Singular integral operators on Triebel-Lizorkin spaces of para-accretive type
The remarkable \(T1\) theorem given by David-Journé provides a general criterion for the \(L^2\) boundedness of singular integral operators. There are many generalization of this theorem. \textit{G. David, J. L. Journé} and \textit{S. Semmes} [Rev. Mat. Iberoam. 1, No. 4, 1--56 (1985; Zbl 0604.42014)] gave \(Tb\) theorem for para-accretive functions \(b\). \textit{K. Wang} [Math. Nachr. 197, 103--133 (1999; Zbl 0933.42006)] considered \(Tb\) theorem on Triebel-Lizorkin spaces. \textit{C.-C. Lin} and \textit{K. Wang} [J. Geom. Anal. 19, No. 3, 667--694 (2009; Zbl 1169.42308)] obtained \(Tb\) theorem on Triebel-Lizorkin spaces of para-accretive type \(\dot{F}^{0,q}_{b,p} = b^{-1}\dot{F}^{0}_{p,q}\) where \(p \leq 1\) and \(q \leq 2\). The authors consider the cases \(1<p\) or \(q>2\). One of their results are the following. A standard singular integral operator \(T\) is bounded from \(\dot{F}^{0,q}_{1,p}\) to \(\dot{F}^{0,q}_{b,p}\) if one of following cases holds: (1) \(1 \leq q <2, 1<p< \infty, Tb \in \dot{F}^{0,q}_{1, \infty}\) and \(T^{*}b \in \bigcup_{0<\delta <1} \dot{F}^{\delta,\infty}_{b, n/\delta}\); (2) \(2 < q \leq \infty, 1<p< \infty, Tb \in \bigcup_{0<\delta <1} \dot{F}^{\delta,\infty}_{b, n/\delta}\) and \(T^{*}b \in \dot{F}^{0,q'}_{b, \infty}.\)
scientific article; zbMATH DE number 5673230
scientific article