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Finite Atomic Decomposition Characterization of Variable Anisotropic Hardy Spaces

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Abstract

In 2011, Dekel et al. developed highly geometric Hardy spaces Hp(Θ), for the full range 0 < p ≤ 1, which are constructed by continuous multi-level ellipsoid cover Θ of ℝn with high anisotropy in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. The authors obtain the finite atomic decomposition characterization of Hardy spaces Hp(Θ) and as an application, the authors prove that given an admissible triplet (p, q, l) with 1 ≤ q ≤ ∞, if T is a sublinear operator and uniformly bounded elements of some quasi-Banach space \({\cal B}\) for maps all (p, q, l)-atoms with q < ∞ (or all continuous (p, q, l)-atoms with q = ∞), then T uniquely extends to a bounded sublinear operator from Hp(Θ) to \({\cal B}\). These results generalize the known results on the anisotropic Hardy spaces of Bownik et al.

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Acknowledgements

The authors would like to express their deep thanks to the referees for their very careful reading and useful comments which do improve the presentation of this article.

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Authors and Affiliations

  1. College of Mathematics and System Science, Xinjiang University, Urumqi, 830046, P. R. China

    An Kang Yu, Ya Juan Yang & Bao De Li

  2. School of Mathematics and Statistics, Qinhai Nationalities University, Xining, 810007, P. R. China

    Ai Ting Wang

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  1. An Kang Yu
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  2. Ya Juan Yang
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Corresponding author

Correspondence to Bao De Li.

Additional information

Supported by the Xinjiang Training of Innovative Personnel Natural Science Foundation of China (Grant No. 2020D01C048) and the National Natural Science Foundation of China (Grant No. 11861062)

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Yu, A.K., Yang, Y.J., Li, B.D. et al. Finite Atomic Decomposition Characterization of Variable Anisotropic Hardy Spaces. Acta. Math. Sin.-English Ser. 38, 571–590 (2022). https://doi.org/10.1007/s10114-022-0648-y

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