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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
Article

Abstract

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 \(\natural \).

Keywords

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.

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Copyright information

© Hebrew University Magnes Press 2013

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael

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