Journal d'Analyse Mathématique

, Volume 121, Issue 1, pp 141–161 | Cite as

On an estimate of Calderón-Zygmund operators by dyadic positive operators

  • Andrei K. LernerEmail author


Given a general dyadic grid D and a sparse family of cubes S = {Q j k D, define a dyadic positive operator A D,S by
$${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$
. Given a Banach function space X(ℝ n ) and the maximal Calderón-Zygmund operator \({T_\natural }\), we show that
$${\left\| {{T_\natural}f} \right\|_X} \leqslant c(T,n)\mathop {\sup }\limits_{D,S} {\left\| {{A_{D,S}}|f|} \right\|_X}$$
This result is applied to weighted inequalities. In particular, it implies (i) the “twoweight conjecture” by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the “A 2 conjecture”; (iii) an extension of certain mixed A p A r estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A 1 estimates (known for T ) to the maximal Calderón-Zygmund operator \(\natural \).


Banach Function Space Young Function Dyadic Cube Weighted Inequality Zygmund Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988.zbMATHGoogle Scholar
  2. [2]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    D. Cruz-Uribe, J. M. Martell, C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Birkhäuser/Springer, Basel, AG, 2011.CrossRefzbMATHGoogle Scholar
  4. [4]
    D. Cruz-Uribe, J. M. Martell and C. Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2012), 408–441.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Cruz-Uribe, A. Reznikov and A. Volberg, Logarithmic bump conditions and the two-weight boundedness of Calderón-Zygmund operators, arXiv: 1112.0676 [math. AP].Google Scholar
  6. [6]
    N. Fujii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), 175–190.MathSciNetzbMATHGoogle Scholar
  7. [7]
    T. P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), 1473–1506.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. P. Hytönen, Representation of singular integrals by dyadic operators, and the A 2 theorem, arXiv: 1108.5119 [math. CA].Google Scholar
  9. [9]
    T. P. Hytönen and M. Lacey, The A p-A inequality for general Calderón-Zygmund operators, Indiana Univ. Math. J., to appear; arXiv: 1108.4797 [math. CA].Google Scholar
  10. [10]
    T. P. Hytönen, M. T. Lacey, H. Martikainen, T. Orponen, M. C. Reguera, E. T. Sawyer, and I. Uriarte-Tuero, Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on Ap weighted spaces, J. Anal. Math. 118 (2012), 177–220.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    T. P. Hytönen, M. T. Lacey, and C. Pérez, Non-probabilistic proof of the A 2 theorem, and sharp weighted bounds for the q-variation of singular integrals, arXiv: 1202.2229 [math. CA].Google Scholar
  12. [12]
    T. P. Hytönen and C. Pérez, Sharp weighted bounds involving A , Anal. PDE 6 (2013), 777–818.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    T. P. 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: 1010.0755 [math. CA].Google Scholar
  14. [14]
    M. T. Lacey, An A p-A inequality for the Hilbert transform, Houston Math. J. 38 (2012), 799–814.MathSciNetzbMATHGoogle Scholar
  15. [15]
    M. T. Lacey, On the A 2 inequality for Calderón-Zygmund operators, arXiv:1106.4802 [math. CA].Google Scholar
  16. [16]
    M. T. Lacey, E. T. Sawyer, and I. Uriarte-Tuero, Two weight inequalities for discrete positive operators, arXiv:0911.3437 [math. CA].Google Scholar
  17. [17]
    A. K. Lerner, A pointwise estimate for the local sharp maximal function with applications to singular integrals, Bull. London Math. Soc. 42 (2010), 843–856.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. K. Lerner, A simple proof of the A 2 conjecture, Int. Math. Res. Not. IMRN 2012. doi: 10.1093/imrn/rns145.Google Scholar
  19. [19]
    A. K. Lerner, Mixed Ap-Ar inequalities for classical singular integrals and Littlewood-Paley operators, J. Geom. Anal. 23 (2013), 1343–1354.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    A. K. Lerner, S. Ombrosi, and C. Pérez, A 1 bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden, Math. Res. Lett. 16 (2009), 149–156.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    F. Nazarov, A. Reznikov, S. Treil, and A. Volberg, A Bellman function proof of the L 2 conjecture, J. Anal. Math. 121 (2013), 255–277.CrossRefMathSciNetGoogle Scholar
  22. [22]
    F. Nazarov, A. Reznikov, V. Vasuynin, and A. Volberg, A 1 conjecture: weak norm estimates of weighted singular operators and Bellman functions, preprint.Google Scholar
  23. [23]
    C. Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J. 43 (1994), 663–683.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    C. Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted L p-spaces with different weights, Proc. LondonMath. Soc.(3) 71 (1995), 135–157.CrossRefzbMATHGoogle Scholar
  25. [25]
    E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1–11.MathSciNetzbMATHGoogle Scholar
  26. [26]
    S. Treil, Sharp A 2 estimates of Haar shifts via Bellman function, arXiv:1105.2252 [math. CA].Google Scholar

Copyright information

© Hebrew University Magnes Press 2013

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael

Personalised recommendations