A reprise of the NTV conjecture for the Hilbert transform

We give a somewhat different organization of the proof of the NTV conjecture for the Hilbert transform that was proved by T. Hyt\"onen, M. Lacey, E.T. Sawyer, C.-Y. Shen and I. Uriarte-Tuero, building on previous work of F. Nazarov, S. Treil and A. Volberg. After modifying the decomposition of the main bilinear form, the proof alters the bottom-up corona construction, the size functional, the straddling lemmas, and the use of recursion of admissible collections of pairs of intervals, from M. Lacey [Lac]. However, the essence of control of the stopping form remains as in the fundamental work of M. Lacey [Lac]. The alternate approach of Sawyer and Wick [SaWi] to controlling functional energy is also discussed, which differs from that previously in the literature, [LaSaShUr3] and [Hyt2].