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15 July 2006 Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces
Assaf Naor, Yuval Peres, Oded Schramm, Scott Sheffield
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Duke Math. J. 134(1): 165-197 (15 July 2006). DOI: 10.1215/S0012-7094-06-13415-4


A metric space X has Markov-type 2 if for any reversible finite-state Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(Dt2)K2tE(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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Assaf Naor. Yuval Peres. Oded Schramm. Scott Sheffield. "Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces." Duke Math. J. 134 (1) 165 - 197, 15 July 2006.


Published: 15 July 2006
First available in Project Euclid: 4 July 2006

zbMATH: 1108.46012
MathSciNet: MR2239346
Digital Object Identifier: 10.1215/S0012-7094-06-13415-4

Primary: 46B09 , 46B20 , 60J99

Rights: Copyright © 2006 Duke University Press


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Vol.134 • No. 1 • 15 July 2006
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