On $A_2$ conjecture and corona decomposition of weights

We consider here a problem of finding the sharp estimate for the boundedness of an arbitrary Calder'on-Zygmund operator in $L^2(w)$, $w\in A_2$. We first prove that for $A_2$ weight $w$ one has that the norm a Calderon–Zygmund operator $T$ in $L^2(w)$ is bounded by the sum of its weak norm, the weak norm of its adjoint, and the $A_2$ norm of the weight. From this result we derive that $\mathrm{\parallel }T{\mathrm{\parallel }}_{L^2(w)\to L^2(w)}\le C[w{]}_{A_2}\log (1+[w{]}_{A_2})$. We believe that the logarithmic factor is superflous. The approach is based on $2$-weight estimates technique and, hence, on non-homogeneous harmonic analysis.