Summability in anisotropic mixed-norm Hardy spaces

Nan Li; Nan Li

doi:10.3934/era.2022171

Electronic Research Archive

2022, Volume 30, Issue 9: 3362-3376. doi: 10.3934/era.2022171

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Summability in anisotropic mixed-norm Hardy spaces

Nan Li ^,

Graduate School, Beijing Sport University, Beijing 100084, China

Academic Editor: Arran Fernandez

Received: 14 May 2022 Revised: 01 July 2022 Accepted: 08 July 2022 Published: 15 July 2022

Let $ H_A^{\vec{p}}(\mathbb{R}^n) $ be the anisotropic mixed-norm Hardy space, where $ \vec{p}\in(0, \infty)^n $ and $ A $ is a general expansive matrix on $ \mathbb{R}^n $. In this paper, a general summability method, the so-called $ \theta $-summability is considered for multi-dimensional Fourier transforms in $ H_A^{\vec{p}}(\mathbb{R}^n) $. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $ \theta $-means, from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. As applications, some norm and almost everywhere convergence results of the $ \theta $-means are presented. Finally, the corresponding conclusions of two well-known specific summability methods, namely, Bochner–Riesz and Weierstrass means, are also obtained.
- mixed-norm Hardy space,
- expansive matrix,
- $ \theta $-summability,
- maximal operator
Citation: Nan Li. Summability in anisotropic mixed-norm Hardy spaces[J]. Electronic Research Archive, 2022, 30(9): 3362-3376. doi: 10.3934/era.2022171

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Abstract

Let $ H_A^{\vec{p}}(\mathbb{R}^n) $ be the anisotropic mixed-norm Hardy space, where $ \vec{p}\in(0, \infty)^n $ and $ A $ is a general expansive matrix on $ \mathbb{R}^n $. In this paper, a general summability method, the so-called $ \theta $-summability is considered for multi-dimensional Fourier transforms in $ H_A^{\vec{p}}(\mathbb{R}^n) $. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $ \theta $-means, from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. As applications, some norm and almost everywhere convergence results of the $ \theta $-means are presented. Finally, the corresponding conclusions of two well-known specific summability methods, namely, Bochner–Riesz and Weierstrass means, are also obtained.

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