A two weight local Tb theorem for the Hilbert transform

• Eric T. Sawyer ^[1] ; Chun-Yen Shen ^[2] ; Ignacio Uriarte-Tuero ^[3]
  1. [1] McMaster University

     McMaster University

     Canadá

     
  2. [2] National Taiwan University

     National Taiwan University

     Taiwán

     
  3. [3] Michigan State University

     Michigan State University

     City of East Lansing, Estados Unidos

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 37, Nº 2, 2021, págs. 415-641
• Idioma: inglés
• DOI: 10.4171/rmi/1209
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• Resumen

  • We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator Tα on the real line R, and any pair of locally finite positive Borel measures (σ,ω) on R. The Hilbert transform is included in the case α=0, and is bounded from L2(σ) to L2(ω) if and only if the Muckenhoupt and energy conditions hold, as well as bQ and b∗Q testing conditions over intervals Q, where the families {bQ} 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.