Para-Accretive Functions, the Weak Boundedness Property and the $Tb$ Theorem

Abstract

G. David, J.-L. Journé and S. Semmes have shown that if $b_1$ and $b_2$ are para-accretive functions on $\mathbb R^n$, then the «$Tb$ Theorem» holds: A linear operator $T$ with Calderón-Zygmund kernel is bounded on $L^2$ if and only if $Tb_1 \in \mathrm BMO, T*b_2 \in \mathrm {BMO}$ and $M_{b_2} TM_{b_1}$ has the weak boundedness property. Conversely they showed that when $b_1 = b_2 = b$, para-accretivity of $b$ is necessary for the $Tb$ Theorem to hold. In this paper we show that para-accretivity of both $b_1$ and $b_2$ 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.