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.

中文翻译：

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函数集的对偶元素和Triebel-Lizorkin函数通过小波的费弗曼-斯坦分解

令\（D \ in \ mathbb {N} \），\（q \ in [2，\ infty} \）和\（（\ mathbb {R} ^ D，| \ cdot |，dx）\）为配备D维Lebesgue测度的欧几里德空间。在这篇文章中，我们建立TRIEBEL-Lizorkin空间的Fefferman斯坦分解\（\点{F} ^ 0 _ {\ infty，\ Q'}（\ mathbb {R} ^ d）\）用的帮助具有特殊拓扑结构的双重功能集。Triebel–Lizorkin空间\（\ dot {F} ^ 0 _ {\ infty，\，q'}（\ mathbb {R} ^ D）\）中的函数可以写为\（D + 1 \ ）在\（\ dot {F} ^ 0 _ {\ infty，\，q'}（\ mathbb {R} ^ D）\ cap L ^ {\ infty}（\ mathbb {R} ^ D）\）中起作用。为了得到这样的分解，首先，一些辅助函数空间\（\ mathrm {WE} ^ {1，\，q}（{\ mathbb {R}} ^ D）\）和\（\ mathrm {WE} ^ { \ infty，\，q'}（\ mathbb {R} ^ D）\）是通过小波展开定义的。结果表明$$ \ begin {aligned} {\ dot {F} ^ 0_ {1，\，q}（{\ mathbb {R}} ^ D）} \ subsetneqq L ^ {1}（{\ mathbb { R}} ^ D）\ cup {\ dot {F} ^ 0_ {1，\，q}（{\ mathbb {R}} ^ D）} \ subset \ mathrm {WE} ^ {1，\，q} （{\ mathbb {R}} ^ D）\子集L ^ {1}（{\ mathbb {R}} ^ D）+ {\ dot {F} ^ 0_ {1，\，q}（{\ mathbb { R}} ^ D）} \ end {aligned} $$和\（\ mathrm {WE} ^ {\ infty，\，q'}（\ mathbb {R} ^ D）\）严格包含在\（\点{F} ^ 0 _ {\ infty，\，q'}（\ mathbb {R} ^ D）\）。接下来，对Triebel–Lizorkin空间\（\ dot {F} ^ 0_ {1，\，q}（\ mathbb {R} ^ D）\）进行Riesz变换表征通过功能集\（\ mathrm {WE} ^ {1，\，q}（{\ mathbb {R}} ^ D）\）建立。然后考虑\（\ mathrm {WE} ^ {1，\，q}（{\ mathbb {R}} ^ D）\）的对偶。以上结果的结果是，Triebel–Lizorkin空间\（\ dot {F} ^ 0_ {1，\，q}（\ mathbb {R} ^ D）\）的Riesz变换特征由Banach空间\（L获得^ {1}（{\ mathbb {R}} ^ D）+ {\ dot {F} ^ 0_ {1，\，q}（{\ mathbb {R}} ^ D）} \\）。尽管当Lin等人获得\（D = 1 \）时Fefferman–Stein型分解。（Mich Math J 62：691–703，2013），正如Lin等人指出的那样，案例\（D = 1 \）中使用的方法无法应用于案例\（D \ ge 2 \）。在后一种情况下，将开发一些与Riesz变换有关的新技能。