Discrete Riesz transforms and sharp metric $X_p$ inequalities

Abstract

For $p\in [2,\infty)$, the metric $X_p$ inequality with sharp scaling parameter is proven here to hold true in $L_p$. The geometric consequences of this result include the following sharp statements about embeddings of $L_q$ into $L_p$ when $2\lt q\lt p\lt \infty$: the maximal $\theta\in (0,1]$ for which $L_q$ admits a bi-$\theta$-Hölder embedding into $L_p$ equals $q/p$, and for $m,n\in \mathbb{N}$, the smallest possible bi-Lipschitz distortion of any embedding into $L_p$ of the grid $\{1,\ldots,m\}^n\subset \ell_q^n$ is bounded above and below by constant multiples (depending only on $p,q$) of the quantity $\min\{n^{(p-q)(q-2)/(q^2(p-2))}, m^{(q-2)/q}\}$.

  • [Bal13] K. Ball, "The Ribe programme," in Séminaire Bourbaki. Vol. 2011/2012, Paris: Math. Soc. France, 2013, vol. 352, p. exp. no. 1047, viii, 147-159.
    @INCOLLECTION{Bal13, mrkey = {3087345},
      issn = {0303-1179},
      author = {Ball, Keith},
      mrclass = {46B85 (46B07 60G46)},
      series = {Astérisque},
      address = {Paris},
      isbn = {978-2-85629-371-3},
      publisher = {Math. Soc. France},
      zblnumber = {1303.46019},
      volume = {352},
      mrnumber = {3087345},
      booktitle = {S{é}minaire Bourbaki. Vol. 2011/2012},
      mrreviewer = {Jes{ú}s A. Jaramillo},
      title = {The {R}ibe programme},
      year = {2013},
      pages = {Exp. No. 1047, viii, 147--159},
      }
  • [Ban86] Go to document R. Bañuelos, "Martingale transforms and related singular integrals," Trans. Amer. Math. Soc., vol. 293, iss. 2, pp. 547-563, 1986.
    @ARTICLE{Ban86, mrkey = {0816309},
      number = {2},
      issn = {0002-9947},
      author = {Ba{ñ}uelos, Rodrigo},
      mrclass = {60G44 (42B20 60G46 60H05)},
      doi = {10.2307/2000021},
      journal = {Trans. Amer. Math. Soc.},
      zblnumber = {0591.60045},
      volume = {293},
      mrnumber = {0816309},
      fjournal = {Transactions of the American Mathematical Society},
      mrreviewer = {Martin L. Silverstein},
      coden = {TAMTAM},
      title = {Martingale transforms and related singular integrals},
      year = {1986},
      pages = {547--563},
      }
  • [EL08] Go to document B. L. Efraim and F. Lust-Piquard, "Poincaré type inequalities on the discrete cube and in the CAR algebra," Probab. Theory Related Fields, vol. 141, iss. 3-4, pp. 569-602, 2008.
    @ARTICLE{EL08, mrkey = {2391165},
      number = {3-4},
      issn = {0178-8051},
      author = {Efraim, L. Ben and Lust-Piquard, F.},
      mrclass = {46L51 (43A70 46B99 46E30 46N30 60E15)},
      doi = {10.1007/s00440-007-0094-x},
      journal = {Probab. Theory Related Fields},
      zblnumber = {1141.60005},
      volume = {141},
      mrnumber = {2391165},
      fjournal = {Probability Theory and Related Fields},
      coden = {PTRFEU},
      title = {Poincaré type inequalities on the discrete cube and in the {CAR} algebra},
      year = {2008},
      pages = {569--602},
      }
  • [BL00] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, Provvidence, RI: Amer. Math. Soc., 2000, vol. 48.
    @BOOK{BL00, mrkey = {1727673},
      author = {Benyamini, Yoav and Lindenstrauss, Joram},
      mrclass = {46-02 (46Bxx 46T99 47-02)},
      series = {Amer. Math. Soc. Colloq. Publ.},
      isbn = {0-8218-0835-4},
      address = {Provvidence, RI},
      publisher = {Amer. Math. Soc.},
      zblnumber = {0946.46002},
      volume = {48},
      mrnumber = {1727673},
      mrreviewer = {Gilles Godefroy},
      title = {Geometric Nonlinear Functional Analysis. {V}ol. 1},
      year = {2000},
      pages = {xii+488},
      }
  • [Bon68] A. Bonami, "Construction d’opérateurs de convolution sur le groupe $D^{\infty }$," C. R. Acad. Sci. Paris Sér. A-B, vol. 266, p. a864-a866, 1968.
    @ARTICLE{Bon68, mrkey = {0231131},
      volume = {266},
      mrnumber = {0231131},
      author = {Bonami, Aline},
      mrclass = {42.50},
      mrreviewer = {J. H. Neuwirth},
      title = {Construction d'opérateurs de convolution sur le groupe {$D\sp{\infty }$}},
      pages = {A864--A866},
      year = {1968},
      journal = {C. R. Acad. Sci. Paris Sér. A-B},
      zblnumber = {0159.42302},
     }
  • [Bou87] Go to document J. Bourgain, "Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms," in Geometrical Aspects of Functional Analysis (1985/86), New York: Springer-Verlag, 1987, vol. 1267, pp. 157-167.
    @INCOLLECTION{Bou87, mrkey = {0907692},
      author = {Bourgain, J.},
      mrclass = {46B20 (46B25 46E30)},
      series = {Lecture Notes in Math.},
      address = {New York},
      publisher = {Springer-Verlag},
      doi = {10.1007/BFb0078143},
      volume = {1267},
      mrnumber = {0907692},
      booktitle = {Geometrical Aspects of Functional Analysis (1985/86)},
      mrreviewer = {G. Schechtman},
      title = {Remarks on the extension of {L}ipschitz maps defined on discrete sets and uniform homeomorphisms},
      pages = {157--167},
      year = {1987},
      zblnumber = {0633.46018},
      }
  • [Bou-hilbert] J. Bourgain, "Vector-valued singular integrals and the $H^1$-BMO duality," in Probability Theory and Harmonic Analysis, New York: Dekker, 1986, vol. 98, pp. 1-19.
    @INCOLLECTION{Bou-hilbert, mrkey = {0830227},
      author = {Bourgain, Jean},
      mrclass = {42B20 (46E40 47B10)},
      series = {Monogr. Textbooks Pure Appl. Math.},
      address = {New York},
      publisher = {Dekker},
      zblnumber = {0602.42015},
      volume = {98},
      mrnumber = {0830227},
      booktitle = {Probability Theory and Harmonic Analysis},
      mrreviewer = {Mario Milman},
      venue = {{C}leveland, {O}hio, 1983},
      title = {Vector-valued singular integrals and the {$H\sp 1$}-{BMO} duality},
      pages = {1--19},
      year = {1986},
      }
  • [CW76] R. R. Coifman and G. Weiss, Transference Methods in Analysis, Providence, RI: Amer. Math. Soc., 1976, vol. 31.
    @BOOK{CW76, mrkey = {0481928},
      author = {Coifman, Ronald R. and Weiss, Guido},
      mrclass = {43A22 (28A65 42-02 43A85)},
      series = {Conf. Board Math. Sci. Reg. Conf. Ser. Math.},
      isbn = {0-8218-1681-0},
      address = {Providence, RI},
      publisher = {Amer. Math. Soc.},
      volume = {31},
      mrnumber = {0481928},
      mrreviewer = {Alberto Torchinsky},
      title = {Transference Methods in Analysis},
      year = {1976},
      pages = {ii+59},
      zblnumber = {0377.43001},
      }
  • [DR85] J. Duoandikoetxea and J. L. Rubio de Francia, "Estimations indépendantes de la dimension pour les transformées de Riesz," C. R. Acad. Sci. Paris Sér. I Math., vol. 300, iss. 7, pp. 193-196, 1985.
    @ARTICLE{DR85, mrkey = {0780616},
      number = {7},
      issn = {0249-6291},
      author = {Duoandikoetxea, Javier and Rubio de Francia, Jos{é} L.},
      mrclass = {42B20},
      journal = {C. R. Acad. Sci. Paris Sér. I Math.},
      zblnumber = {0577.42015},
      volume = {300},
      mrnumber = {0780616},
      fjournal = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
      mrreviewer = {Gerald B. Folland},
      coden = {CASMEI},
      title = {Estimations indépendantes de la dimension pour les transformées de {R}iesz},
      year = {1985},
      pages = {193--196},
      }
  • [FJS88] Go to document T. Figiel, W. B. Johnson, and G. Schechtman, "Random sign embeddings from $l^n_r,\;2<r<\infty$," Proc. Amer. Math. Soc., vol. 102, iss. 1, pp. 102-106, 1988.
    @ARTICLE{FJS88, mrkey = {0915724},
      number = {1},
      issn = {0002-9939},
      author = {Figiel, T. and Johnson, W. B. and Schechtman, G.},
      mrclass = {46B25 (47B10 47B37 47D30)},
      doi = {10.2307/2046039},
      journal = {Proc. Amer. Math. Soc.},
      zblnumber = {0655.46016},
      volume = {102},
      mrnumber = {0915724},
      fjournal = {Proceedings of the American Mathematical Society},
      coden = {PAMYAR},
      title = {Random sign embeddings from {$l\sp n\sb r,\;2   year = {1988},
      pages = {102--106},
      }
  • [GNS12] Go to document O. Giladi, A. Naor, and G. Schechtman, "Bourgain’s discretization theorem," Ann. Fac. Sci. Toulouse Math., vol. 21, iss. 4, pp. 817-837, 2012.
    @ARTICLE{GNS12, mrkey = {3052032},
      number = {4},
      issn = {0240-2963},
      author = {Giladi, Ohad and Naor, Assaf and Schechtman, Gideon},
      mrclass = {46B20 (46-02)},
      journal = {Ann. Fac. Sci. Toulouse Math.},
      zblnumber = {1283.46015},
      volume = {21},
      mrnumber = {3052032},
      fjournal = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
      title = {Bourgain's discretization theorem},
      year = {2012},
      pages = {817--837},
      doi = {10.5802/afst.1352},
     }
  • [GPP80] E. D. Gluskin, A. Pietsch, and J. Puhl, "A generalization of Khintchine’s inequality and its application in the theory of operator ideals," Studia Math., vol. 67, iss. 2, pp. 149-155, 1980.
    @ARTICLE{GPP80, mrkey = {0583295},
      number = {2},
      issn = {0039-3223},
      author = {Gluskin, E. D. and Pietsch, A. and Puhl, J.},
      mrclass = {47B10 (47D30)},
      journal = {Studia Math.},
      zblnumber = {0441.47032},
      volume = {67},
      mrnumber = {0583295},
      fjournal = {Polska Akademia Nauk. Institut Matematyczny. Studia Mathematica},
      mrreviewer = {Bernd Rosenberger},
      coden = {SMATAZ},
      title = {A generalization of {K}hintchine's inequality and its application in the theory of operator ideals},
      year = {1980},
      pages = {149--155},
      }
  • [Gun86] R. F. Gundy, "Sur les transformations de Riesz pour le semi-groupe d’Ornstein-Uhlenbeck," C. R. Acad. Sci. Paris Sér. I Math., vol. 303, iss. 19, pp. 967-970, 1986.
    @ARTICLE{Gun86, mrkey = {0877182},
      number = {19},
      issn = {0249-6291},
      author = {Gundy, Richard F.},
      mrclass = {60H07 (47A35 60J60)},
      journal = {C. R. Acad. Sci. Paris Sér. I Math.},
      zblnumber = {0606.60063},
      volume = {303},
      mrnumber = {0877182},
      fjournal = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
      coden = {CASMEI},
      title = {Sur les transformations de {R}iesz pour le semi-groupe d'{O}rnstein-{U}hlenbeck},
      year = {1986},
      pages = {967--970},
      }
  • [GV79] R. F. Gundy and N. T. Varopoulos, "Les transformations de Riesz et les intégrales stochastiques," C. R. Acad. Sci. Paris Sér. A-B, vol. 289, iss. 1, p. a13-a16, 1979.
    @ARTICLE{GV79, mrkey = {0545671},
      number = {1},
      issn = {0151-0509},
      author = {Gundy, Richard F. and Varopoulos, Nicolas Th.},
      mrclass = {60H05 (60J65)},
      journal = {C. R. Acad. Sci. Paris Sér. A-B},
      zblnumber = {0413.60003},
      volume = {289},
      mrnumber = {0545671},
      fjournal = {Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B},
      mrreviewer = {Maurizio Pratelli},
      coden = {CHASAP},
      title = {Les transformations de {R}iesz et les intégrales stochastiques},
      year = {1979},
      pages = {A13--A16},
      }
  • [Hei80] Go to document S. Heinrich, "Ultraproducts in Banach space theory," J. Reine Angew. Math., vol. 313, pp. 72-104, 1980.
    @ARTICLE{Hei80, mrkey = {0552464},
      issn = {0075-4102},
      author = {Heinrich, Stefan},
      mrclass = {46Bxx (03C20 03H20 46-02)},
      doi = {10.1515/crll.1980.313.72},
      journal = {J. Reine Angew. Math.},
      zblnumber = {0412.46017},
      volume = {313},
      mrnumber = {0552464},
      fjournal = {Journal für die Reine und Angewandte Mathematik},
      mrreviewer = {C. Ward Henson},
      coden = {JRMAA8},
      title = {Ultraproducts in {B}anach space theory},
      year = {1980},
      pages = {72--104},
      }
  • [JMST79] Go to document W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, "Symmetric structures in Banach spaces," Mem. Amer. Math. Soc., vol. 19, iss. 217, p. v, 1979.
    @ARTICLE{JMST79, mrkey = {0527010},
      number = {217},
      issn = {0065-9266},
      author = {Johnson, W. B. and Maurey, B. and Schechtman, G. and Tzafriri, L.},
      mrclass = {46B25 (46E30 47B38)},
      doi = {10.1090/memo/0217},
      journal = {Mem. Amer. Math. Soc.},
      zblnumber = {0421.46023},
      volume = {19},
      mrnumber = {0527010},
      fjournal = {Memoirs of the American Mathematical Society},
      mrreviewer = {Z. Chanturia},
      coden = {MAMCAU},
      title = {Symmetric structures in {B}anach spaces},
      year = {1979},
      pages = {v+298},
      }
  • [JSZ85] Go to document W. B. Johnson, G. Schechtman, and J. Zinn, "Best constants in moment inequalities for linear combinations of independent and exchangeable random variables," Ann. Probab., vol. 13, iss. 1, pp. 234-253, 1985.
    @ARTICLE{JSZ85, mrkey = {0770640},
      number = {1},
      issn = {0091-1798},
      author = {Johnson, W. B. and Schechtman, G. and Zinn, J.},
      mrclass = {60E15 (60G50)},
      url = {http://www.jstor.org/stable/2243635},
      journal = {Ann. Probab.},
      zblnumber = {0564.60020},
      volume = {13},
      mrnumber = {0770640},
      fjournal = {The Annals of Probability},
      mrreviewer = {Olav Kallenberg},
      coden = {APBYAE},
      title = {Best constants in moment inequalities for linear combinations of independent and exchangeable random variables},
      year = {1985},
      pages = {234--253},
      }
  • [Kah64] J. Kahane, "Sur les sommes vectorielles $\sum \pm u_{n}$," C. R. Acad. Sci. Paris, vol. 259, pp. 2577-2580, 1964.
    @ARTICLE{Kah64, mrkey = {0169279},
      volume = {259},
      mrnumber = {0169279},
      author = {Kahane, Jean-Pierre},
      mrclass = {60.08},
      mrreviewer = {J. Bass},
      title = {Sur les sommes vectorielles {$\sum \pm u\sb{n}$}},
      pages = {2577--2580},
      year = {1964},
      journal = {C. R. Acad. Sci. Paris},
      zblnumber = {0127.35504},
      }
  • [LT77] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I, New York: Springer-Verlag, 1977, vol. 92.
    @BOOK{LT77, mrkey = {0500056},
      author = {Lindenstrauss, Joram and Tzafriri, Lior},
      mrclass = {46-02 (46A45 46BXX)},
      series = {Ergeb. Math. Grenzgeb.},
      address = {New York},
      isbn = {3-540-08072-4},
      publisher = {Springer-Verlag},
      zblnumber = {0362.46013},
      volume = {92},
      mrnumber = {0500056},
      note = {Sequence spaces},
      mrreviewer = {S. V. Kisljakov},
      title = {Classical {B}anach spaces. {I}},
      year = {1977},
      pages = {xiii+188},
      }
  • [Lus86] F. Lust-Piquard, "Inégalités de Khintchine dans $C_p\;(1<p<\infty)$," C. R. Acad. Sci. Paris Sér. I Math., vol. 303, iss. 7, pp. 289-292, 1986.
    @ARTICLE{Lus86, mrkey = {0859804},
      number = {7},
      issn = {0249-6291},
      author = {Lust-Piquard, Fran{ç}oise},
      mrclass = {47B10 (47D25 60B11)},
      journal = {C. R. Acad. Sci. Paris Sér. I Math.},
      zblnumber = {0592.47038},
      volume = {303},
      mrnumber = {0859804},
      fjournal = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
      mrreviewer = {Werner Linde},
      coden = {CASMEI},
      title = {Inégalités de {K}hintchine dans {$C\sb p\;(1   year = {1986},
      pages = {289--292},
      }
  • [Lus98] Go to document F. Lust-Piquard, "Riesz transforms associated with the number operator on the Walsh system and the fermions," J. Funct. Anal., vol. 155, iss. 1, pp. 263-285, 1998.
    @ARTICLE{Lus98, mrkey = {1623158},
      number = {1},
      issn = {0022-1236},
      author = {Lust-Piquard, Fran{ç}oise},
      mrclass = {81S05 (42C10 46N50 47A30 47N50)},
      doi = {10.1006/jfan.1997.3217},
      journal = {J. Funct. Anal.},
      zblnumber = {0908.47027},
      volume = {155},
      mrnumber = {1623158},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Fumio Hiai},
      coden = {JFUAAW},
      title = {Riesz transforms associated with the number operator on the {W}alsh system and the fermions},
      year = {1998},
      pages = {263--285},
      }
  • [Mau03] Go to document B. Maurey, "Type, cotype and $K$-convexity," in Handbook of the Geometry of Banach Spaces, Vol. 2, Amsterdam: North-Holland, 2003, pp. 1299-1332.
    @INCOLLECTION{Mau03, mrkey = {1999197},
      author = {Maurey, Bernard},
      mrclass = {46B03 (46B07 46B08 46B20)},
      address = {Amsterdam},
      publisher = {North-Holland},
      doi = {10.1016/S1874-5849(03)80037-2},
      zblnumber = {1074.46006},
      mrnumber = {1999197},
      booktitle = {Handbook of the Geometry of {B}anach Spaces, {V}ol. 2},
      mrreviewer = {Fran{ç}oise Lust-Piquard},
      title = {Type, cotype and {$K$}-convexity},
      pages = {1299--1332},
      year = {2003},
      }
  • [MP76] B. Maurey and G. Pisier, "Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach," Studia Math., vol. 58, iss. 1, pp. 45-90, 1976.
    @ARTICLE{MP76, mrkey = {0443015},
      number = {1},
      issn = {0039-3223},
      author = {Maurey, Bernard and Pisier, Gilles},
      mrclass = {60B05 (46B05)},
      journal = {Studia Math.},
      zblnumber = {0344.47014},
      volume = {58},
      mrnumber = {0443015},
      fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
      mrreviewer = {J. Hoffmann-Jorgensen},
      title = {Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de {B}anach},
      year = {1976},
      pages = {45--90},
      }
  • [McC67] Go to document C. A. McCarthy, "$c_{p}$," Israel J. Math., vol. 5, pp. 249-271, 1967.
    @ARTICLE{McC67, mrkey = {0225140},
      issn = {0021-2172},
      author = {McCarthy, Charles A.},
      mrclass = {46.10},
      doi = {10.1007/BF02771613},
      journal = {Israel J. Math.},
      volume = {5},
      mrnumber = {0225140},
      fjournal = {Israel Journal of Mathematics},
      mrreviewer = {T. L. Hicks},
      title = {{$c\sb{p}$}},
      year = {1967},
      pages = {249--271},
      zblnumber = {0156.37902},
      }
  • [MN04] Go to document M. Mendel and A. Naor, "Euclidean quotients of finite metric spaces," Adv. Math., vol. 189, iss. 2, pp. 451-494, 2004.
    @ARTICLE{MN04, mrkey = {2101227},
      number = {2},
      issn = {0001-8708},
      author = {Mendel, Manor and Naor, Assaf},
      mrclass = {54E35 (46B20 46C05)},
      doi = {10.1016/j.aim.2003.12.001},
      journal = {Adv. Math.},
      zblnumber = {1088.46007},
      volume = {189},
      mrnumber = {2101227},
      fjournal = {Advances in Mathematics},
      mrreviewer = {Hermann Render},
      coden = {ADMTA4},
      title = {Euclidean quotients of finite metric spaces},
      year = {2004},
      pages = {451--494},
      }
  • [MN06] Go to document M. Mendel and A. Naor, "Some applications of Ball’s extension theorem," Proc. Amer. Math. Soc., vol. 134, iss. 9, pp. 2577-2584, 2006.
    @ARTICLE{MN06, mrkey = {2213735},
      number = {9},
      issn = {0002-9939},
      author = {Mendel, Manor and Naor, Assaf},
      mrclass = {46B20 (51F99)},
      doi = {10.1090/S0002-9939-06-08298-0},
      journal = {Proc. Amer. Math. Soc.},
      zblnumber = {1108.46052},
      volume = {134},
      mrnumber = {2213735},
      fjournal = {Proceedings of the American Mathematical Society},
      mrreviewer = {Mikhail Ostrovski{\u\i}},
      coden = {PAMYAR},
      title = {Some applications of {B}all's extension theorem},
      year = {2006},
      pages = {2577--2584},
      }
  • [MN08] Go to document M. Mendel and A. Naor, "Metric cotype," Ann. of Math., vol. 168, iss. 1, pp. 247-298, 2008.
    @ARTICLE{MN08, mrkey = {2415403},
      number = {1},
      issn = {0003-486X},
      author = {Mendel, Manor and Naor, Assaf},
      mrclass = {46B07 (05C12 46B20 54E35 90C27)},
      doi = {10.4007/annals.2008.168.247},
      journal = {Ann. of Math.},
      zblnumber = {1187.46014},
      volume = {168},
      mrnumber = {2415403},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Mikhail Ostrovskii},
      coden = {ANMAAH},
      title = {Metric cotype},
      year = {2008},
      pages = {247--298},
      }
  • [Mey84] Go to document P. -A. Meyer, "Transformations de Riesz pour les lois gaussiennes," in Seminar on Probability, XVIII, New York: Springer-Verlag, 1984, vol. 1059, pp. 179-193.
    @INCOLLECTION{Mey84, mrkey = {0770960},
      author = {Meyer, P.-A.},
      mrclass = {60H07},
      series = {Lecture Notes in Math.},
      address = {New York},
      publisher = {Springer-Verlag},
      doi = {10.1007/BFb0100043},
      volume = {1059},
      mrnumber = {0770960},
      booktitle = {Seminar on Probability, {XVIII}},
      mrreviewer = {Naresh C. Jain},
      title = {Transformations de {R}iesz pour les lois gaussiennes},
      pages = {179--193},
      year = {1984},
      zblnumber = {0543.60078},
      }
  • [Nao12] Go to document A. Naor, "An introduction to the Ribe program," Jpn. J. Math., vol. 7, iss. 2, pp. 167-233, 2012.
    @ARTICLE{Nao12, mrkey = {2995229},
      number = {2},
      issn = {0289-2316},
      author = {Naor, Assaf},
      mrclass = {46-02 (46B07 46B80 46B85)},
      doi = {10.1007/s11537-012-1222-7},
      journal = {Jpn. J. Math.},
      zblnumber = {1261.46013},
      volume = {7},
      mrnumber = {2995229},
      fjournal = {Japanese Journal of Mathematics},
      title = {An introduction to the {R}ibe program},
      year = {2012},
      pages = {167--233},
      }
  • [NS02] Go to document A. Naor and G. Schechtman, "Remarks on non linear type and Pisier’s inequality," J. Reine Angew. Math., vol. 552, pp. 213-236, 2002.
    @ARTICLE{NS02, mrkey = {1940437},
      issn = {0075-4102},
      author = {Naor, Assaf and Schechtman, Gideon},
      mrclass = {46B03 (46B20)},
      doi = {10.1515/crll.2002.092},
      journal = {J. Reine Angew. Math.},
      zblnumber = {1033.46013},
      volume = {552},
      mrnumber = {1940437},
      fjournal = {Journal für die Reine und Angewandte Mathematik},
      mrreviewer = {Vania D. Mascioni},
      coden = {JRMAA8},
      title = {Remarks on non linear type and {P}isier's inequality},
      year = {2002},
      pages = {213--236},
      }
  • [NS14] Go to document A. Naor and G. Schechtman, "Metric $X_p$ inequalities," Forum Math. Pi, vol. 4, p. 3, 2016.
    @ARTICLE{NS14, mrkey = {3456183},
      issn = {2050-5086},
      author = {Naor, Assaf and Schechtman, Gideon},
      mrclass = {46B80 (46B25 46B85 47B10)},
      doi = {10.1017/fmp.2016.1},
      journal = {Forum Math. Pi},
      zblnumber = {06539068},
      volume = {4},
      mrnumber = {3456183},
      fjournal = {Forum of Mathematics. Pi},
      title = {Metric {$X\sb p$} inequalities},
      year = {2016},
      pages = {e3, 81},
      }
  • [Ost13] Go to document M. I. Ostrovskii, Metric Embeddings, Berlin: De Gruyter, 2013.
    @BOOK{Ost13, mrkey = {3114782},
      number = {49},
      author = {Ostrovskii, Mikhail I.},
      mrclass = {46-01 (46-02 46B20 46B85)},
      series = {De Gruyter Stud. Math.},
      isbn = {978-3-11-026340-4; 978-3-11-026401-2},
      address = {Berlin},
      publisher = {De Gruyter},
      doi = {10.1515/9783110264012},
      zblnumber = {1279.46001},
      mrnumber = {3114782},
      note = {Bilipschitz and coarse embeddings into Banach spaces},
      mrreviewer = {Florent Baudier},
      title = {Metric Embeddings},
      year = {2013},
      pages = {xii+372},
      }
  • [Pal36] Go to document R. E. A. C. Paley, "Some theorems on abstract spaces," Bull. Amer. Math. Soc., vol. 42, iss. 4, pp. 235-240, 1936.
    @ARTICLE{Pal36, mrkey = {1563277},
      number = {4},
      issn = {0002-9904},
      author = {Paley, R. E. A. C.},
      mrclass = {Contributed Item},
      doi = {10.1090/S0002-9904-1936-06277-4},
      journal = {Bull. Amer. Math. Soc.},
      zblnumber = {62.0464.03},
      volume = {42},
      mrnumber = {1563277},
      fjournal = {Bulletin of the American Mathematical Society},
      title = {Some theorems on abstract spaces},
      year = {1936},
      pages = {235--240},
      }
  • [PZ30] Go to document R. ~E. A. ~C. Paley and A. Zygmund, "On some series of functions, (1)," Math. Proc. Cambridge Philos. Soc., vol. 26, pp. 337-357, 1930.
    @ARTICLE{PZ30, zblnumber = {56.0254.01},
      volume = {26},
      author = {Paley, R.~E. A.~C. and Zygmund, A.},
      title = { On some series of functions, (1)},
      doi = {10.1017/S0305004100016078},
      pages = {337--357},
      year = {1930},
      journal = {Math. Proc. Cambridge Philos. Soc.},
      }
  • [Pis82] Go to document G. Pisier, "Holomorphic semigroups and the geometry of Banach spaces," Ann. of Math., vol. 115, iss. 2, pp. 375-392, 1982.
    @ARTICLE{Pis82, mrkey = {0647811},
      number = {2},
      issn = {0003-486X},
      author = {Pisier, Gilles},
      mrclass = {46B20 (47D05)},
      doi = {10.2307/1971396},
      journal = {Ann. of Math.},
      zblnumber = {0487.46008},
      volume = {115},
      mrnumber = {0647811},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {T. Figiel},
      coden = {ANMAAH},
      title = {Holomorphic semigroups and the geometry of {B}anach spaces},
      year = {1982},
      pages = {375--392},
      }
  • [Pis88] Go to document G. Pisier, "Riesz transforms: a simpler analytic proof of P.-A. Meyer’s inequality," in Séminaire de Probabilités, XXII, New York: Springer-Verlag, 1988, vol. 1321, pp. 485-501.
    @INCOLLECTION{Pis88, mrkey = {960544},
      author = {Pisier, Gilles},
      mrclass = {60J35 (42B20)},
      series = {Lecture Notes in Math.},
      address = {New York},
      publisher = {Springer-Verlag},
      doi = {10.1007/BFb0084154},
      zblnumber = {0645.60061},
      volume = {1321},
      mrnumber = {960544},
      booktitle = {Séminaire de {P}robabilités, {XXII}},
      mrreviewer = {R. F. Gundy},
      title = {Riesz transforms: a simpler analytic proof of {P}.-{A}. {M}eyer's inequality},
      pages = {485--501},
      year = {1988},
      }
  • [Ros70] Go to document H. P. Rosenthal, "On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables," Israel J. Math., vol. 8, pp. 273-303, 1970.
    @ARTICLE{Ros70, mrkey = {0271721},
      issn = {0021-2172},
      author = {Rosenthal, Haskell P.},
      mrclass = {46.35},
      doi = {10.1007/BF02771562},
      journal = {Israel J. Math.},
      zblnumber = {0213.19303},
      volume = {8},
      mrnumber = {0271721},
      fjournal = {Israel Journal of Mathematics},
      mrreviewer = {A. Pietsch},
      title = {On the subspaces of {$L\sp{p}$} {$(p>2)$} spanned by sequences of independent random variables},
      year = {1970},
      pages = {273--303},
      }
  • [Sch38] Go to document I. J. Schoenberg, "Metric spaces and positive definite functions," Trans. Amer. Math. Soc., vol. 44, iss. 3, pp. 522-536, 1938.
    @ARTICLE{Sch38, mrkey = {1501980},
      number = {3},
      issn = {0002-9947},
      author = {Schoenberg, I. J.},
      mrclass = {42A82 (43A35 46B99 54E35)},
      doi = {10.2307/1989894},
      journal = {Trans. Amer. Math. Soc.},
      zblnumber = {64.0617.02},
      volume = {44},
      mrnumber = {1501980},
      fjournal = {Transactions of the American Mathematical Society},
      coden = {TAMTAM},
      title = {Metric spaces and positive definite functions},
      year = {1938},
      pages = {522--536},
      }
  • [Ste83] Go to document E. M. Stein, "Some results in harmonic analysis in ${\bf R}^{n}$, for $n\rightarrow \infty $," Bull. Amer. Math. Soc., vol. 9, iss. 1, pp. 71-73, 1983.
    @ARTICLE{Ste83, mrkey = {0699317},
      number = {1},
      issn = {0273-0979},
      author = {Stein, E. M.},
      mrclass = {42B25 (42B20)},
      doi = {10.1090/S0273-0979-1983-15157-1},
      journal = {Bull. Amer. Math. Soc.},
      volume = {9},
      mrnumber = {0699317},
      fjournal = {American Mathematical Society. Bulletin. New Series},
      coden = {BAMOAD},
      title = {Some results in harmonic analysis in {${\bf R}^{n}$},
      for {$n\rightarrow \infty $}},
      year = {1983},
      pages = {71--73},
      zblnumber = {0515.42018},
      }

Authors

Assaf Naor

Princeton University, Princeton, NJ