The duality about function set and Fefferman-Stein Decomposition

Let $D\in \mathbb{N}$, $q\in [2,\mathrm{\infty })$ and $({\mathbb{R}}^D,\mathrm{\mid }\cdot \mathrm{\mid },dx)$ 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 ${\dot{F}}_{\mathrm{\infty },q^{\mathrm{\prime }}}^0({\mathbb{R}}^D)$ on basis of the dual on function set which has special topological structure. The function in Triebel-Lizorkin spaces ${\dot{F}}_{\mathrm{\infty },q^{\mathrm{\prime }}}^0({\mathbb{R}}^D)$ can be written as the certain combination of $D+1$ functions in ${\dot{F}}_{\mathrm{\infty },q^{\mathrm{\prime }}}^0({\mathbb{R}}^D)\bigcap L^{\mathrm{\infty }}({\mathbb{R}}^D)$. To get such decomposition, {\bf (i),} The authors introduce some auxiliary function space ${\mathrm{W}\mathrm{E}}^{1,q}({\mathbb{R}}^D)$ and ${\mathrm{W}\mathrm{E}}^{\mathrm{\infty },q^{\mathrm{\prime }}}({\mathbb{R}}^D)$ 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$$\mathrm{W}\mathrm{E}$^{\infty,,q’}($\mathbb{R}$^D)$isstrictlycontainedin$\dot{F}^0{\infty,,q’}($\mathbb{R}$^D)$. {\bf (ii),} The authors establish the Riesz transform characterization of Triebel-Lizorkin spaces ${\dot{F}}_{1,q}^0({\mathbb{R}}^D)$ by function set $$\mathrm{W}\mathrm{E}$^{1,,q}(\mathbb R^D)$\mathrm{.}(\mathbf{i}\mathbf{i}\mathbf{i}),Wealsoconsiderthedualof$$\mathrm{W}\mathrm{E}$^{1,,q}(\mathbb R^D)$. As a consequence of the above results, the authors get also Riesz transform characterization of Triebel-Lizorkin spaces ${\dot{F}}_{1,q}^0({\mathbb{R}}^D)$ by Banach space $L^1+{\dot{F}}_{1,q}^0$. 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 $D\ge 2$, which needs some new methodology.