Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

 Access to full text

 Full (PDF)

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] 

1. Assaf Naor, Comparison of Metric Spectral Gaps