On A₂ conjecture and corona decomposition of weights

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We consider here a problem of finding the sharp estimate for the boundedness of an arbitrary Calderón-Zygmund operator in L²(w), w∈ A₂. We first prove that for A₂ weight w one has that the norm a Calderon--Zygmund operator T in L²(w) is bounded by the sum of its weak norm, the weak norm of its adjoint, and the A₂ norm of the weight. From this result we derive that Tₗ₂₍w₎→ ₗ₂₍w₎ ≤ C[w]ₐ₂ (1+[w]ₐ₂). We believe that the logarithmic factor is superflous. The approach is based on 2-weight estimates technique and, hence, on non-homogeneous harmonic analysis.