Riesz Transform Characterization and Fefferman-Stein Decomposition of Triebel-Lizorkin Spaces

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, 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 $Dge2$, which needs some new skills.