论文标题
多项式生长组的定量不舒适性成均匀凸出的空间
Quantitative nonembeddability of groups of polynomial growth into uniformly convex spaces
论文作者
论文摘要
非亚伯语简单地连接了尼洛氏谎言基团，而不是几乎有限生成的多项式生长的群体不会准确地嵌入均匀凸出的banach空间中。我们通过表明上述组中的半径$ r \ ge 2 $的球必须产生bilipschitz扭曲至少$（\ log r）^{1/q} $的常数倍数，以量化这一事实。对于$ l^p $（$ 1 <p <\ infty $）空间，此界限是锋利的。我们通过使用矢量价值的Littlewood-paley--paley--paley--paley-stein理论方法将上述群体的功能建立``垂直与水平不等式''来证明这一点。这些不平等是定量的无易于性陈述，将上述组映射到沿着某些中央亚组沿数量折叠的均匀凸空空间中的任何Lipschitz映射。 在Carnot组映射到$ l^p $（$ 1 <p <\ infty $）空间的特殊情况下，我们证明定量崩溃发生在换向器子组上；这与Cheeger and Kleiner（2006）和Lee and Naor（2006）给出的定性Pansu-Semmes无核能论点一致。我们通过在Carnot组上建立一个古典Dorronsoro定理来证明这一点。以前，在海森伯格集团的环境中，法斯勒和奥普宁（Fässler）和奥普宁（Orponen，2019年）在laplacian的指数$α$范围内建立了一个单方面的dorronsoro定理$ 0 <α<2 $；这种限制不会出现在交换环境中，并且是由于它们使用水平多项式作为近似值而引起的。我们确定了正确的近似多项式类别的正确类别，并证明了在Carnot组的一般环境中的整个范围$ 0 <α<\ indy $的指标，从而增强Fässler和Orponen的工作。
Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces. We quantify this fact by showing that a ball of radius $r\ge 2$ in the aforementioned groups must incur bilipschitz distortion at least a constant multiple of $(\log r)^{1/q}$ into a $q(\ge 2)$-uniformly convex Banach space. This bound is sharp for the $L^p$ ($1<p<\infty$) spaces. We prove this by establishing ``vertical versus horizontal inequalities'' for functions from the aforementioned groups into uniformly convex spaces, using the vector-valued Littlewood--Paley--Stein theory approach of Lafforgue and Naor (2012). These inequalities are quantitative nonembeddability statements that any Lipschitz mapping from the aforementioned groups into a uniformly convex space quantitatively collapses along certain central subgroups. In the special case of mappings of Carnot groups into the $L^p$ ($1<p<\infty$) spaces, we prove that the quantitative collapse occurs on the commutator subgroup; this is in line with the qualitative Pansu--Semmes nonembeddability argument given by Cheeger and Kleiner (2006) and Lee and Naor (2006). We prove this by establishing a version of the classical Dorronsoro theorem on Carnot groups. Previously, in the setting of Heisenberg groups, Fässler and Orponen (2019) established a one-sided Dorronsoro theorem with a restriction $0<α<2$ on the range of exponents $α$ of the Laplacian; this restriction does not appear in the commutative setting and is caused by their use of horizontal polynomials as approximants. We identify the correct class of approximant polynomials and prove the two-sided Dorronsoro theorem with the full range $0<α<\infty$ of exponents in the general setting of Carnot groups, thus strengthening and extending the work of Fässler and Orponen.
