Abstract
Given a general dyadic grid D and a sparse family of cubes S = {Q k j ∈ D, define a dyadic positive operator A D,S by
. Given a Banach function space X(ℝn) and the maximal Calderón-Zygmund operator \({T_\natural }\), we show that
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 \(\natural \).
Similar content being viewed by others
References
C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988.
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.
D. Cruz-Uribe, J. M. Martell, C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Birkhäuser/Springer, Basel, AG, 2011.
D. Cruz-Uribe, J. M. Martell and C. Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2012), 408–441.
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].
N. Fujii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), 175–190.
T. P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), 1473–1506.
T. P. Hytönen, Representation of singular integrals by dyadic operators, and the A 2 theorem, arXiv: 1108.5119 [math. CA].
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].
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.
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].
T. P. Hytönen and C. Pérez, Sharp weighted bounds involving A ∞, Anal. PDE 6 (2013), 777–818.
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].
M. T. Lacey, An A p-A ∞ inequality for the Hilbert transform, Houston Math. J. 38 (2012), 799–814.
M. T. Lacey, On the A 2 inequality for Calderón-Zygmund operators, arXiv:1106.4802 [math. CA].
M. T. Lacey, E. T. Sawyer, and I. Uriarte-Tuero, Two weight inequalities for discrete positive operators, arXiv:0911.3437 [math. CA].
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.
A. K. Lerner, A simple proof of the A 2 conjecture, Int. Math. Res. Not. IMRN 2012. doi: 10.1093/imrn/rns145.
A. K. Lerner, Mixed Ap-Ar inequalities for classical singular integrals and Littlewood-Paley operators, J. Geom. Anal. 23 (2013), 1343–1354.
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.
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.
F. Nazarov, A. Reznikov, V. Vasuynin, and A. Volberg, A 1 conjecture: weak norm estimates of weighted singular operators and Bellman functions, preprint.
C. Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J. 43 (1994), 663–683.
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.
E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1–11.
S. Treil, Sharp A 2 estimates of Haar shifts via Bellman function, arXiv:1105.2252 [math. CA].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lerner, A.K. On an estimate of Calderón-Zygmund operators by dyadic positive operators. JAMA 121, 141–161 (2013). https://doi.org/10.1007/s11854-013-0030-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-013-0030-1