The duality about function set and Fefferman-Stein Decomposition

Let $D\in\mathbb{N}$, $q\in[2,\infty)$ and $(\mathbb{R}^D,|\cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, the authors establish the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$ on basis of the dual on function set which has special topological structure. The function in Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$ can be written as the certain combination of $D+1$ functions in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D) \bigcap L^{\infty}(\mathbb{R}^D)$. To get such decomposition, {\bf (i),} The authors introduce some auxiliary function space $\mathrm{WE}^{1,\,q}(\mathbb R^D)$ and $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ defined via wavelet expansions. The authors proved $\dot{F}^{0}_{1,q} \subsetneqq L^{1} \bigcup \dot{F}^{0}_{1,q}\subset {\rm WE}^{1,\,q}\subset L^{1} + \dot{F}^{0}_{1,q}$ and $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ is strictly contained in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$. {\bf (ii),} The authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ by function set $\mathrm{WE}^{1,\,q}(\mathbb R^D)$. {\bf (iii),} We also consider the dual of $\mathrm{WE}^{1,\,q}(\mathbb R^D)$. As a consequence of the above results, the authors get also Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ by Banach space $L^{1} + \dot{F}^{0}_{1,q}$. Although Fefferman-Stein type decomposition when $D=1$ was obtained by C.-C. Lin et al. [Michigan Math. J. 62 (2013), 691-703], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the cases $D\ge2$, which needs some new methodology.