The Dual Elements of Function Sets and Fefferman–Stein Decomposition of Triebel–Lizorkin Functions via Wavelets

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, we establish the Fefferman–Stein decomposition of Triebel–Lizorkin spaces $\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D)$ with the help of the dual on function sets which have special topological structure. A function in Triebel–Lizorkin spaces $\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D)$ can be written as a specific combination of $D+1$ functions in $\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D) \cap L^{\infty }(\mathbb {R}^D)$. To get such a decomposition, first, some auxiliary function spaces $\mathrm {WE}^{1,\,q}({\mathbb {R}}^D)$ and $\mathrm {WE}^{\infty ,\,q'}(\mathbb {R}^D)$ are defined via wavelet expansions. It is shown that

$$\begin{aligned} {\dot{F}^0_{1,\,q}({\mathbb {R}}^D)}\subsetneqq L^{1}({\mathbb {R}}^D) \cup {\dot{F}^0_{1,\,q}({\mathbb {R}}^D)}\subset \mathrm{WE}^{1,\,q}({\mathbb {R}}^D)\subset L^{1}({\mathbb {R}}^D) + {\dot{F}^0_{1,\,q}({\mathbb {R}}^D)}\end{aligned}$$

and $\mathrm {WE}^{\infty ,\,q'}(\mathbb {R}^D)$ is strictly contained in $\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D)$. Next, the Riesz transform characterization of Triebel–Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb {R}^D)$ by the function set $\mathrm {WE}^{1,\,q}({\mathbb {R}}^D)$ is established. Then the dual of $\mathrm {WE}^{1,\,q}({\mathbb {R}}^D)$ is considered. As a consequence of the above results, a Riesz transform characterization of Triebel–Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb {R}^D)$ by Banach space $L^{1}({\mathbb {R}}^D) + {\dot{F}^0_{1,\,q}({\mathbb {R}}^D)}$ is obtained. Although Fefferman–Stein type decompositions when $D=1$ was obtained by Lin et al. (Mich Math J 62:691–703, 2013), as was pointed out by Lin et al., the approach used in the case $D=1$ cannot be applied to the cases $D\ge 2$. In the latter cases, some new skills related to Riesz transforms are to be developed.