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Probability Theory and Related Fields

, Volume 152, Issue 1–2, pp 179–206 | Cite as

Interpolation and Φ-moment inequalities of noncommutative martingales

Article

Abstract

This paper is devoted to the study of Φ-moment inequalities for noncommutative martingales. In particular, we prove the noncommutative Φ-moment analogues of martingale transformations, Stein’s inequalities, Khintchine’s inequalities for Rademacher’s random variables, and Burkholder–Gundy’s inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper.

Keywords

τ-Measurable operators Noncommutative martingale Interpolation Φ-Moment martingale inequality Noncommutative Orlicz space 

Mathematics Subject Classification (2000)

46L53 46L52 60G42 

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References

  1. 1.
    Akemann C.A., Anderson J., Pedersen G.K.: Triangle inequalities in operator algebras. Linear Multilinear Algebra 11, 167–178 (1982)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Attal S., Coquio A.: Quantum stopping times and quasi-left continuity. Ann. Inst. H. Poincaré Probab. Stat. 40, 497–512 (2004)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bekjan T.N.: Φ-Inequalities of non-commutative martingales. Rocky Mt. J. Math. 36, 401–412 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bekjan T.N., Chen Z., Perrin M., Yin Z.: Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales. J. Funct. Anal. 258, 2483–2505 (2010)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bekjan T.N., Xu Q.: Riesz and Szegö type factorizations for noncommutative Hardy spaces. J. Oper. Theory 62(1), 101–117 (2009)MathSciNetGoogle Scholar
  6. 6.
    Blecher D.P., Labuschagne L.E.: Applications of the Fuglede-Kadison determinant: Szegö’s theorem and outers for noncommutative H p. Trans. Am. Math. Soc. 360(11), 6131–6147 (2008)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bourgain, J.: Vector valued singualr integrals and H1-BMO duality. In: Probability Theory and Harmonic Analysis, pp. 1–19. Dekker, New York (1986)Google Scholar
  8. 8.
    Burkholder D.L.: Distribution function inequalities for martingales. Ann. Probab. 1(1), 19–42 (1973)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Burkholder D.L.: A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9(6), 997–1011 (1981)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Burkholder, D.L., Davis, B., Gundy, R.: Integral inequalities for convex functions operators on martingales. In: Proc. 6th Berkley Symp. II, pp. 223–240 (1972)Google Scholar
  11. 11.
    Burkholder D.L., Gundy R.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124, 249–304 (1970)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Cuculescu I.: Martingales on von Neumann algebras. J. Multivar. Anal. 1, 17–27 (1971)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Dodds P.G., Dodds T.K., de Pager B.: Fully symmetric operator spaces. Integ. Equ. Oper. Theory 15, 942–972 (1992)CrossRefMATHGoogle Scholar
  14. 14.
    Fack T., Kosaki H.: Generalized s-numbers of τ-measure operators. Pac. J. Math. 123, 269–300 (1986)MATHMathSciNetGoogle Scholar
  15. 15.
    Garsia A.M.: On a convex function inequality for martingales. Ann. Probab. 1, 171–174 (1973)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Hu, Y.: Théorèmes ergodiques et théorèmes d’extrapolation non commutatifs. Thesis, Université de Franche-Comté (2007)Google Scholar
  17. 17.
    Hu Y.: Noncommutative extrapolation theorems and applications. Ill. J. Math. 53, 463–482 (2009)MATHGoogle Scholar
  18. 18.
    Junge M.: Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549, 149–190 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Junge M., Xu Q.: Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31, 948–995 (2003)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Junge M., Xu Q.: Noncommutative Burkholder/Rosenthal inequalities II: applications. Isr. J. Math. 167, 227–282 (2008)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Junge M., Xu Q.: On the best constants in some noncommutative martingale inequalities. Bull. Lond. Math. Soc. 37, 243–253 (2005)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Lust-Piquard F.: Inégalites de Khintchine dans c p  (1 < p < ∞). C. R. Acad. Sci. Paris 303, 289–292 (1986)MATHMathSciNetGoogle Scholar
  23. 23.
    Lust-Piquard F.: A Grothendieck factorization theorem on 2-convex Schatten spaces. Isr. J. Math. 79, 331–365 (1992)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Lust-Piquard F., Pisier G.: Noncommutative Khintchine and Paley inequalities. Arkiv för Mat. 29, 241–260 (1991)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Lust-Piquard F., Xu Q.: The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators. J. Funct. Anal. 244, 488–503 (2007)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Maligranda, L.: Indices and interpolation. Dissert. Math., vol. 234. Polska Akademia Nauk, Inst. Mat. (1985)Google Scholar
  27. 27.
    Maligranda, L.: Orlicz spaces and interpolation. In: Seminars in Mathematics, Departamento de Matemática, Universidade Estadual de Campinas, Brasil (1989)Google Scholar
  28. 28.
    Marsalli M., West G.: Noncommutative H p spaces. J. Oper. Theory 40, 339–355 (1997)MathSciNetGoogle Scholar
  29. 29.
    Le Merdy Ch., Sukochev F.: Rademacher averages on noncommutative symmetric spaces. J. Funct. Anal. 255, 3329–3355 (2008)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Orlicz W.: On a class of operators over the space of integrable functions. Studia Math. 14, 302–309 (1954)MathSciNetGoogle Scholar
  31. 31.
    Parcet J., Randrianantoanina N.: Gundy’s decomposition for noncommutative martingales and applications. Proc. Lond. Math. Soc. 93(3), 227–252 (2006)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Perrin M.: A noncommutative Davis’ decomposition for martingales. J. Lond. Math. Soc. (2) 80(3), 627–648 (2009)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Pisier, G.: Les inégalités de Khintchine-Kahane. Séminaire sur la Géométrie des Espaces de Banach (1977–1978), Exosé n°, Ec. Polytechnique, Palaiseau (1978)Google Scholar
  34. 34.
    Pisier G., Xu Q.: Non-commutative martingale inequalities. Commun. Math. Phys. 189, 667–698 (1997)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Pisier, G., Xu, Q.: Noncommutative L p-spaces. In: Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517 (2003)Google Scholar
  36. 36.
    Randrianantoanina N.: Non-commutative martingale transforms. J. Funct. Anal. 194, 181–212 (2002)MATHMathSciNetGoogle Scholar
  37. 37.
    Randrianantoanina N.: A weak-type inequality for non-commutative martingales and applications. Proc. Lond. Math. Soc. 91(3), 509–544 (2005)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Randrianantoanina N.: Conditional square functions for noncommutative martingales. Ann. Probab. 35, 1039–1070 (2007)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Xu Q.: Analytic functions with values in lattices and symmetric spaces of measurable operators. Math. Proc. Camb. Phil. Soc. 109, 541–563 (1991)CrossRefMATHGoogle Scholar
  40. 40.
    Xu, Q.: Noncommutative L p-Spaces and Martingale Inequalities. (2007, Book manuscript)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceXinjiang UniversityUrumqiChina
  2. 2.Wuhan Institute of Physics and MathematicsChinese Academy of SciencesWuhanChina

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