### Abstract

Let ℍ = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d<inf>W</inf>(·, ·) associated to the generating set {a, b, a<sup>−1</sup>, b<sup>−1</sup>}. Letting B<inf>n</inf> = {x ∈ ℍ: d<inf>W</inf>(x, e<inf>ℍ</inf>) ⩽ n} denote the corresponding closed ball of radius n ∈ ℕ, and writing c = [a, b] = aba<sup>−1</sup>b<sup>−1</sup>, 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 (Formula Presented). It follows that for every n ∈ ℕ the bi-Lipschitz distortion of every f: B<inf>n</inf> → X is at least a constant multiple of (log n)<sup>1/q</sup>, an asymptotically optimal estimate as n → ∞.

Original language | English (US) |
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Pages (from-to) | 309-339 |

Number of pages | 31 |

Journal | Israel Journal of Mathematics |

Volume | 203 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*203*(1), 309-339. https://doi.org/10.1007/s11856-014-1088-x

**Vertical versus horizontal Poincaré inequalities on the Heisenberg group.** / Lafforgue, Vincent; Naor, Assaf.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 203, no. 1, pp. 309-339. https://doi.org/10.1007/s11856-014-1088-x

}

TY - JOUR

T1 - Vertical versus horizontal Poincaré inequalities on the Heisenberg group

AU - Lafforgue, Vincent

AU - Naor, Assaf

PY - 2014

Y1 - 2014

N2 - Let ℍ = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric dW(·, ·) associated to the generating set {a, b, a−1, b−1}. Letting Bn = {x ∈ ℍ: dW(x, eℍ) ⩽ n} denote the corresponding closed ball of radius n ∈ ℕ, and writing c = [a, b] = aba−1b−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 (Formula Presented). It follows that for every n ∈ ℕ the bi-Lipschitz distortion of every f: Bn → X is at least a constant multiple of (log n)1/q, an asymptotically optimal estimate as n → ∞.

AB - Let ℍ = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric dW(·, ·) associated to the generating set {a, b, a−1, b−1}. Letting Bn = {x ∈ ℍ: dW(x, eℍ) ⩽ n} denote the corresponding closed ball of radius n ∈ ℕ, and writing c = [a, b] = aba−1b−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 (Formula Presented). It follows that for every n ∈ ℕ the bi-Lipschitz distortion of every f: Bn → X is at least a constant multiple of (log n)1/q, an asymptotically optimal estimate as n → ∞.

UR - http://www.scopus.com/inward/record.url?scp=84939877546&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939877546&partnerID=8YFLogxK

U2 - 10.1007/s11856-014-1088-x

DO - 10.1007/s11856-014-1088-x

M3 - Article

VL - 203

SP - 309

EP - 339

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -