Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Liu, Jun; Huang, Long; Yue, Chenlong

doi:10.3390/math9182216

Open AccessArticle

Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

by

Jun Liu

^1,*,

Long Huang

² and

Chenlong Yue

¹

School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China

²

School of Mathematics and Information Science, Key Laboratory of Mathematics and Interdisciplinary Sciences of the Guangdong Higher Education Institute, Guangzhou University, Guangzhou 510006, China

^*

Author to whom correspondence should be addressed.

Mathematics 2021, 9(18), 2216; https://doi.org/10.3390/math9182216

Submission received: 19 August 2021 / Revised: 4 September 2021 / Accepted: 6 September 2021 / Published: 9 September 2021

(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)

Download Versions Notes

Abstract

Let

\vec{p} \in {(0, \infty)}^{n}

be an exponent vector and A be a general expansive matrix on

R^{n}

. Let

H_{A}^{\vec{p}} (R^{n})

be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of

H_{A}^{\vec{p}} (R^{n})

, the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from

H_{A}^{\vec{p}} (R^{n})

to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on

H_{A}^{\vec{p}} (R^{n})

. In addition, the boundedness of anisotropic Calderón–Zygmund operators from

H_{A}^{\vec{p}} (R^{n})

to the mixed-norm Lebesgue space

L^{\vec{p}} (R^{n})

is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on

R^{n}

.

Keywords: expansive matrix; (mixed-norm) Hardy space; molecule; Calderón–Zygmund operator

Share and Cite

MDPI and ACS Style

Liu, J.; Huang, L.; Yue, C. Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications. Mathematics 2021, 9, 2216. https://doi.org/10.3390/math9182216

AMA Style

Liu J, Huang L, Yue C. Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications. Mathematics. 2021; 9(18):2216. https://doi.org/10.3390/math9182216

Chicago/Turabian Style

Liu, Jun, Long Huang, and Chenlong Yue. 2021. "Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications" Mathematics 9, no. 18: 2216. https://doi.org/10.3390/math9182216

APA Style

Liu, J., Huang, L., & Yue, C. (2021). Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications. Mathematics, 9(18), 2216. https://doi.org/10.3390/math9182216

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI