2004/10/19 by Assaf Naor, Yuval Peres, Naor, Assaf +5
Computer Science · Mathematics · #46B99 (primary) #60B99 (secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.math/0410422
openalex publication_date 2004/10/19 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
A metric space X has em Markov type 2, if for any reversible\nfinite-state Markov chain Zt (with Z0 chosen according to the\nstationary distribution) and any map f from the state space to X, the\ndistance Dt from f(Z0) to f(Zt) satisfies E(Dt2) \≤ K2 t\n E(D12) for some K=K(X)<\∞. This notion is due to K. Ball (1992), who\nshowed its importance for the Lipschitz extension problem. However until now,\nonly Hilbert space (and its bi-Lipschitz equivalents) were known to have Markov\ntype 2. We show that every Banach space with modulus of smoothness of power\ntype 2 (in particular, Lp for p>2) has Markov type 2; this proves a\nconjecture of Ball. We also show that trees, hyperbolic groups and simply\nconnected Riemannian manifolds of pinched negative curvature have Markov type\n2. Our results are applied to settle several conjectures on Lipschitz\nextensions and embeddings. In particular, we answer a question posed by Johnson\nand Lindenstrauss in 1982, by showing that for 1<q<2<p<\∞, any Lipschitz\nmapping from a subset of Lp to Lq has a Lipschitz extension defined on\nall of Lp.\n