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On Pisier’s inequality for UMD targets

Part of: Normed linear spaces and Banach spaces; Banach lattices Nontrigonometric harmonic analysis Stochastic processes

Published online by Cambridge University Press:  15 June 2020

Alexandros Eskenazis
Alexandros Eskenazis*
Affiliation:
Institut de Mathématiques de Jussieu, Sorbonne Université, 4, Place Jussieu, 75252 Paris Cedex 05, France
*
e-mail: alexandros.eskenazis@imj-prg.fr
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Abstract

We prove an extension of Pisier’s inequality (1986) with a dimension-independent constant for vector-valued functions whose target spaces satisfy a relaxation of the UMD property.

Keywords

Pisier’s inequalityBanach space valued martingaleUMD Banach space

MSC classification

Primary: 46B07: Local theory of Banach spaces
Secondary: 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 60G46: Martingales and classical analysis
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 282 - 291
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author was supported by a postdoctoral fellowship of the Fondation Sciences Mathématiques de Paris.

References

Bourgain, J., Vector-valued singular integrals and the ${H}^1$ -BMO duality. In: Probability theory and harmonic analysis (Cleveland, OH, 1983), Monogr. Textbooks Pure Appl. Math., 98, Dekker, New York, 1986, pp. 1–19.Google Scholar
Eskenazis, A., Geometric inequalities and advances in the ribe program. Ph.D. thesis, Princeton University, 2019.Google Scholar
Garling, D. J. H., Random martingale transform inequalities. In: Probability in Banach spaces 6 (Sandbjerg, 1986), Progr. Probab., 20, Birkhäuser Boston, Boston, MA, 1990, pp. 101–119.CrossRefGoogle Scholar
Hytönen, T. and Naor, A., Pisier’s inequality revisited. Studia Math. 215(2013), 221–235.CrossRefGoogle Scholar
Ivanisvili, P., van Handel, R., and Volberg, A., Rademacher type and Enflo type coincide. Ann. of Math. Preprint, 2020. https://arxiv.org/abs/2003.06345, 2020.Google Scholar
Khintchine, A., Über dyadische Brüche. Math. Z. 18(1923), 109–116.CrossRefGoogle Scholar
Naor, A. and Schechtman, G., Remarks on non linear type and Pisier’s inequality. J. Reine Angew. Math. 552(2002), 213–236.Google Scholar
Pisier, G., Sur les espaces de Banach qui ne contiennent pas uniformément de ${l}_n^1$ . C. R. Acad. Sci. Paris Sér. A-B 277(1973), A991–A994.Google Scholar
Pisier, G., Holomorphic semigroups and the geometry of Banach spaces. Ann. of Math. (2) 115(1982), 375–392.CrossRefGoogle Scholar
Pisier, G., Probabilistic methods in the geometry of Banach spaces. In: Probability and analysis (Varenna, 1985), Lecture Notes in Math., 1206, Springer, Berlin, 1986, pp. 167–241.CrossRefGoogle Scholar
Pisier, G., Martingales in Banach spaces. Cambridge Studies in Advanced Mathematics, 155, Cambridge University Press, Cambridge, 2016.Google Scholar
Qiu, Y., On the UMD constants for a class of iterated ${L}_p\left({L}_q\right)$ spaces. J. Funct. Anal. 263(2012), 2409–2429.CrossRefGoogle Scholar
Stein, E. M., Topics in harmonic analysis related to the Littlewood-Paley theory. Ann. Math. Studies, 60 (Princeton University Press, Princeton, N.J), University of Tokyo Press, Tokyo, 1970.Google Scholar
Talagrand, M., Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis’ graph connectivity theorem. Geom. Funct. Anal. 1993(3), 295–314.Google Scholar