TY - JOUR

T1 - Vertical versus horizontal Poincaré inequalities on the Heisenberg group

AU - Lafforgue, Vincent

AU - Naor, Assaf

N1 - Funding Information:
∗∗ A. N. was supported in part by NSF grant CCF-0832795, BSF grant 2010021, the Packard Foundation and the Simons Foundation. Part of this work was completed while A. N. was visiting Universitéde Paris Est Marne-la-Vallée. Received December 13, 2012 and in revised form July 18, 2013
Funding Information:
∗ V. L. was supported in part by ANR grant KInd.
Publisher Copyright:
© 2014, Hebrew University of Jerusalem.

PY - 2014/10

Y1 - 2014/10

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

AN - SCOPUS:84939877546

VL - 203

SP - 309

EP - 339

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -