JournalsrmiVol. 6, No. 1pp. 17–41

Para-Accretive Functions, the Weak Boundedness Property and the TbTb Theorem

  • Yongsheng Han

    Auburn University, USA
  • Eric T. Sawyer

    McMaster University, Hamilton, Canada
Para-Accretive Functions, the Weak Boundedness Property and the $Tb$ Theorem cover
Download PDF

Abstract

G. David, J.-L. Journé and S. Semmes have shown that if b1b_1 and b2b_2 are para-accretive functions on Rn\mathbb R^n, then the «TbTb Theorem» holds: A linear operator TT with Calderón-Zygmund kernel is bounded on L2L^2 if and only if Tb1BMO,Tb2BMOTb_1 \in \mathrm BMO, T*b_2 \in \mathrm {BMO} and Mb2TMb1M_{b_2} TM_{b_1} has the weak boundedness property. Conversely they showed that when b1=b2=bb_1 = b_2 = b, para-accretivity of bb is necessary for the TbTb Theorem to hold. In this paper we show that para-accretivity of both b1b_1 and b2b_2 is necessary for the TbTb Theorem to hold in general. In addition, we give a characterization of para-accretivity in terms of the weak boundedness property and use this to give a sharp TbTb Theorem for Besov and Triebel-Lizorkin spaces.

Cite this article

Yongsheng Han, Eric T. Sawyer, Para-Accretive Functions, the Weak Boundedness Property and the TbTb Theorem. Rev. Mat. Iberoam. 6 (1990), no. 1, pp. 17–41

DOI 10.4171/RMI/93