TY - JOUR

T1 - Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

AU - Naor, Assaf

AU - Peres, Yuval

AU - Schramm, Oded

AU - Sheffield, Scott

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2006/7/15

Y1 - 2006/7/15

N2 - A metric space X has Markov-type 2 if for any reversible finite-state Markov chain {Z,} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance D t from f(Z0) to f(Zt) satisfies E(D t2) ≤ K2 t E(D12) for some K = K(X) < ∞. This notion is due to K.Ball [2], who showed its importance for the Lipschitz extension problem. Until now, however, only Hilbert space (and metric spaces that embed bi-Lipschitzly into it) was known to have Markov-type 2. We show that every Banach space with modulus of smoothness of power-type 2 (in particular, Lp for p > 2) has Markov-type 2; this proves a conjecture of Ball (see [2, Section 6]). We also show that trees, hyperbolic groups, and simply connected Riemannian manifolds of pinched negative curvature have Markov-type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in [28, Section 2] by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp.

AB - A metric space X has Markov-type 2 if for any reversible finite-state Markov chain {Z,} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance D t from f(Z0) to f(Zt) satisfies E(D t2) ≤ K2 t E(D12) for some K = K(X) < ∞. This notion is due to K.Ball [2], who showed its importance for the Lipschitz extension problem. Until now, however, only Hilbert space (and metric spaces that embed bi-Lipschitzly into it) was known to have Markov-type 2. We show that every Banach space with modulus of smoothness of power-type 2 (in particular, Lp for p > 2) has Markov-type 2; this proves a conjecture of Ball (see [2, Section 6]). We also show that trees, hyperbolic groups, and simply connected Riemannian manifolds of pinched negative curvature have Markov-type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in [28, Section 2] by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp.

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U2 - 10.1215/S0012-7094-06-13415-4

DO - 10.1215/S0012-7094-06-13415-4

M3 - Article

AN - SCOPUS:33746865347

VL - 134

SP - 165

EP - 197

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 1

ER -