Israel Journal of Mathematics

, Volume 203, Issue 1, pp 309–339

# Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Article

## Abstract

Let ℍ = 〈a, b|a[a, b] = [a, b]ab[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d W (·, ·) associated to the generating set {a, b, a −1, b −1}. Letting B n = {x ∈ ℍ: d W (x, e ) ⩽ n} denote the corresponding closed ball of radius n ∈ ℕ, and writing c = [a, b] = aba −1 b −1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ℍ → X satisfies
$$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)}$$
. It follows that for every n ∈ ℕ the bi-Lipschitz distortion of every f: B n X is at least a constant multiple of (log n)1/q , an asymptotically optimal estimate as n → ∞.

## Keywords

Banach Space Heisenberg Group Convex Banach Space Carnot Group Uniform Convexity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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