A two weight local Tb theorem for the Hilbert transform

摘要

We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T-alpha on the real line R, and any pair of locally finite positive Borel measures (sigma, omega) on 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)* testing conditions over intervals Q, where the families {b(Q)} and {b(Q)*} 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.