Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Vincent Lafforgue, Assaf Naor

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)309-339
Number of pages31
JournalIsrael Journal of Mathematics
Volume203
Issue number1
DOIs
StatePublished - 2014

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Uniform Convexity
Generating Set
Heisenberg Group
Discrete Group
Asymptotically Optimal
Lipschitz
Modulus
Ball
Horizontal
Vertical
Radius
Banach space
Denote
Metric
Closed
Invariant
Estimate

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Israel Journal of Mathematics, Vol. 203, No. 1, 2014, p. 309-339.

Research output: Contribution to journalArticle

Lafforgue, Vincent ; Naor, Assaf. / Vertical versus horizontal Poincaré inequalities on the Heisenberg group. In: Israel Journal of Mathematics. 2014 ; Vol. 203, No. 1. pp. 309-339.
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