Abstract
The purpose of this paper is to present a new atomic decomposition for a dense class of weighted Hardy spaces \(H_{w}^{p}(\mathbb R^n)\) via the discrete Calderón-type reproducing formula and the weighted Littlewood–Paley–Stein theory, where \(w\in A_\infty\) is a Muckenhoupt's weight and \(0<p<\infty\). Our results can recover and improve the known ones in the literature by avoiding using the maximal function characterization and the Calderón–Zygmund decomposition. Moreover, we give a new proof of the weighted Hardy spaces estimates for generalized Calderón–Zygmund operators in terms of the atomic decomposition and the vector-valued singular integral operator theory. Although the theory of \(H_{w}^{p}(\mathbb R^n)\) is well known, we give new and simpler proofs, which in turn is amenable to utilization in general and nonclassical settings.
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The author wishes to express his heartfelt thanks to the anonymous referee for the valuable suggestions that improved the paper significantly.
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This research was supported by National Natural Science Foundation of China (Grant No. 11901309), Natural Science Foundation of Jiangsu Province of China (Grant No. BK20180734), Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 18KJB110022).
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Tan, J. A revisit to the atomic decomposition of weighted Hardy spaces. Acta Math. Hungar. 168, 490–508 (2022). https://doi.org/10.1007/s10474-022-01289-0
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DOI: https://doi.org/10.1007/s10474-022-01289-0