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

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 (Formula presented.). Given a Banach function space X(ℝⁿ) and the maximal Calderón-Zygmund operator(Formula presented.) 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 "A₂ 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 T_{{music natural sign}}.