Triebel-Lizorkin Spaces of Para-Accretive Type and a Tb Theorem

Abstract

In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type $\dot{F}^{\alpha,q}_{b,p}$ , which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a necessary and sufficient condition for the $\dot{F}^{0,q}_{1,p}-\dot{F}^{0,q}_{b,p}$ boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M _b TM _b ∈WBP is bounded from $\dot{F}^{0,q}_{1,p}$ to $\dot{F}^{0,q}_{b,p}$ if and only if $Tb\in \dot{F}^{0,q}_{b,\infty}$ and T ^* b=0 for $\frac{n}{n+\varepsilon}<p\le1$ and $\frac{n}{n+\varepsilon}<q\le 2$ , where ε is the regularity exponent of the kernel of T.