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
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.
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The authors would like to thank Dachun Yang and Xing Fu who contributed some useful suggestions to this study.
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Communicated by Doron Lubinsky.
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This project is supported by the National Natural Science Foundation of China (Grant no. 11571261), Macao Government FDCT099 and The Science and Technology Development Fund, Macau SAR (File no. 0123/2018/A3).
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Yang, Q., Qian, T. The Dual Elements of Function Sets and Fefferman–Stein Decomposition of Triebel–Lizorkin Functions via Wavelets. Comput. Methods Funct. Theory 20, 185–216 (2020). https://doi.org/10.1007/s40315-020-00309-w
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DOI: https://doi.org/10.1007/s40315-020-00309-w