Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Vincent Lafforgue, Assaf Naor

Research output: Contribution to journalArticlepeer-review


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, e) ⩽ 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 → ∞.

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

ASJC Scopus subject areas

  • General Mathematics


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