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^{\alpha} on the real line \mathbb{R} , and any pair of locally finite positive Borel measures (\sigma,\omega) on \mathbb{R} . The Hilbert transform is included in the case \alpha = 0 , and is bounded from L^{2}(\sigma) to L^{2}(\omega) if and only if the Muckenhoupt and energy conditions hold, as well as b_{Q} and b_{Q}^{\ast} testing conditions over intervals Q , where the families \{b_{Q}\} and \{b_{Q}^{\ast}\} are p -weakly accretive for some p > 2 . A number of new ideas are needed to accommodate weak goodness, including a new method for handling the stubborn nearby form, and an additional corona construction to deal with the stopping form. In a sense, this theorem improves the T1 theorem obtained by the authors and M. Lacey.