G. David, J.-L. Journé and S. Semmes have shown that if and are para-accretive functions on , then the « Theorem» holds: A linear operator with Calderón-Zygmund kernel is bounded on if and only if and has the weak boundedness property. Conversely they showed that when , para-accretivity of is necessary for the Theorem to hold. In this paper we show that para-accretivity of both and is necessary for the 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 Theorem for Besov and Triebel-Lizorkin spaces.
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Yongsheng Han, Eric T. Sawyer, Para-Accretive Functions, the Weak Boundedness Property and the Theorem. Rev. Mat. Iberoam. 6 (1990), no. 1, pp. 17–41DOI 10.4171/RMI/93