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

Abstract : 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 $n\to\infty$.
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Contributor : Vincent Lafforgue <>
Submitted on : Wednesday, April 22, 2015 - 4:37:22 PM
Last modification on : Wednesday, March 24, 2021 - 10:54:04 AM

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Vincent Lafforgue, Assaf Naor. Vertical versus horizontal Poincar\'e inequalities on the Heisenberg group. Israël Journal of Mathematics, Hebrew University Magnes Press, 2014, 203 (1), pp.309-339. ⟨10.1007/s11856-014-1088-x⟩. ⟨hal-01144788⟩



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