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Weak Musielak-Orlicz Hardy Spaces

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Real-Variable Theory of Musielak-Orlicz Hardy Spaces

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2182))

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Abstract

In this chapter, we introduce the weak Musielak-Orlicz Hardy space \(W\!H^{\varphi }(\mathbb{R}^{n})\) via the grand maximal function and then obtain its vertical or its non-tangential maximal function characterizations. We also establish other real-variable characterizations of \(W\!H^{\varphi }(\mathbb{R}^{n})\), respectively, by means of the atom, the molecule, the Lusin area function, the Littlewood-Paley g-function or the g λ ∗-function. As an application, the boundedness of Calderón-Zygmund operators from \(H^{\varphi }(\mathbb{R}^{n})\) to \(W\!H^{\varphi }(\mathbb{R}^{n})\) in the critical case is presented.

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Notes

  1. 1.

    See, for example, [69, p. 400, Theorem 2.8], [74, Theorem 9.1.9], [181, p. 5, Theorem 9] or [190, p. 233, Theorem 4.1].

  2. 2.

    See also [27, Lemma 7.13].

  3. 3.

    See, for example, [69, p. 400, Theorem 2.8], [74, Theorem 9.1.9], [181, p. 5, Theorem 9] or [190, p. 233, Theorem 4.1].

  4. 4.

    See, for example, [73, p. 463].

  5. 5.

    See, for example, [163, Theorem  3.17].

  6. 6.

    See, for example, [40, p. 624].

  7. 7.

    See, for example, [183, p. 82].

  8. 8.

    See, for example, [181, p. 87, Theorems 2 and 3] and [181, p. 5, Theorem  9].

  9. 9.

    See [61, p. 50, Theorem 1.64].

  10. 10.

    See, for example, [40, p. 624].

  11. 11.

    See, for example, [7, Theorem  3.1].

  12. 12.

    See, for example, [69, p. 400, Theorem 2.8], [74, Theorem 9.1.9], [181, p. 5, Theorem 9] or [190, p. 233, Theorem 4.1].

  13. 13.

    See, for example, [69, p. 400, Theorem 2.8], [74, Theorem 9.1.9] or [181, p. 5, Theorem 9]. or [190, p. 233, Theorem 4.1].

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, People’s Republic of China

    Dachun Yang

  2. Department of Mathematics, School of Sciences, Beijing Jiaotong University, Beijing, People’s Republic of China

    Yiyu Liang

  3. Department of Mathematics, University of Quy Nhon, Quy Nhon, Vietnam

    Luong Dang Ky

Authors
  1. Dachun Yang
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  2. Yiyu Liang
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  3. Luong Dang Ky
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Yang, D., Liang, Y., Ky, L.D. (2017). Weak Musielak-Orlicz Hardy Spaces. In: Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics, vol 2182. Springer, Cham. https://doi.org/10.1007/978-3-319-54361-1_7

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