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

Abstract

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 ${\dot{F}}_{1,q}^0({\mathbb{R}}^D)$. As a consequence, the authors obtain the Fefferman-Stein decomposition of Triebel-Lizorkin spaces ${\dot{F}}_{\infty ,q^{\prime }}^0({\mathbb{R}}^D)$. Finally, the authors give an explicit example to show that ${\dot{F}}_{1,q}^0({\mathbb{R}}^D)$ is strictly contained in ${\mathrm{WE}}^{1,q}({\mathbb{R}}^D)$ and, by duality, ${\mathrm{WE}}^{\infty ,q^{\prime }}({\mathbb{R}}^D)$ is strictly contained in ${\dot{F}}_{\infty ,q^{\prime }}^0({\mathbb{R}}^D)$. 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 $D\ge 2$, which needs some new skills.

This paper has been withdrawn by the author due to existsing a gap in the proof of the main theorem