Vertical versus horizontal Poincar\'e inequalities on the Heisenberg group

Let $\H=$ 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 $n\to\infty$

Uniform embeddings, homeomorphisms and quotient maps between Banach spaces (A short survey)

AbstractThe paper is a short survey dealing with questions of the following type: For which pair of Banach spaces X and Y is X Lipschitz equivalent or uniform homeomorphic to a subset of Y? To what extent does the uniform structure of a Banach space or its unit ball determine the linear structure of the space? What is the right notion of a nonlinear quotient space