Let $\H= < a,b | a[a,b]=[a,b]a \wedge b[a,b]=[a,b]b>$ 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 $n\in \N$, 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 { 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 $n\in \N$ 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 $n\to\infty$.
Categories: math.MG math.FA math.GR