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October, 2018 Variable Anisotropic Hardy Spaces and Their Applications
Jun Liu, Ferenc Weisz, Dachun Yang, Wen Yuan
Taiwanese J. Math. 22(5): 1173-1216 (October, 2018). DOI: 10.11650/tjm/171101


Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the variable anisotropic Hardy space $H_A^{p(\cdot)}(\mathbb{R}^n)$ associated with $A$, via the non-tangential grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $H_A^{p(\cdot)}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\lambda}^{\ast}$-function. As applications, the authors first establish a criterion on the boundedness of sublinear operators from $H^{p(\cdot)}_A(\mathbb{R}^n)$ into a quasi-Banach space. Then, applying this criterion, the authors show that the maximal operators of the Bochner-Riesz and the Weierstrass means are bounded from $H^{p(\cdot)}_A(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$ and, as consequences, some almost everywhere and norm convergences of these Bochner-Riesz and Weierstrass means are also obtained. These results on the Bochner-Riesz and the Weierstrass means are new even in the isotropic case.


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Jun Liu. Ferenc Weisz. Dachun Yang. Wen Yuan. "Variable Anisotropic Hardy Spaces and Their Applications." Taiwanese J. Math. 22 (5) 1173 - 1216, October, 2018.


Received: 5 August 2017; Accepted: 13 November 2017; Published: October, 2018
First available in Project Euclid: 29 November 2017

zbMATH: 06965415
MathSciNet: MR3859372
Digital Object Identifier: 10.11650/tjm/171101

Primary: 42B35
Secondary: 42B08 , 42B25 , 42B30 , 46E30

Keywords: (finite) atom , (variable) Hardy space , Bochner-Riesz means , expansive matrix , grand maximal function , Littlewood-Paley function , Weierstrass means

Rights: Copyright © 2018 The Mathematical Society of the Republic of China


Vol.22 • No. 5 • October, 2018
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