The engulfing property for sections of convex functions in the Heisenberg group and the associated quasi--metric

Markov convexity and nonembeddability of the Heisenberg group

We compute the Markov convexity invariant of the continuous infinite dimensional Heisenberg group $\mathbb{H}_\infty$ to show that it is Markov 4-convex and cannot be Markov $p$-convex for any $p < 4$. As Markov convexity is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a different proof of the classical theorem of Pansu and Semmes that the Heisenberg group does not admit a biLipschitz embedding into any Euclidean space. The Markov convexity lower bound will follow from exhibiting an explicit embedding of Laakso graphs $G_n$ into $\mathbb{H}_\infty$ that has distorti…

关于Heisenberg群中Fenchel变换的注记

Note on the Fenchel transform in the Heisenberg group

给定一个定义在海森堡群上的实值函数，我们给出了一个抽象凸性的定义和Fenchel变换，它考虑了群的次黎曼结构。在我们的主要结果中，我们证明了，同样的欧几里德的情况下，一个凸函数可以通过其迭代Fenchel变换的特点，H-次微分的性质起着至关重要的作用。

关于Heisenberg群中的Minty定理

On Minty's theorem in the Heisenberg group

本文将凸分析的一些经典结果推广到Heisenberg群的次黎曼情形。特别是，我们提供了一个水平版本的Minty定理的最大H-单调算子定义在海森堡群的值在其李代数的第一层。

Subelliptic geometric Hardy type inequalities on half-spaces and convex domains

In this paper we present $L^2$ and $L^p$ versions of the geometric Hardy inequalities in half-spaces and convex domains on stratified (Lie) groups. As a consequence, we obtain the geometric uncertainty principles. We give examples of the obtained results for the Heisenberg and the Engel groups.

Estimates on the Markov Convexity of Carnot Groups and Quantitative Nonembeddability

We show that every graded nilpotent Lie group $G$ of step $r$, equipped with a left invariant metric homogeneous with respect to the dilations induced by the grading, (this includes all Carnot groups with Carnot-Caratheodory metric) is Markov $p$-convex for all $p \in [2r,\infty)$. We also show that this is sharp whenever $G$ is a Carnot group with $r \leq 3$, a free Carnot group, or a jet space group; such groups are not Markov $p$-convex for any $p \in (0,2r)$. This continues a line of research started by Li who proved this sharp result when $G$ is the Heisenberg group. As corollaries, we ob…

Heisenberg群在超自反巴拿赫空间中的精确定量不可嵌入性

Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

令$\H$表示离散海森堡群，配备有与某个有限对称生成集相关联的字度量$d_W$。我们表明，如果$(X,\|\cdot\|)$是一个$p$凸Banach空间，那么对于任何Lipschitz函数$f:\H\to X$存在$x,y\in \H$与$d_W(x,y)$任意大和\begin{equation}\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim \left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}. \end{equation}我们还表明，任何嵌入到$X$的球半径$R\ge 4$在$\H$招致双Lipschitz失真增长至少作为一个常数倍的\begin{equation}\label{eq:dist abs} \left(\frac{\log R}{\log\log R}\right)^{1/p}. \end{equation}这两个~ \eqref{eq:comp abs}和~ \eqref{eq:dist abs}是尖锐的迭代对数项。当$X$是Hilbert空间时，我们得到了一个表示论证明，给出了~ \eqref{eq:comp abs}和~ \eqref{eq:dist abs}的上界，这两个上界都是一个普适常数.