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We obtain a two weight local theorem for any elliptic and gradient elliptic fractional singular integral operator on the real line , and any pair of locally finite positive Borel measures on . The Hilbert transform is included in the case , and is bounded from to if and only if the Muckenhoupt and energy conditions hold, as well as and testing conditions over intervals , where the families and are -weakly accretive for some . 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 theorem obtained by the authors and M. Lacey.
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Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero, A two weight local theorem for the Hilbert transform. Rev. Mat. Iberoam. 37 (2021), no. 2, pp. 415–641DOI 10.4171/RMI/1209