### Abstract

A metric space X has Markov-type 2 if for any reversible finite-state Markov chain {Z,} (with Z_{0} chosen according to the stationary distribution) and any map f from the state space to X, the distance D _{t} from f(Z_{0}) to f(Z_{t}) satisfies E(D _{t}^{2}) ≤ K^{2} t E(D_{1}^{2}) 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, L_{p} 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 L_{p} to L_{q} has a Lipschitz extension defined on all of L_{p}.

Original language | English (US) |
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Pages (from-to) | 165-197 |

Number of pages | 33 |

Journal | Duke Mathematical Journal |

Volume | 134 |

Issue number | 1 |

DOIs | |

State | Published - Jul 15 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*134*(1), 165-197. https://doi.org/10.1215/S0012-7094-06-13415-4