Advertisement

Geometric & Functional Analysis GAFA

, Volume 2, Issue 2, pp 137–172 | Cite as

Markov chains, Riesz transforms and Lipschitz maps

Article

Abstract

it is shown that a version of Maurey's extension theorem holds for Lipschitz maps between metric spaces satisfying certain geometric conditions, analogous to type and cotype. As a consequence, a classical Theorem of Kirszbraun can be generalised to include maps intoL p , 1<p<2. These conditions describe the wandering of symmetric Markov processes in the spaces in question. Estimates are obtained for the root-mean-square wandering of such processes in theL p spaces. The duality theory for these geometric conditions (in normed spaces) is shown to be closely related to the behavior of the Riesz transforms associated to Markov chains. Several natural open problems are collected in the final chapter.

Keywords

Markov Chain Open Problem Markov Process Normed Space Duality Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Be]W. Beckner, Inequalities in Fourier Analysis, Ann. of Math. 102 (1975), 159–182.Google Scholar
  2. [B]J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Arkiv für Math. 21 (1983), 163–168.Google Scholar
  3. [BMW]J. Bourgain, V.D. Milman, H. Wolfson, On the type of metric spaces, Trans. Amer. Math. Soc. 294 (1986), 295–317.Google Scholar
  4. [Bu]D. Burkholder, A geometrical condition that implies the existence of certain singular integrals of Banach-space-valued functions, Proc. Conf. Harmonic Analysis (in honor of A. Zygmund), Univeristy of Chicago, 1981.Google Scholar
  5. [BG]Bui-Minh-Chi, V.I. Gurarii, Some characteristics of normed spaces and their applications to the generalisation of Parseval's inequality for Banach spaces, Sbor. Theor. Funct. 8 (1969), 74–91 (Russian).Google Scholar
  6. [E]T. Enflo, Uniform homeomorphisms between Banach spaces, Séminaire Maurey-Schwartz 75–76. Exposé no. 18 Ecole Polytechnique, Paris.Google Scholar
  7. [F]T. Figiel, On the moduli of convexity and smoothness, Studia Math. 56 (1976) 121–155.Google Scholar
  8. [G]M. Gromov, Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1–147.Google Scholar
  9. [JL]W.B. Johnson, J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Conference in modern analysis and probability, Contemp. Math. 26 Amer. Math. Soc. (1984).Google Scholar
  10. [JLS]W.B. Johnson, J. Lindenstrauss, G. Schechtman, On Lipschitz embeddings of finite metric spaces into low dimensional normed spaces, Israel Seminar on G.A.F.A., Springer-Verlag, Lecture notes 1267, (1987).Google Scholar
  11. [L1]J. Lindenstrauss, On non-linear projections in Banach spaces, Michigan Math. J. 11 (1964), 263–287.Google Scholar
  12. [L2]J. Lindenstrauss, On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J. 10 (1963), 241–252.Google Scholar
  13. [LT]J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II, Ergebnisse 97, Springer-Verlag (1979).Google Scholar
  14. [MarP]M.B. Marcus, G. Pisier, Characterizations of almost surely continuousp-stable random Fourier series and strongly stationary processes, Acta Math. 152 (1984), 245–301.Google Scholar
  15. [M]B. Maurey, Un théorème de prolongement, C.R. Acad. Sci. Paris 279 (1974), 329–332.Google Scholar
  16. [MP]B. Maurey, G. Pisier, Séries de variables aléatoires vectorielles indépendentes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), 45–90.Google Scholar
  17. [MS]V.D. Milman, G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Lecture Notes 1200, Springer-Verlag (1986).Google Scholar
  18. [P1]G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326–350.Google Scholar
  19. [P2]G. Pisier, Holomorphic semi-groups and the geometry of Banach spaces, Ann. Math. 115 (1982), 375–392.Google Scholar
  20. [S]I. Schoenberg, Metric spaces and completely monotonic functions, Ann. of Math. 39 (1938), 811–841.Google Scholar
  21. [WW]J.H. Wells, L.R. Williams, Embeddings and Extensions in Analysis, Ergebnisse 84, Springer-Verlag (1975).Google Scholar

Copyright information

© Birkhäuser Verlag 1992

Authors and Affiliations

  • K. Ball
    • 1
    • 2
  1. 1.Department of MathematicsT.A.M.U.College StationUSA
  2. 2.Department of MathematicsU.C.L.LondonEngland

Personalised recommendations