Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces

A metric space X has Markov type 2, if for any reversible flnite-state Markov chain fZtg (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) satisfles E(D 2) • K 2 tE(D 2) for some K = K(X) 2) has Markov type 2; this proves a conjecture of Ball. 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 1982, by showing that for 1 < q < 2 < p < 1, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension deflned on all of Lp.

[1]  M. D. Kirszbraun.  Über die zusammenziehende und Lipschitzsche Transformationen , 1934 .

[2]  Joseph E. Borzellino,et al.  When is a Trigonometric Polynomial Not a Trigonometric Polynomial , 1935 .

[3]  O. Hanner.  On the uniform convexity ofLp andlp , 1956 .

[4]  Joram Lindenstrauss.  On the modulus of smoothness and divergent series in Banach spaces. , 1963 .

[5]  Per Enflo.  Topological Groups in Which Multiplication on One Side is Differentiable or Linear. , 1969 .

[6]  P. Enflo,et al.  Uniform structures and square roots in topological groups , 1970 .

[7]  P. Enflo.  Uniform structures and square roots in topological groups , 1970 .

[8]  Stanisław Kwapień,et al.  Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients , 1972 .

[9]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[10]  On Banach spaces X for which $Π_{2}(ℒ_{∞},X)=B(ℒ_{∞},X)$ , 1972 .

[11]  Joram Lindenstrauss,et al.  Classical Banach spaces , 1973 .

[12]  Jim Pitman.  ONE-DIMENSIONAL BROWNIAN MOTION AND THE THREE-DIMENSIONAL BESSEL PROCESS , 1974 .

[13]  J. Hoffmann-jorgensen.  Sums of independent Banach space valued random variables , 1974 .

[14]  Type et cotype dans les espaces munis de structures locales inconditionnelles , 1974 .

[15]  G. Pisier.  Martingales with values in uniformly convex spaces , 1975 .

[16]  J. Wells,et al.  Embeddings and Extensions in Analysis , 1975 .

[17]  T. Figiel.  On the moduli of convexity and smoothness , 1976 .

[18]  G. Pisier,et al.  Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach , 1976 .

[19]  R. C. James.  Nonreflexive spaces of type 2 , 1978 .

[20]  P. Enflo.  On infinite-dimensional topological groups , 1978 .

[21]  Nguyễn Tố Như,et al.  Lipschitz extensions and Lipschitz retractions in metric spaces , 1981 .

[22]  Length Functions and Free Products with Amalgamation of Groups , 1981 .

[23]  Gilles Pisier,et al.  Holomorphic semi-groups and the geometry of Banach spaces , 1982 .

[24]  G. Pisier,et al.  Characterizations of almost surely continuousp-stable random Fourier series and strongly stationary processes , 1984 .

[25]  W. B. Johnson,et al.  Extensions of Lipschitz mappings into Hilbert space , 1984 .

[26]  Peter B. Shalen,et al.  Valuations, Trees, and Degenerations of Hyperbolic Structures, I , 1984 .

[27]  G. Pisier.  Probabilistic methods in the geometry of Banach spaces , 1986 .

[28]  Jean Bourgain,et al.  On type of metric spaces , 1986 .

[29]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[30]  G. Pisier,et al.  Random series in the real interpolation spaces between the spaces v p , 1987 .

[31]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[32]  R. Durrett.  Probability: Theory and Examples , 1993 .

[33]  É. Ghys,et al.  Sur Les Groupes Hyperboliques D'Apres Mikhael Gromov , 1990 .

[34]  P. Wojtaszczyk.  Banach Spaces For Analysts: Preface , 1991 .

[35]  Keith Ball,et al.  Markov chains, Riesz transforms and Lipschitz maps , 1992 .

[36]  K. Ball,et al.  Sharp uniform convexity and smoothness inequalities for trace norms , 1994 .

[37]  J. Eschenburg.  Comparison Theorems in Riemannian Geometry , 1994 .

[38]  Terry Lyons,et al.  Decomposition of Dirichlet Processes and its Application , 1994 .

[39]  J. Diestel,et al.  Absolutely Summing Operators , 1995 .

[40]  R. Latala.  Estimation of moments of sums of independent real random variables , 1997 .

[41]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[42]  J. Lindenstrauss,et al.  Geometric Nonlinear Functional Analysis , 1999 .

[43]  M. Gromov.  Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[44]  T. Laakso.  Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality , 2000 .

[45]  Oded Schramm,et al.  Embeddings of Gromov Hyperbolic Spaces , 2000 .

[46]  J. Heinonen.  Lectures on Analysis on Metric Spaces , 2000 .

[47]  D. Burkholder.  Chapter 6 - Martingales and Singular Integrals in Banach Spaces , 2001 .

[48]  J. Lindenstrauss,et al.  Handbook of geometry of Banach spaces , 2001 .

[49]  David Preiss.  GEOMETRIC NONLINEAR FUNCTIONAL ANALYSIS, Volume 1 (American Mathematical Society Colloquium Publications 48) By Y OAV B ENYAMINI and J ORAM L INDENSTRAUSS : 488 pp., US$65.00, ISBN 0-8218-0835-4 (American Mathematical Society, Providence, RI, 2000). , 2001 .

[50]  U. Lang,et al.  Bilipschitz Embeddings of Metric Spaces into Space Forms , 2001 .

[51]  Assaf Naor,et al.  A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between Lp spaces , 2001 .

[52]  J. Lindenstrauss,et al.  Lipschitz Quotients from Metric Trees and from Banach Spaces Containing ℓ1 , 2002 .

[53]  G. Schechtman,et al.  Remarks on non linear type and Pisier's inequality , 2002 .

[54]  Nathan Linial,et al.  Girth and Euclidean distortion , 2002 .

[55]  Nathan Linial,et al.  Girth and euclidean distortion , 2002, STOC '02.

[56]  J. R. Lee,et al.  Embedding the diamond graph in Lp and dimension reduction in L1 , 2004, math/0407520.

[57]  Nathan Linial,et al.  On metric ramsey-type phenomena , 2003, STOC '03.

[58]  V. Schroeder,et al.  Embedding of Hyperbolic Spaces in the Product of Trees , 2003 .

[59]  B. Maurey,et al.  Chapter 30 - Type, Cotype and K-Convexity , 2003 .

[60]  P. Assouad.  Plongements lipschitziens dans Rn , 2003 .

[61]  James R. Lee,et al.  Metric Structures in L1: Dimension, Snowflakes, and Average Distortion , 2004, LATIN.

[62]  Robert Krauthgamer,et al.  Measured descent: a new embedding method for finite metrics , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[63]  Jussi Väisälä,et al.  Gromov hyperbolic spaces , 2005 .

[64]  James R. Lee,et al.  Extending Lipschitz functions via random metric partitions , 2005 .

[65]  H. Hanche-Olsen.  On the uniform convexity of L^p , 2005, math/0502021.