Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

By Long Huang, Jun Liu, Dachun Yang, Wen Yuan. Journal of Geometric Analysis, 2018

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Let $$\vec {a}:=(a_1,\ldots ,a_n)\in [1,\infty )^n$$a→:=(a1,…,an)∈[1,∞)n, $$\vec {p}:=(p_1,\ldots ,p_n)\in (0,\infty )^n$$p→:=(p1,…,pn)∈(0,∞)n and $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$Ha→p→(Rn) be the anisotropic mixed-norm Hardy space associated with $$\vec {a}$$a→ defined via the non-tangential grand maximal function. In this article, via first establishing a Calderón–Zygmund decomposition and a discrete Calderón reproducing formula, the authors then characterize $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$Ha→p→(Rn), respectively, by means of atoms, the Lusin area function, the Littlewood–Paley g-function or $$g_{\lambda }^*$$gλ∗-function. The obtained Littlewood–Paley g-function characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$Ha→p→(Rn) coincidentally confirms a conjecture proposed by Hart et al. (Trans Am Math Soc, https://doi.org/10.1090/tran/7312, 2017). Applying the aforementioned Calderón–Zygmund decomposition as well as the atomic characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$Ha→p→(Rn), the authors establish a finite atomic characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$Ha→p→(Rn), which further induces a criterion on the boundedness of sublinear operators from $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$Ha→p→(Rn) into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of anisotropic Calderón–Zygmund operators from $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$Ha→p→(Rn) to itself [or to the mixed-norm Lebesgue space $$L^{\vec {p}}(\mathbb {R}^n)$$Lp→(Rn)]. The obtained atomic characterizations of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$Ha→p→(Rn) and boundedness of anisotropic Calderón–Zygmund operators on these Hardy-type spaces positively answer two questions mentioned by Cleanthous et al. (J Geom Anal 27:2758–2787, 2017). All these results are new even for the isotropic mixed-norm Hardy spaces on $$\mathbb {R}^n$$Rn.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/577174d32a29cd534ed1bea759d3fca3f2f63821

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In Applied Mathematics

  1. Dachun Yang
  2. Wen Yuan
  3. Long Huang
  4. Yangyang Zhang
  5. Songbai Wang
  6. T. Nogayama
  7. Yoshihiro Sawano
  8. Jun Liu

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