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Discrete Littlewood-Paley-Stein Characterization and L2 Atomic Decomposition of Local Hardy Spaces

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Abstract

Usually, the condition that T is bounded on L2(ℝn) is assumed to prove the boundedness of an operator T on a Hardy space. With this assumption, one only needs to prove the uniformly boundness of T on atoms, since T(f)= ∑i λiT(ai), provided that f = ∑i λiai in L2 (ℝn), where ai is an L2 atom of this Hardy space. So far, the L2 atomic decomposition of local Hardy spaces hp(ℝn), 0 > p ≤ 1, hasn’t been established. In this paper, we will solve this problem, and also show that hp(ℝn) can also be characterized by discrete Littlewood-Paley functions.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Sciences, Nantong University, Nantong, 226007, P. R. China

    Wei Ding

  2. Department of Mathematics, Nantong Normal College, Nantong, 226010, P. R. China

    Li Xin Jiang & Yue Ping Zhu

Authors
  1. Wei Ding
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  2. Li Xin Jiang
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  3. Yue Ping Zhu
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Corresponding authors

Correspondence to Wei Ding, Li Xin Jiang or Yue Ping Zhu.

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Supported by NNSF of China (Grant Nos. 11501308 and 11771223)

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Ding, W., Jiang, L.X. & Zhu, Y.P. Discrete Littlewood-Paley-Stein Characterization and L2 Atomic Decomposition of Local Hardy Spaces. Acta. Math. Sin.-English Ser. 35, 1681–1695 (2019). https://doi.org/10.1007/s10114-019-8532-0

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  • DOI: https://doi.org/10.1007/s10114-019-8532-0

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