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 $T{b}_1\in \mathrm{B}MO,T\ast b_2\in \mathrm{BMO}$ and $M_{b_2}T{M}_{b_1}$ 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.