Probability Theory and Related Fields

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

Interpolation and Φ-moment inequalities of noncommutative martingales



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.


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

Mathematics Subject Classification (2000)

46L53 46L52 60G42 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations