论文标题
H型Carnot组的稳定定理
Stability theorems for H-type Carnot groups
论文作者
论文摘要
我们介绍了第二步Carnot group $ {\ Mathbb G} $的H型偏差$δ（{\ Mathbb G}）$，该$ {\ Mathbb G} $，它测量了该组与海森伯格型组的偏差。我们表明$δ（{\ Mathbb g}）= 0 $，仅当$ {\ Mathbb G} $带有垂直度量标准，该垂直度量赋予了H-type组的结构。我们计算了第二步组的几个天然发生的家庭的H型偏差。此外，我们提供与H型偏差相当的分析表达式。结果，我们为H型组类别建立了新的分析特征。例如，用$ n（g）=（|| x || _h^4+16 || t || t || t || _v^2）^{1/4} $，$ g = \ exp（x+t）$，规范的kaplan-type type quasi-type quasi-norm在第二步中$ { g_v $，我们证明$ {\ mathbb g} $是h-type，并且仅当$ || \ nabla_0 n（g）|| _h^2 = || x || x || x || _h^2/n（g）^2 $ for All $ G \ ne 0 $。同样，我们表明$ {\ mathbb g} $是h-type，并且仅当$ n^{2-q} $是$ {\ Mathcal l} $ - in $ {\ Mathbb g} \ setMinus \ setMinus \ {0 \} $中的$ {\ mathcal l} $ - 谐波。这里$ \ nabla_0 $表示水平差异操作员，$ {\ Mathcal l} $规范的子拉普拉斯式，$ q = \ dim {\ mathfrak v} _1+2 \ 2 \ 2 \ dim {\ dim {\ mathfrak v} _2 V} _1 \ oplus {\ Mathfrak V} _2 $是Lie代数的分层。众所周知，H型组满足了这两个分析结论。这些结果的新内容在于相反的方向。这项工作的动机来自关于可极化的卡诺群体的长期猜想。我们在第两个卡诺组的第两个卡诺群体上为亚拉普拉斯的基本解决方案制定了定量稳定性的猜想。它的有效性将暗示所有可极化组的所有步骤都允许H型组结构。我们确认了一系列各向异性海森堡组的猜想。
We introduce the H-type deviation $δ({\mathbb G})$ of a step two Carnot group ${\mathbb G}$, which measures the deviation of the group from the class of Heisenberg-type groups. We show that $δ({\mathbb G})=0$ if and only if ${\mathbb G}$ carries a vertical metric which endows it with the structure of an H-type group. We compute the H-type deviation for several naturally occurring families of step two groups. In addition, we provide analytic expressions which are comparable to the H-type deviation. As a consequence, we establish new analytic characterizations for the class of H-type groups. For instance, denoting by $N(g)=(||x||_h^4+16||t||_v^2)^{1/4}$, $g=\exp(x+t)$, the canonical Kaplan-type quasi-norm in a step two group ${\mathbb G}$ with taming Riemannian metric $g_h\oplus g_v$, we show that ${\mathbb G}$ is H-type if and only if $||\nabla_0 N(g)||_h^2=||x||_h^2/N(g)^2$ for all $g\ne 0$. Similarly, we show that ${\mathbb G}$ is H-type if and only if $N^{2-Q}$ is ${\mathcal L}$-harmonic in ${\mathbb G} \setminus \{0\}$. Here $\nabla_0$ denotes the horizontal differential operator, ${\mathcal L}$ the canonical sub-Laplacian, and $Q = \dim{\mathfrak v}_1+2\dim{\mathfrak v}_2$ the homogeneous dimension of ${\mathbb G}$, where ${\mathfrak v}_1\oplus{\mathfrak v}_2$ is the stratification of the Lie algebra. It is well-known that H-type groups satisfy both of these analytic conclusions. The new content of these results lies in the converse directions. Motivation for this work comes from a longstanding conjecture regarding polarizable Carnot groups. We formulate a quantitative stability conjecture regarding the fundamental solution for the sub-Laplacian on step two Carnot groups. Its validity would imply that all step two polarizable groups admit an H-type group structure. We confirm this conjecture for a sequence of anisotropic Heisenberg groups.
