Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Weak type inequality for noncommutative differentially subordinated martingales
Download PDF
Download PDF
  • Published: 04 May 2007

Weak type inequality for noncommutative differentially subordinated martingales

  • Adam Osȩkowski1 

Probability Theory and Related Fields volume 140, pages 553–568 (2008)Cite this article

  • 113 Accesses

  • 6 Citations

  • Metrics details

Abstract

In the paper we focus on self-adjoint noncommutative martingales. We provide an extension of the notion of differential subordination, which is due to Burkholder in the commutative case. Then we show that there is a noncommutative analogue of the Burkholder method of proving martingale inequalities, which allows us to establish the weak type (1,1) inequality for differentially subordinated martingales. Moreover, a related sharp maximal weak type (1,1) inequality is proved.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Burkholder D.L. (1966). Martingale transforms. Ann. Math. Statist. 37: 1494–1504

    MathSciNet  Google Scholar 

  2. Burkholder, D.L.: Sharp inequalities formartingales and stochastic integrals.Astésque , 75–94 (1988)

  3. Fack T., Kosaki H. (1986). Generalized s-numbers of τ-measurable operators. Pacific J. Math. 123(2): 269–300

    MATH  MathSciNet  Google Scholar 

  4. Junge M. (2002). Doob’s inequality for noncommutative martingales. J. Reine Angew. Math 549: 149–190

    MATH  MathSciNet  Google Scholar 

  5. Junge M., Musat M. (2007). A noncommutative version of the John–Nirenberg Theorem. Trans. Am. Math. Soc. 359(1): 115–142

    Article  MATH  MathSciNet  Google Scholar 

  6. Junge M., Xu Q. (2003). Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31(2): 948–995

    Article  MATH  MathSciNet  Google Scholar 

  7. Musat M. (2003). Interpolation between noncommutative BMO and noncommutative L p spaces. J. Funct. Anal. 202: 195–225

    Article  MATH  MathSciNet  Google Scholar 

  8. Parcet, J., Randrianantoanina, N.: Gundy’s decomposition for noncommutative martingales and applications. Proc. Lond. Math. Soc. (3) 93(1), 227–252 (2006)

    Google Scholar 

  9. Pisier G., Xu Q. (1997). Noncommutative martingale inequalities. Commun. Math. Phys. 189: 667–698

    Article  MATH  MathSciNet  Google Scholar 

  10. Pisier, G., Xu, Q.: Noncommutative L p -spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces II. North Holland, Amsterdam, pp. 1459–1517 (2003)

  11. Randrianantoanina N. (2002). Noncommutative martingale transforms. J. Funct. Anal. 194: 181–212

    MATH  MathSciNet  Google Scholar 

  12. Randianantoanina N. (2004). Square function inequalities for noncommutative martingales. Israel J. Math. 140: 333–365

    MathSciNet  Google Scholar 

  13. Randrianantoanina N. (2005). A weak-type inequality for noncommutative martingales and applications. Proc. Lond. Math. Soc. 91(3): 509–544

    Article  MATH  MathSciNet  Google Scholar 

  14. Randrianantoanina, N.: Conditioned square functions for noncommutative martingales, Preprint

  15. Takesaki M. (1979). Theory of operator algebras I. Springer, New York

    MATH  Google Scholar 

  16. Xu, Q.: Recent development on noncommutative martingale inequalities. Functional Space Theory and its Applications. In: Proceedings of International Conference & 13th Academic Symposium in China, Wuhan, Research Information Ltd, UK, pp. 283–314 (2003)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Adam Osȩkowski

Authors
  1. Adam Osȩkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adam Osȩkowski.

Additional information

Research supported by MEN Grant 1 PO3A 012 29.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Osȩkowski, A. Weak type inequality for noncommutative differentially subordinated martingales. Probab. Theory Relat. Fields 140, 553–568 (2008). https://doi.org/10.1007/s00440-007-0075-0

Download citation

  • Received: 25 August 2006

  • Revised: 30 January 2007

  • Published: 04 May 2007

  • Issue Date: March 2008

  • DOI: https://doi.org/10.1007/s00440-007-0075-0

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Noncommutative probability space
  • Martingale
  • Weak type (1,1) inequality
  • Differentially subordinated martingales

Mathematics Subject Classification (2000)

  • Primary: 46L53
  • Secondary: 60G42
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature