Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

Manor Mendel; Assaf Naor

Analysis and Geometry in Metric Spaces (2013)

  • Volume: 1, page 163-199
  • ISSN: 2299-3274

Abstract

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The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.

How to cite

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Manor Mendel, and Assaf Naor. "Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces." Analysis and Geometry in Metric Spaces 1 (2013): 163-199. <http://eudml.org/doc/266565>.

@article{ManorMendel2013,
abstract = {The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.},
author = {Manor Mendel, Assaf Naor},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Markov cotype; Lipschitz extension; CAT (0) metric spaces; nonlinear spectral gaps; CAT(0) metric spaces},
language = {eng},
pages = {163-199},
title = {Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces},
url = {http://eudml.org/doc/266565},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Manor Mendel
AU - Assaf Naor
TI - Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 163
EP - 199
AB - The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.
LA - eng
KW - Markov cotype; Lipschitz extension; CAT (0) metric spaces; nonlinear spectral gaps; CAT(0) metric spaces
UR - http://eudml.org/doc/266565
ER -

References

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  1. [1] A. Andoni, A. Naor, and O. Neiman. Snowflake universality of Wasserstein spaces. Preprint, (2010). 
  2. [2] A. Andoni, A. Naor, and O. Neiman. On isomorphic dimension reduction in `1. Preprint, (2011). 
  3. [3] K. Ball. Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal., 2(2):137-172, (1992).[Crossref] Zbl0788.46050
  4. [4] K. Ball. The Ribe programme. Séminaire Bourbaki, exposé 1047, (2012). 
  5. [5] K. Ball, E. A. Carlen, and E. H. Lieb. Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math., 115(3):463-482, (1994).[Crossref] Zbl0803.47037
  6. [6] W. Ballmann. Lectures on spaces of nonpositive curvature, volume 25 of DMV Seminar. Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. 
  7. [7] Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, (2000). Zbl0946.46002
  8. [8] J. Bourgain. A counterexample to a complementation problem. Compositio Math., 43(1):133-144, (1981). Zbl0437.46016
  9. [9] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, (1999). Zbl0988.53001
  10. [10] B. Brinkman, A. Karagiozova, and J. R. Lee. Vertex cuts, random walks, and dimension reduction in series-parallel graphs. In STOC’07-Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 621-630. ACM, New York, (2007). Zbl1232.68163
  11. [11] A. Brudnyi and Y. Brudnyi. Methods of geometric analysis in extension and trace problems. Volume 2, volume 103 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, (2012). Zbl1253.46001
  12. [12] T. Christiansen and K. T. Sturm. Expectations and martingales in metric spaces. Stochastics, 80(1):1-17, (2008). Zbl1216.60041
  13. [13] J. Ding, J. R. Lee, and Y. Peres. Markov type and threshold embeddings. Preprint available at http://arxiv.org/abs/1208.6088,(2012). Zbl1279.46013
  14. [14] S. Doss. Moyennes conditionnelles et martingales dans un espace métrique. C. R. Acad. Sci. Paris, 254:3630-3632, (1962). Zbl0113.33302
  15. [15] A. Dvoretzky. Some results on convex bodies and Banach spaces. In Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pages 123-160. Jerusalem Academic Press, Jerusalem, (1961). 
  16. [16] M. Émery. Stochastic calculus in manifolds. Universitext. Springer-Verlag, Berlin, 1989. With an appendix by P.-A. Meyer. Zbl0697.60060
  17. [17] A. Es-Sahib and H. Heinich. Barycentre canonique pour un espace métrique à courbure négative. In Séminaire de Probabilités, XXXIII, volume 1709 of Lecture Notes in Math., pages 355-370. Springer, Berlin, (1999). Zbl0952.60010
  18. [18] T. Figiel. On the moduli of convexity and smoothness. Studia Math., 56:121-155, (1976). Zbl0344.46052
  19. [19] T. Figiel, W. B. Johnson, and G. Schechtman. Factorizations of natural embeddings of lnp into Lr . I. Studia Math., 89(1):79-103, (1988). Zbl0671.46009
  20. [20] M. Gromov. Random walk in random groups. Geom. Funct. Anal., 13(1):73-146, (2003).[Crossref] Zbl1122.20021
  21. [21] A. Grothendieck. Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo, 8:1-79, (1953). 
  22. [22] S. Heinrich. Ultraproducts in Banach space theory. J. Reine Angew. Math., 313:72-104, (1980). Zbl0412.46017
  23. [23] W. B. Johnson and J. Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Conference in modern analysis and probability (New Haven, Conn., 1982), volume 26 of Contemp. Math., pages 189-206. Amer. Math. Soc., Providence, RI, (1984). Zbl0539.46017
  24. [24] W. B. Johnson, J. Lindenstrauss, and G. Schechtman. Extensions of Lipschitz maps into Banach spaces. Israel J. Math., 54(2):129-138, (1986). Zbl0626.46007
  25. [25] W. B. Johnson, H. P. Rosenthal, and M. Zippin. On bases, finite dimensional decompositions and weaker structures in Banach spaces. Israel J. Math., 9:488-506, (1971). Zbl0217.16103
  26. [26] J. Jost. Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, (1997). Zbl0896.53002
  27. [27] N. J. Kalton. Spaces of Lipschitz and Hölder functions and their applications. Collect. Math., 55(2):171-217, (2004). Zbl1069.46004
  28. [28] N. J. Kalton. Lipschitz and uniform embeddings into `1. Fund. Math., 212(1):53-69, (2011). Zbl1220.46014
  29. [29] N. J. Kalton. The uniform structure of Banach spaces. Math. Ann., 354(4):1247-1288, (2012). Zbl1268.46018
  30. [30] M. Kapovich and B. Leeb. On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds. Geom. Funct. Anal., 5(3):582-603, (1995).[Crossref] Zbl0829.57006
  31. [31] M. D. Kirszbraun. Über die zusammenziehenden und Lipschitzchen Transformationen. Fundam. Math., 22:77-108, (1934). Zbl0009.03904
  32. [32] U. Lang. Extendability of large-scale Lipschitz maps. Trans. Amer. Math. Soc., 351(10):3975-3988, (1999). Zbl1010.54016
  33. [33] U. Lang, B. Pavlovic, and V. Schroeder. Extensions of Lipschitz maps into Hadamard spaces. Geom. Funct. Anal., 10(6):1527-1553, (2000).[Crossref] Zbl0990.53070
  34. [34] U. Lang and T. Schlichenmaier. Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not., (58):3625-3655, (2005).[Crossref] Zbl1095.53033
  35. [35] U. Lang and V. Schroeder. Kirszbraun’s theorem and metric spaces of bounded curvature. Geom. Funct. Anal., 7(3):535-560, (1997).[Crossref] Zbl0891.53046
  36. [36] J. R. Lee and A. Naor. Extending Lipschitz functions via random metric partitions. Invent. Math., 160(1):59-95, (2005). Zbl1074.46004
  37. [37] J. Lindenstrauss and A. Pełczynski. Absolutely summing operators in Lp-spaces and their applications. Studia Math., 29:275-326, (1968). Zbl0183.40501
  38. [38] J. Lindenstrauss and H. P. Rosenthal. The Lp spaces. Israel J. Math., 7:325-349, (1969). Zbl0205.12602
  39. [39] K. Makarychev and Y. Makarychev. Metric extension operators, vertex sparsifiers and Lipschitz extendability. In 51th Annual IEEE Symposium on Foundations of Computer Science, pages 255-264, (2010). 
  40. [40] B. Maurey. Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces Lp. Société Mathématique de France, Paris, 1974. With an English summary, Astérisque, No. 11. Zbl0278.46028
  41. [41] B. Maurey. Type, cotype and K-convexity. In Handbook of the geometry of Banach spaces, Vol. 2, pages 1299-1332. North-Holland, Amsterdam, (2003). Zbl1074.46006
  42. [42] M. Mendel and A. Naor. Metric cotype. Ann. of Math. (2), 168(1):247-298, (2008). Zbl1187.46014
  43. [43] M. Mendel and A. Naor. Nonlinear spectral calculus and super-expanders. To appear in Inst. Hautes Études Sci. Publ. Math., available at http://arxiv.org/abs/1207.4705,(2012). 
  44. [44] M. Mendel and A. Naor. Expanders with respect to Hadamard spaces and random graphs. Preprint, (2013). Zbl1316.05109
  45. [45] M. Mendel and A. Naor. Markov convexity and local rigidity of distorted metrics. J. Eur. Math. Soc. (JEMS), 15(1):287-337, (2013).[Crossref] Zbl1266.46016
  46. [46] V. D. Milman and G. Schechtman. Asymptotic theory of finite-dimensional normed spaces, volume 1200 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. Zbl0606.46013
  47. [47] G. J. Minty. On the extension of Lipschitz, Lipschitz-Hölder continuous, and monotone functions. Bull. Amer. Math. Soc., 76:334-339, (1970).[Crossref] Zbl0191.34603
  48. [48] A. Naor. A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between Lp spaces. Mathematika, 48(1-2):253-271 (2003), (2001). Zbl1059.46059
  49. [49] A. Naor. An introduction to the Ribe program. Jpn. J. Math., 7(2):167-233, (2012). Zbl1261.46013
  50. [50] A. Naor, Y. Peres, O. Schramm, and S. Sheffield. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1):165-197, (2006). Zbl1108.46012
  51. [51] A. Naor and G. Schechtman. Remarks on non linear type and Pisier’s inequality. J. Reine Angew. Math., 552:213-236, (2002). Zbl1033.46013
  52. [52] A. Naor and L. Silberman. Poincaré inequalities, embeddings, and wild groups. Compos. Math., 147(5):1546-1572, (2011). Zbl1267.20057
  53. [53] A. Navas. An L1 ergodic theorem with values in a non-positively curved space via a canonical barycenter map. Ergodic Theory Dynam. Systems, FirstView:1-15. 
  54. [54] S.-i. Ohta. Extending Lipschitz and Hölder maps between metric spaces. Positivity, 13(2):407-425, (2009).[Crossref] Zbl1198.54048
  55. [55] S.-i. Ohta. Markov type of Alexandrov spaces of non-negative curvature. Mathematika, 55(1-2):177-189, (2009).[Crossref] Zbl1195.46019
  56. [56] A. Pietsch. Absolut p-summierende Abbildungen in normierten Räumen. Studia Math., 28:333-353, (1966/1967). Zbl0156.37903
  57. [57] G. Pisier. Martingales with values in uniformly convex spaces. Israel J. Math., 20(3-4):326-350, (1975). Zbl0344.46030
  58. [58] G. Schechtman. More on embedding subspaces of Lp in lnr . Compositio Math., 61(2):159-169, (1987). Zbl0659.46021
  59. [59] K.-T. Sturm. Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. Ann. Probab., 30(3):1195-1222, (2002). Zbl1017.60050
  60. [60] K.-T. Sturm. Probability measures on metric spaces of nonpositive curvature. In Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), volume 338 of Contemp. Math., pages 357-390. Amer. Math. Soc., Providence, RI, (2003). 
  61. [61] M. Talagrand. Embedding subspaces of L1 into lN1 . Proc. Amer. Math. Soc., 108(2):363-369, (1990).[Crossref] 

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