Abstract
G. David, J.-L. Journe and S. Semmes have shown that if b1 and b2 are para-accretive functions on Rn, then the Tb theorem holds: A linear operator T with Calderon-Zygmund kernel is bounded on L2 if and only if Tb1 I BMO, T*b2 I BMO and Mb2TMb1 has the weak boundedness property. Conversely they showed that when b1 = b2 = b, para-accretivity of b is necessary for Tb Theorem to hold. In this paper we show that para-accretivity of both b1 and b2 is necessary for the Tb 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 Tb Theorem for Besov and Triebel-Lizorkin spaces.
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