Duality of multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators (Q2796087)

Let \(T_1\) and \(T_2\) be Calderón-Zygmund operators associated with two kinds of anisotropic homogeneities on \({\mathbb R}^N\), respectively: NEWLINE\[NEWLINE \delta\circ_1 (x_1, \ldots, x_m) = ( \delta^{a_1}x_1, \ldots \delta^{a_m}x_m) \;\text{and} \;\delta\circ_2 (x_1, \ldots, x_m) = ( \delta^{b_1}x_1, \ldots \delta^{b_m}x_m). NEWLINE\]NEWLINENEWLINENEWLINE\textit{D. H. Phong} and \textit{E. M. Stein} [Am. J. Math. 104, 141--172 (1982; Zbl 0526.35079)] gave a necessary and sufficient condition for the composition operator \(T_1 \circ T_2\) to be of weak-type \((1,1)\). \textit{Y. Han} et al. [Rev. Mat. Iberoam. 29, No. 4, 1127--1157 (2013; Zbl 1291.42018)] introduced a new Hardy space \(H^p_{hom}\) associated with the composition of these two different homogeneities and proved that \(T_1 \circ T_2\) is bounded on such spaces. \textit{W. Ding} [Acta Math. Sin., Engl. Ser. 30, No. 6, 933--948 (2014; Zbl 1433.42008)] developed the theory of Triebel-Lizorkin spaces \(F_p^{\alpha,q}\) associated with the composition of these two different homogeneities.NEWLINENEWLINEThe authors identify the dual space of \(F_p^{\alpha,q}\). They prove that if \(1<p<\infty\), then NEWLINE\[NEWLINE (F_p^{\alpha,q})^{*} = F_{p'}^{-\alpha,q'}, NEWLINE\]NEWLINE and if \(0<p \leq 1\), then NEWLINE\[NEWLINE (F_p^{\alpha,q})^{*} = CMO_p^{-\alpha, q'}. NEWLINE\]