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

Abstract

In this article, we use a discrete Calderon-type reproducing formula and Plancherel-Polya-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 MbTMb∈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 e is the regularity exponent of the kernel of T.