  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, via an auxiliary function space $\mathrm{WE}^{1,\,q}(\mathbb R^D)$
defined via wavelet expansions, the authors establish the Riesz transform
characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$.
As a consequence, the authors obtain the Fefferman-Stein decomposition of
Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$. Finally, the
authors give an explicit example to show that $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$
is strictly contained in $\mathrm{WE}^{1,\,q}(\mathbb{R}^D)$ and, by duality,
$\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ is strictly contained in
$\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$. Although all results when $D=1$ were
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 case $D\ge2$, which needs some new skills.
