A two weight local 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
Download PDF

A subscription is required to access this article.

Abstract

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.

Cite this article

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. 415–641

DOI 10.4171/RMI/1209