A two weight local Tb theorem for the Hilbert transform

We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T on the real line, and any pair of locally finite positive Borel measures on the line. This includes the Hilbert transform and in a sense improves on the T1 theorem by the authors and M. Lacey.Comment: 121 pages, 3 figures, 50 pages of appendices. We correct three gaps in the treatment of the stopping form in v12: the proof of Lemma 9.3 there requires a larger size functional, a collection of pairs is missing from the decomposition at the bottom of page 149, and an error was made in the definition of restricted norm of a stopping form. Main results unchange