The duality about function set and Fefferman-Stein Decomposition

Let $Dinmathbb{N}$, $qin[2,infty)$ 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 $Dge2$, which needs some new methodology.