_{n}$ contains a measure zero set $N$ such that every Lipschitz map $f\colon \mathbb{E}_{n}\to \mathbb{R}$ is differentiable at some point of $N$. Model filiform groups are a class of Carnot groups which can have arbitrarily high step. Essential to our work is the question of whether existence of an (almost) maximal directional derivative $Ef(x)$ in a Carnot group implies differentiability of a Lipschitz map $f$ at $x$. We show that such an implication is valid in model Filiform groups except for a one-dimensional subspace of horizontal direct…

Carnot群上的Poincaré不等式及Schrödinger算子的谱间隙

Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

本文研究了Carnot群上概率测度下的Poincaré不等式条件，并证明了相关Schrödinger算子具有谱间隙。

Carnot-Carathéodory空间中SBV函数的内蕴周长、紧性和Poincaré不等式

Intrinsic perimeter, compactness and Poincaré inequality for SBV functions in Carnot-Carathéodory spaces

通过引入内蕴可数可求积集的内蕴周长测度，在满足性质$\mathcal R$的等则Carnot-Carathéodory空间中证明了有界变差特殊函数的紧性结果和Poincaré不等式。

卡诺群上的强制不等式：驯服奇异性

Coercive Inequalities on Carnot Groups: Taming Singularities

本文在卡诺群上提出通过引入自然奇异性来获得强制不等式的方法，构造了保证庞加莱和对数索博列夫不等式的概率测度，并应用于海森堡群上的 Dyson-Ornstein-Uhlenbeck 模型及谱离散性研究。

Approximations of Sobolev norms in Carnot groups

This paper deals with a notion of Sobolev space $W^{1,p}$ introduced by J.Bourgain, H.Brezis and P.Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by A.Ponce to obtain a Poincaré-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressd in terms of the intrinsic distance is equivalent to the $L^p$ norm of the intrinsic gradient, and provide a Poincaré-type inequality on Carnot groups by means of a const…

Coercive Inequalities and U-Bounds

We prove Poincaré and Log$^β$-Sobolev inequalities for probability measures on step-two Carnot groups.