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2024 Summability in Anisotropic Musielak–Orlicz Hardy Spaces
Jiashuai Ruan
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Taiwanese J. Math. Advance Publication 1-16 (2024). DOI: 10.11650/tjm/240503


Let $\varphi \colon \mathbb{R}^{n} \times [0,\infty) \to [0,\infty)$ be an anisotropic growth function and $A$ a general expansive matrix on $\mathbb{R}^{n}$. Let $H_{A}^{\varphi}(\mathbb{R}^{n})$ be the anisotropic Musielak–Orlicz Hardy space associated with $A$. In this paper, a general summability method, the so-called $\theta$-summability is considered for multi-dimensional Fourier transforms in $H_{A}^{\varphi}(\mathbb{R}^{n})$. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $\theta$-means, from $H_{A}^{\varphi}(\mathbb{R}^{n})$ to the Musielak–Orlicz space $L^{\varphi}(\mathbb{R}^{n})$. As applications, some norm and almost everywhere convergence results of the $\theta$-means, which generalize the well-known Lebesgue's theorem, are presented. Finally, the corresponding conclusions of two well-known specific summability methods, that is, Bochner–Riesz and Weierstrass means, are also obtained.


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Jiashuai Ruan. "Summability in Anisotropic Musielak–Orlicz Hardy Spaces." Taiwanese J. Math. Advance Publication 1 - 16, 2024.


Published: 2024
First available in Project Euclid: 30 May 2024

Digital Object Identifier: 10.11650/tjm/240503

Primary: 42B08
Secondary: 42A24 , 42B25 , 42B30

Keywords: anisotropic Musielak–Orlicz Hardy space , Bochner–Riesz summation , Maximal operator , summability , Weierstrass summation

Rights: Copyright © 2024 The Mathematical Society of the Republic of China

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