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Equivalent characterizations of Hardy spaces with variable exponent via wavelets

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Abstract

Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, ℋp(·)(ℝn), into Lebesgue spaces with variable exponent, Lp(·)(ℝn), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of ℋp(·)(ℝn) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of ℋp(·)(ℝn) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on Lp(·)(ℝn), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of ℋp(·)(ℝn) by a known idea.

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Acknowledgements

The author would like to express his deep thanks to the anonymous referees for their several enlightening comments on this article, which do improve the presentation of this article. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11701160, 11871100).

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  1. Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Xing Fu

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Fu, X. Equivalent characterizations of Hardy spaces with variable exponent via wavelets. Front. Math. China 14, 737–759 (2019). https://doi.org/10.1007/s11464-019-0777-5

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