Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Abstract

Let $\stackrel{˝}{=}<a,b|a[a,b]=[a,b]a\wedge b[a,b]=[a,b]b>$ be the discrete Heisenberg group, equipped with the left-invariant word metric $d_W(\cdot ,\cdot )$ associated to the generating set $a,b,a^{-1},b^{-1}$. Letting $B_n= {x\in \H: d_W(x,e_\H)\le n}$ denote the corresponding closed ball of radius $n\in \mathbb{N}$, and writing $c=[a,b]=ab{a}^{-1}{b}^{-1}$, we prove that if $(X,|\cdot {|}_X)$ is a Banach space whose modulus of uniform convexity has power type $q\in [2,\infty )$ then there exists $K\in (0,\infty )$ such that every $f:\stackrel{˝}{\to }X$ satisfies {multline*} \sum_{k=1}^{n^2}\sum_{x\in B_n}\frac{|f(xc^k)-f(x)|_X^q}{k^{1+q/2}}\le K\sum_{x\in B_{21n}} \Big(|f(xa)-f(x)|^q_X+\|f(xb)-f(x)\|^q_X\Big). {multline*} It follows that for every $n\in \mathbb{N}$ the bi-Lipschitz distortion of every $f:B_n\to X$ is at least a constant multiple of $(\log n{)}^{1/q}$, an asymptotically optimal estimate as $n\to \infty$.

References in corpus (2)

• Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces
• L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry