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

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

$${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$

. Given a Banach function space X(ℝn) 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 \(\natural \).

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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

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Keywords

  • Banach Function Space
  • Young Function
  • Dyadic Cube
  • Weighted Inequality
  • Zygmund Operator