Skip to main content
SpringerLink
Log in

A Bellman function proof of the L 2 bump conjecture

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We approach the problem of finding the sharp sufficient condition for boundedness of all two weight Calderón-Zygmund operators. We solve this problem in L 2 by writing a formula for a Bellman function of the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Cruz-Uribe, J. M. Martell, and C. Pérez, Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture, Adv. Math. 216 (2007), 647–676.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Cruz-Uribe, J. M. Martell, and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia, Birkhäuser/Springer Basel AG, Basel, 2011.

    Book  MATH  Google Scholar 

  3. D. Cruz-Uribe, J. M. Martell, and C. Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2012), 408–441.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Cruz-Uribe and C. Pérez, Sharp two-weight, weak-type norm inequalities for singular integral operators, Math. Res. Lett. 6(1989), 417–427.

    Article  Google Scholar 

  5. D. Cruz-Uribe and C. Pérez, Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators, Indiana Univ. Math. J. 49 (2000), 697–721.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Cruz-Uribe and C. Pérez, On the two-weight problem for singular integral operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), 821–849.

    MathSciNet  MATH  Google Scholar 

  7. D. Cruz-Uribe, A. Reznikov, and A. Volberg, Logarithmic bump conditions and the two-weight boundedness of Calderón-Zygmund operators, arXiv:1112. 0676v4, [math. AP].

  8. C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129–206.

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Hytönen,, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2), 175 (2012), 1473–1506.

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Hytönen, C. Pérez, S. Treil, and A. Volberg,, Sharp weighted estimates for dyadic shifts and the A 2 conjecture, J. Reine Angew. Math., to appear; arXiv:11010. 0755v2, [math. CA].

  11. P. Koosis, Moyennes quadratiques pondérées de fonctions périodiques et de leurs conjuguées harmoniques, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), A255–A257.

    MathSciNet  Google Scholar 

  12. A. Lerner, A pointwise estimate for local sharp maximal function with applications to singular integrals, Bull. London Math. Soc. 42 (2010), 843–856.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, J. Anal. Math. 121 (2013), 141–161.

    Article  MathSciNet  Google Scholar 

  14. C. Liaw and S. Treil, Regularizations of general singular integral operators, Rev. Mat. Iberoam. 29 (2013), 53–74.

    Article  MathSciNet  MATH  Google Scholar 

  15. F. Nazarov, A. Reznikov, S. Treil, and A. Volberg, The sharp bump condition for the two-weight problem for classical singular integral operator: the Bellman function approach, preprint, October, 2011.

  16. F. Nazarov, A. Reznikov, S. Treil, and A. Volberg, A solution of the bump conjecture for all Calderón-Zygmund operators: the Bellman function approach, sashavolberg. wordpress. com, Feb. 11, 2012.

  17. F. Nazarov, A. Reznikov, and A. Volberg, The proof of A 2 conjecture in a geometrically doubling metric space, Indiana Univ. Math. J. to appear. arXiv:1106. 1342v2 [math. CA].

  18. F. Nazarov, S. Treil, and A. Volberg, Bellman function and two-weight inequality for martingale transform, J. Amer. Math. Soc. 12 (1999), 909–928.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. J. Neugebauer, Inserting A p-weights, Proc. Amer. Math. Soc. 87 (1983), 644–648.

    MathSciNet  MATH  Google Scholar 

  20. C. Pérez, Weighted norm inequalities for singular integral operators, J. LondonMath. Soc.(2) 49 (1994), 296–308.

    Article  MATH  Google Scholar 

  21. C. Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp-spaces with different weights, Proc. LondonMath. Soc.(3) 71 (1995), 135–157.

    Article  MATH  Google Scholar 

  22. C. Pérez and R. Wheeden, Uncertainty principle estimates for vector fields, J. Funct. Anal. 181 (2001), 146–188.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. L. Rubio de Francia, Boundedness of maximal functions and singular integrals in weighted Lp spaces, Proc. Amer. Math. Soc. 83 (1981), 673–679.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Treil, Sharp A 2 estimates of Haar shifts via Bellman function, arXiv:1105. 2252v1 [math. CA].

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kent State University, Kent, OH, USA

    Fedor Nazarov

  2. Department of Mathematics, Michigan State University, East Lansing, MI, USA

    Alexander Reznikov & ALexander Volberg

  3. Department of Mathematics, Brown University, Providence, RI, USA

    Sergei Treil

Authors
  1. Fedor Nazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Reznikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergei Treil
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. ALexander Volberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work of F. Nazarov, S. Treil, and A. Volberg is supported by the National Science Foundation under the grant DMS-0758552, DMS-1265549.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nazarov, F., Reznikov, A., Treil, S. et al. A Bellman function proof of the L 2 bump conjecture. JAMA 121, 255–277 (2013). https://doi.org/10.1007/s11854-013-0035-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-013-0035-9

Keywords

Search

Navigation