JournalsrmiVol. 37, No. 2pp. 415–641

A two weight local TbTb theorem for the Hilbert transform

  • Eric T. Sawyer

    McMaster University, Hamilton, Canada
  • Chun-Yen Shen

    National Taiwan University, Taipei, Taiwan
  • Ignacio Uriarte-Tuero

    Michigan State University, East Lansing, USA
A two weight local $Tb$ theorem for the Hilbert transform cover

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Abstract

We obtain a two weight local TbTb theorem for any elliptic and gradient elliptic fractional singular integral operator TαT^{\alpha} on the real line R\mathbb{R}, and any pair of locally finite positive Borel measures (σ,ω)(\sigma,\omega) on R\mathbb{R}. The Hilbert transform is included in the case α=0\alpha = 0, and is bounded from L2(σ)L^{2}(\sigma) to L2(ω)L^{2}(\omega) if and only if the Muckenhoupt and energy conditions hold, as well as bQb_{Q} and bQb_{Q}^{\ast} testing conditions over intervals QQ, where the families {bQ}\{b_{Q}\} and {bQ}\{b_{Q}^{\ast}\} are pp-weakly accretive for some p>2p > 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 T1T1 theorem obtained by the authors and M. Lacey.

Cite this article

Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero, A two weight local TbTb theorem for the Hilbert transform. Rev. Mat. Iberoam. 37 (2020), no. 2, pp. 415–641

DOI 10.4171/RMI/1209