METRIC $X_{p}$ INEQUALITIES

ASSAF NAOR ; GIDEON SCHECHTMAN

Forum of Mathematics, Pi, Tome 4 (2016)

Résumé

For every $p\in (0,\infty )$ we associate to every metric space $(X,d_{X})$ a numerical invariant $\mathfrak{X}_{p}(X)\in [0,\infty ]$ such that if $\mathfrak{X}_{p}(X)\infty$ and a metric space $(Y,d_{Y})$ admits a bi-Lipschitz embedding into $X$ then also $\mathfrak{X}_{p}(Y)\infty$ . We prove that if $p,q\in (2,\infty )$ satisfy $q$ then $\mathfrak{X}_{p}(L_{p})\infty$ yet $\mathfrak{X}_{p}(L_{q})=\infty$ . Thus, our new bi-Lipschitz invariant certifies that $L_{q}$ does not admit a bi-Lipschitz embedding into $L_{p}$ when $2$ . This completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the embeddability of $L_{p}$ spaces into each other, the previously understood cases of which were metric notions of type and cotype, which however fail to certify the nonembeddability of $L_{q}$ into $L_{p}$ when $2$ . Among the consequences of our results are new quantitative restrictions on the bi-Lipschitz embeddability into $L_{p}$ of snowflakes of $L_{q}$ and integer grids in $\ell _{q}^{n}$ , for $2$ . As a byproduct of our investigations, we also obtain results on the geometry of the Schatten $p$ trace class $S_{p}$ that are new even in the linear setting.

Détail

DOI : 10.1017/fmp.2016.1

Bibliographie

[1] Akemann, C. A., Anderson, J. and Pedersen, G. K., ‘Triangle inequalities in operator algebras’, Linear Multilinear Algebra 11(2) (1982), 167–178.Google Scholar | DOI

[2] Albiac, F. and Baudier, F., ‘Embeddability of snowflaked metrics with applications to the nonlinear geometry of the spaces L and ℓ for 0 < p < ∞ ’, J. Geom. Anal. 25(1) (2015), 1–24.Google Scholar | DOI

[3] Ando, T. and Zhan, X., ‘Norm inequalities related to operator monotone functions’, Math. Ann. 315(4) (1999), 771–780.Google Scholar | DOI

[4] Andoni, A., Naor, A. and Neiman, O., ‘Snowflake universality of Wasserstein spaces’, Preprint, 2015, arXiv:1509.08677.Google Scholar

[5] Aronszajn, N., ‘Differentiability of Lipschitzian mappings between Banach spaces’, Studia Math. 57(2) (1976), 147–190.Google Scholar | DOI

[6] Aujla, J. S. and Bourin, J.-C., ‘Eigenvalue inequalities for convex and log-convex functions’, Linear Algebra Appl. 424(1) (2007), 25–35.Google Scholar | DOI

[7] Austin, T. and Naor, A., ‘On the bi-Lipschitz structure of Wasserstein spaces’, Preprint, 2015.Google Scholar

[8] Austin, T., Naor, A. and Peres, Y., ‘The wreath product of ℤ with ℤ has Hilbert compression exponent 2∕3’, Proc. Amer. Math. Soc. 137(1) (2009), 85–90.Google Scholar | DOI

[9] Ball, K., ‘Markov chains, Riesz transforms and Lipschitz maps’, Geom. Funct. Anal. 2(2) (1992), 137–172.Google Scholar | DOI

[10] Ball, K., ‘The Ribe programme’, Astérisque (352): Exp. No. 1047, viii, 147, 159, 2013, Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058.Google Scholar

[11] Ball, K., Carlen, E. A. and Lieb, E. H., ‘Sharp uniform convexity and smoothness inequalities for trace norms’, Invent. Math. 115(3) (1994), 463–482.Google Scholar | DOI

[12] Banach, S., Théorie des opérations linéaires. Éditions Jacques Gabay, Sceaux, 1993. Reprint of the 1932 original.Google Scholar

[13] Bartal, Y., Linial, N., Mendel, M. and Naor, A., ‘On metric Ramsey-type phenomena’, Ann. of Math. (2) 162(2) (2005), 643–709.Google Scholar | DOI

[14] Baudier, F., ‘Quantitative nonlinear embeddings into Lebesgue sequence spaces’, J. Topol. Anal., to appear, Preprint, 2012, arXiv:1210.0588.Google Scholar

[15] Benyamini, Y. and Lindenstrauss, J., Geometric Nonlinear Functional Analysis, Vol. 1, American Mathematical Society Colloquium Publications, 48 (American Mathematical Society, Providence, RI, 2000).Google Scholar

[16] Bhatia, R., Matrix Analysis, Graduate Texts in Mathematics, 169 (Springer, New York, 1997).Google Scholar | DOI

[17] Bourgain, J., ‘The metrical interpretation of superreflexivity in Banach spaces’, Israel J. Math. 56(2) (1986), 222–230.Google Scholar | DOI

[18] Bourgain, J., ‘Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms’, inGeometrical Aspects of Functional Analysis (1985/86), Lecture Notes in Mathematics, 1267 (Springer, Berlin, 1987), 157–167.Google Scholar | DOI

[19] Bourgain, J., Milman, V. and Wolfson, H., ‘On type of metric spaces’, Trans. Amer. Math. Soc. 294(1) (1986), 295–317.Google Scholar | DOI

[20] Bourin, J.-C. and Uchiyama, M., ‘A matrix subadditivity inequality for f (A + B) and f (A) + f (B)’, Linear Algebra Appl. 423(2–3) (2007), 512–518.Google Scholar | DOI

[21] Bretagnolle, J., Dacunha-Castelle, D. and Krivine, J.-L., ‘Fonctions de type positif sur les espaces L p ’, C. R. Acad. Sci. Paris 261 (1965), 2153–2156.Google Scholar

[22] Brieussel, J. and Zheng, T., ‘Speed of random walks, isoperimetry and compression of finitely generated groups’, Preprint, 2015, arXiv:1510.08040.Google Scholar

[23] Carlen, E., ‘Trace inequalities and quantum entropy: an introductory course’, inEntropy and the Quantum, Contemporary Mathematics, 529 (American Mathematical Society, Providence, RI, 2010), 73–140.Google Scholar

[24] Carlen, E. A. and Lieb, E. H., ‘A Minkowski type trace inequality and strong subadditivity of quantum entropy. II. Convexity and concavity’, Lett. Math. Phys. 83(2) (2008), 107–126.Google Scholar | DOI

[25] Christensen, J. P. R., ‘Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings’, Publ. Dép. Math. (Lyon) 10(2) (1973), 29–39. Actes du Deuxième Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1973), I, pp. 29–39.Google Scholar

[26] David, G. and Semmes, S., Fractured Fractals and Broken Dreams, Oxford Lecture Series in Mathematics and its Applications, 7 (The Clarendon Press, Oxford University Press, New York, 1997), Self-similar geometry through metric and measure.Google Scholar | DOI

[27] Ding, J., Lee, J. R. and Peres, Y., ‘Markov type and threshold embeddings’, Geom. Funct. Anal. 23(4) (2013), 1207–1229.Google Scholar | DOI

[28] Dixmier, J., ‘Formes linéaires sur un anneau d’opérateurs’, Bull. Soc. Math. France 81 (1953), 9–39.Google Scholar | DOI

[29] Enflo, P., ‘On the nonexistence of uniform homeomorphisms between L -spaces’, Ark. Mat. 8 (1969), 103–105.Google Scholar | DOI

[30] Enflo, P., ‘Uniform structures and square roots in topological groups. I, II’, Israel J. Math. 8 (1970), 230–252. ibid., (1970), 253–272.Google Scholar | DOI

[31] Enflo, P., ‘Uniform homeomorphisms between Banach spaces’, inSéminaire Maurey-Schwartz (1975–1976), Espaces, L p , applications radonifiantes et géométrie des espaces de Banach, Exp. No. 18 (Centre Math., École Polytech., Palaiseau, 1976), 7.Google Scholar

[32] Epstein, H., ‘Remarks on two theorems of E. Lieb’, Comm. Math. Phys. 31 (1973), 317–325.Google Scholar | DOI

[33] Figiel, T., Johnson, W. B. and Schechtman, G., ‘Random sign embeddings from l n ,  2 < r < ∞ ’, Proc. Amer. Math. Soc. 102(1) (1988), 102–106.Google Scholar

[34] Giladi, O., Mendel, M. and Naor, A., ‘Improved bounds in the metric cotype inequality for Banach spaces’, J. Funct. Anal. 260(1) (2011), 164–194.Google Scholar | DOI

[35] Giladi, O. and Naor, A., ‘Improved bounds in the scaled Enflo type inequality for Banach spaces’, Extracta Math. 25(2) (2010), 151–164.Google Scholar

[36] Giladi, O., Naor, A. and Schechtman, G., ‘Bourgain’s discretization theorem’, Ann. Fac. Sci. Toulouse Math. (6) 21(4) (2012), 817–837.Google Scholar | DOI

[37] Gluskin, E. D., Pietsch, A. and Puhl, J., ‘A generalization of Khintchine’s inequality and its application in the theory of operator ideals’, Studia Math. 67(2) (1980), 149–155.Google Scholar | DOI

[38] Gromov, M., ‘Filling Riemannian manifolds’, J. Differential Geom. 18(1) (1983), 1–147.Google Scholar | DOI

[39] Hytönen, T. and Naor, A., ‘Pisier’s inequality revisited’, Studia Math. 215(3) (2013), 221–235.Google Scholar | DOI

[40] Johnson, W. B., Maurey, B., Schechtman, G. and Tzafriri, L., ‘Symmetric structures in Banach spaces’, Mem. Amer. Math. Soc. 19(217) (1979), v+298.Google Scholar

[41] Johnson, W. B., Schechtman, G. and Zinn, J., ‘Best constants in moment inequalities for linear combinations of independent and exchangeable random variables’, Ann. Probab. 13(1) (1985), 234–253.Google Scholar | DOI

[42] Kadeć, M. Ĭ., ‘Linear dimension of the spaces L and l ’, Uspehi Mat. Nauk 13(6(84)) (1958), 95–98.Google Scholar

[43] Kadec, M. I. and Pełczyński, A., ‘Bases, lacunary sequences and complemented subspaces in the spaces L ’, Studia Math. 21 (1961/1962), 161–176.Google Scholar | DOI

[44] Kalton, N. J., ‘The nonlinear geometry of Banach spaces’, Rev. Mat. Complut. 21(1) (2008), 7–60.Google Scholar | DOI

[45] Kalton, N. J. and Randrianarivony, N. L., ‘The coarse Lipschitz geometry of l ⊕l ’, Math. Ann. 341(1) (2008), 223–237.Google Scholar | DOI

[46] Ledoux, M. and Talagrand, M., Probability in Banach Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23 (Springer, Berlin, 1991), Isoperimetry and processes.Google Scholar | DOI

[47] Lee, J. R., Naor, A. and Peres, Y., ‘Trees and Markov convexity’, Geom. Funct. Anal. 18(5) (2009), 1609–1659.Google Scholar | DOI

[48] Li, S., ‘Markov convexity and nonembeddability of the Heisenberg group’, Ann. Inst. Fourier, to appear, Preprint, 2014, arXiv:1404.6751.Google Scholar

[49] Lieb, E. H. and Thirring, W. E., ‘Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities’, inStudies in Mathematical Physics (Princeton University Press, Princeton, NJ, 1976), 269–303.Google Scholar

[50] Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces. I (Springer, Berlin, New York, 1977), Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92.Google Scholar | DOI

[51] Linial, N., Magen, A. and Naor, A., ‘Girth and Euclidean distortion’, Geom. Funct. Anal. 12(2) (2002), 380–394.Google Scholar | DOI

[52] Lövblom, G.-M., ‘Uniform homeomorphisms between unit balls in L -spaces’, Math. Scand. 62(2) (1988), 294–302.Google Scholar | DOI

[53] Lust-Piquard, F., ‘Inégalités de Khintchine dans C  (1 < p < ∞)’, C. R. Acad. Sci. Paris Sér. I Math. 303(7) (1986), 289–292.Google Scholar

[54] Mankiewicz, P., ‘On Lipschitz mappings between Fréchet spaces’, Studia Math. 41 (1972), 225–241.Google Scholar | DOI

[55] Matoušek, J., Lectures on Discrete Geometry, Graduate Texts in Mathematics, 212 (Springer, New York, 2002).Google Scholar | DOI

[56] Maurey, B., ‘Type, cotype and K-convexity’, inHandbook of the Geometry of Banach Spaces, Vol. 2 (North-Holland, Amsterdam, 2003), 1299–1332.Google Scholar | DOI

[57] Mccarthy, C. A., ‘ c ’, Israel J. Math. 5 (1967), 249–271.Google Scholar | DOI

[58] Mendel, M. and Naor, A., ‘Euclidean quotients of finite metric spaces’, Adv. Math. 189(2) (2004), 451–494.Google Scholar | DOI

[59] Mendel, M. and Naor, A., ‘Some applications of Ball’s extension theorem’, Proc. Amer. Math. Soc. 134(9) (2006), 2577–2584. (electronic).Google Scholar | DOI

[60] Mendel, M. and Naor, A., ‘Scaled Enflo type is equivalent to Rademacher type’, Bull. Lond. Math. Soc. 39(3) (2007), 493–498.Google Scholar | DOI

[61] Mendel, M. and Naor, A., ‘Metric cotype’, Ann. of Math. (2) 168(1) (2008), 247–298.Google Scholar | DOI

[62] Mendel, M. and Naor, A., ‘Markov convexity and local rigidity of distorted metrics’, J. Eur. Math. Soc. (JEMS) 15(1) (2013), 287–337.Google Scholar | DOI

[63] Mendel, M. and Naor, A., ‘Spectral calculus and Lipschitz extension for barycentric metric spaces’, Anal. Geom. Metr. Spaces 1 (2013), 163–199.Google Scholar | DOI

[64] Mendel, M. and Naor, A., ‘Nonlinear spectral calculus and super-expanders’, Publ. Math. Inst. Hautes Études Sci. 119 (2014), 1–95.Google Scholar | DOI

[65] Mendel, M. and Naor, A., ‘Expanders with respect to Hadamard spaces and random graphs’, Duke Math. J. 164(8) (2015), 1471–1548.Google Scholar | DOI

[66] Naor, A., ‘An introduction to the Ribe program’, Jpn. J. Math. 7(2) (2012), 167–233.Google Scholar | DOI

[67] Naor, A., ‘Comparison of metric spectral gaps’, Anal. Geom. Metr. Spaces 2 (2014), Art. 1.Google Scholar

[68] Naor, A., ‘Discrete Riesz transforms and sharp metric inequalities’, Preprint, 2016,arXiv:1601.03332.Google Scholar | DOI

[69] Naor, A. and Peres, Y., ‘Embeddings of discrete groups and the speed of random walks’, Int. Math. Res. Not. IMRN (2008), pages Art. ID rnn 076, 34.Google Scholar | DOI

[70] Naor, A. and Peres, Y., ‘ L compression, traveling salesmen, and stable walks’, Duke Math. J. 157(1) (2011), 53–108.Google Scholar | DOI

[71] Naor, A., Peres, Y., Schramm, O. and Sheffield, S., ‘Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces’, Duke Math. J. 134(1) (2006), 165–197.Google Scholar | DOI

[72] Naor, A. and Schechtman, G., ‘Remarks on non linear type and Pisier’s inequality’, J. Reine Angew. Math. 552 (2002), 213–236.Google Scholar

[73] Naor, A. and Silberman, L., ‘Poincaré inequalities, embeddings, and wild groups’, Compos. Math. 147(5) (2011), 1546–1572.Google Scholar | DOI

[74] Ohta, S.-I., ‘Markov type of Alexandrov spaces of non-negative curvature’, Mathematika 55(1–2) (2009), 177–189.Google Scholar | DOI

[75] Ostrovskii, M. I., Metric Embeddings, De Gruyter Studies in Mathematics, 49 (De Gruyter, Berlin, 2013), Bilipschitz and coarse embeddings into Banach spaces.Google Scholar | DOI

[76] Paley, R. E. A. C., ‘Some theorems on abstract spaces’, Bull. Amer. Math. Soc. 42(4) (1936), 235–240.Google Scholar | DOI

[77] Paley, R. E. A. C. and Zygmund, A., ‘On some series of functions, (1)’, Math. Proc. Cambridge Philos. Soc. 26(3) (1930), 337–357.Google Scholar | DOI

[78] Pisier, G., ‘Some results on Banach spaces without local unconditional structure’, Compos. Math. 37(1) (1978), 3–19.Google Scholar

[79] Pisier, G., ‘Probabilistic methods in the geometry of Banach spaces’, inProbability and Analysis (Varenna, 1985), Lecture Notes in Mathematics, 1206 (Springer, Berlin, 1986), 167–241.Google Scholar | DOI

[80] Pisier, G., ‘Non-commutative vector valued L -spaces and completely p-summing maps’, Astérisque 247 (1998), vi+131.Google Scholar

[81] Pisier, G. and Xu, Q., ‘Non-commutative L p -spaces’, inHandbook of the Geometry of Banach Spaces, Vol. 2 (North-Holland, Amsterdam, 2003), 1459–1517.Google Scholar | DOI

[82] Ribe, M., ‘On uniformly homeomorphic normed spaces’, Ark. Mat. 14(2) (1976), 237–244.Google Scholar | DOI

[83] Rosenthal, H. P., ‘On the subspaces of L p (p > 2 spanned by sequences of independent random variables’, Israel J. Math. 8 (1970), 273–303.Google Scholar | DOI

[84] Schoenberg, I. J., ‘Metric spaces and positive definite functions’, Trans. Amer. Math. Soc. 44(3) (1938), 522–536.Google Scholar | DOI

[85] Sukochev, F. A., ‘Non-isomorphism of L -spaces associated with finite and infinite von Neumann algebras’, Proc. Amer. Math. Soc. 124(5) (1996), 1517–1527.Google Scholar | DOI

[86] Talagrand, M., ‘Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis’ graph connectivity theorem’, Geom. Funct. Anal. 3(3) (1993), 295–314.Google Scholar | DOI

[87] Veomett, E. and Wildrick, K., ‘Spaces of small metric cotype’, J. Topol. Anal. 2(4) (2010), 581–597.Google Scholar | DOI

[88] Wagner, R., ‘Notes on an inequality by Pisier for functions on the discrete cube’, inGeometric Aspects of Functional Analysis, Lecture Notes in Mathematics, 1745 (Springer, Berlin, 2000), 263–268.Google Scholar | DOI

[89] Wells, J. H. and Williams, L. R., Embeddings and Extensions in Analysis (Springer, New York, Heidelberg, 1975), Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84.Google Scholar | DOI

[90] Wojtaszczyk, P., Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, 25 (Cambridge University Press, Cambridge, 1991).Google Scholar | DOI