This is an announcement for the paper "Vertical versus horizontal Poincar'e inequalities on the Heisenberg group" by Vincent Lafforgue and Assaf Naor.
Abstract: Let $\H= \left\langle a,b,|, a[a,b]=[a,b]a\ \wedge\ b[a,b]=[a,b]b\right\rangle$ 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 $n\in \N$, and writing $c=[a,b]=aba^{-1}b^{-1}$, 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 $f:\H\to X$ satisfies \begin{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). \end{multline*} It follows that for every $n\in \N$ 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 $n\to\infty$.
Archive classification: math.MG math.FA math.GR
Submitted from: naor@cims.nyu.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1212.2107
or