43
Views
1
CrossRef citations to date
0
Altmetric
Research articles

Bourgain discretization using Lebesgue-Bochner spaces

&
Pages 611-621 | Received 17 Dec 2018, Published online: 17 Apr 2019

References

  • S.A. Argyros and R.G. Haydon, A hereditarily indecomposable 𝓛∞-space that solves the scalar-plus-compact problem, Acta Math. 206(1) (2011), 1–54. doi: 10.1007/s11511-011-0058-y
  • B. Begun, A remark on almost extensions of Lipschitz functions, Israel J. Math. 109 (1999), 151–155. doi: 10.1007/BF02775032
  • A. Benedek and R. Panzone, The space LP , with mixed norm, Duke Math. J. 28 (1961), 301–324. doi: 10.1215/S0012-7094-61-02828-9
  • C. Bessaga and A. Pe-lczyński, Banach spaces non-isomorphic to their Cartesian squares, I, Bull. Acad. Polon. Sci. Śer. Sci. Math. Astr. Phys. 8 (1960), 77–80.
  • A. Boccuto, A.V. Bukhvalov, and A.R. Sambucini, Some inequalities in classical spaces with mixed norms, Positivity 6(4) (2002), 393–411. doi: 10.1023/A:1021353215312
  • J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, In: Geometrical aspects of functional analysis, (1985/86), pp. 157–167, Lecture Notes in Math., Vol. 1267, Springer, Berlin, 1987.
  • N. Clavero and J. Soria, Optimal rearrangement invariant Sobolev embeddings in mixed norm spaces, J. Geom. Anal. 26(4) (2016), 2930–2954. doi: 10.1007/s12220-015-9655-x
  • A. Defant, D. Popa, and U. Schwarting, Coordinatewise multiple summing operators in Banach spaces, J. Funct. Anal. 259(1) (2010), 220–242. doi: 10.1016/j.jfa.2010.01.008
  • J. Diestel, An approach to the theory of Orlicz spaces of Lebesgue-Bochner measurable functions and to the theory of Orlicz spaces of finitely additive vector-valued set functions with applications to the representation of multilinear continuous operators, Ph.D. Thesis, The Catholic University of America, Washington, D.C., 1968.
  • J. Diestel and J.J. Uhl, Vector measures, With a foreword by B.J. Pettis, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977.
  • C. Fernández-González, C. Palazuelos, and D. Pérez-García, The natural rearrangement invariant structure on tensor products, J. Math. Anal. Appl. 343(1) (2008), 40–47. doi: 10.1016/j.jmaa.2008.01.016
  • T. Figiel, An example of infinite dimensional reflexive Banach space non-isomorphic to its Cartesian square, Studia Math. 42 (1972), 295–306. doi: 10.4064/sm-42-3-295-306
  • D.P. Giesy and R.C. James, Uniformly non-ℓ(1) and B-convex Banach spaces, Studia Math. 48 (1973), 61–69. doi: 10.4064/sm-48-1-61-69
  • O. Giladi, A. Naor, and G. Schechtman, Bourgain’s discretization theorem, Ann. Fac. Sci. Toulouse Math. (6) 21(4) (2012), 817–837; (See also a later correction in arXiv:1110.5368v2.) doi: 10.5802/afst.1352
  • G. Godefroy and N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159(1) (2003), 121–141. doi: 10.4064/sm159-1-6
  • W. Grey and G. Sinnamon, The inclusion problem for mixed norm spaces, Trans. Amer. Math. Soc. 368(12) (2016), 8715–8736. doi: 10.1090/tran6665
  • G.H. Hardy, J.E. Littlewood, and G. Pólya, Inequalities, Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.
  • S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73(3) (1982), 225–251. doi: 10.4064/sm-73-3-225-251
  • T. Hytönen, S. Li, and A. Naor, Quantitative affine approximation for UMD targets, Discrete Anal. 2016(6) (2016), 1–37.
  • T. Hytönen and A. Naor, Heat flow and quantitative differentiation, J. Eur. Math. Soc. (JEMS), to appear; arXiv:1608.01915.
  • R.C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518–527. doi: 10.2307/1969430
  • W.B. Johnson, B. Maurey, and G. Schechtman, Non-linear factorization of linear operators, Bull. Lond. Math. Soc. 41(4) (2009), 663–668. doi: 10.1112/blms/bdp040
  • H.E. Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York/Heidelberg, 1974.
  • S. Li and A. Naor, Discretization and affine approximation in high dimensions, Israel J. Math. 197(1) (2013), 107–129. doi: 10.1007/s11856-012-0182-1
  • J. Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, Vol. 212, Springer-Verlag, New York, 2002.
  • A. Naor, Metric dimension reduction: a snapshot of the Ribe program, Proc. Int. Cong. of Math. 2018, Rio de Janeiro, to appear; arXiv:1809.0237.
  • A. Naor and G. Schechtman, Planar Earthmover is not in L1, SIAM J. Comput. 37 (2007), 804–826. doi: 10.1137/05064206X
  • M.I. Ostrovskii, Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces, De Gruyter Studies in Mathematics, Vol 49, Walter de Gruyter & Co., Berlin, 2013.
  • Y. Raynaud, Finite representability of ℓp(X) in Orlicz function spaces, Israel J. Math. 65(2) (1989), 197–213. doi: 10.1007/BF02764860
  • C.J. Read, When E and E[E] are isomorphic, In: Geometry of Banach Spaces, Strobl, 1989, London Math. Soc. Lecture Note Ser., Vol. 158, pp. 245–252, Cambridge Univ. Press, Cambridge, 1990.
  • M. Ribe, On uniformly homeomorphic normed spaces, Ark. Mat. 14(2) (1976), 237–244. doi: 10.1007/BF02385837
  • Z. Semadeni, Banach spaces non-isomorphic to their Cartesian squares, II, Bull. Acad. Polon. Sci. Śer. Sci. Math. Astr. Phys. 8 (1960), 81–84.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.