Taiwanese Journal of Mathematics

Variable Anisotropic Hardy Spaces and Their Applications

Jun Liu, Ferenc Weisz, Dachun Yang, and Wen Yuan

Full-text: Open access

Abstract

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.

Article information

Source
Taiwanese J. Math., Volume 22, Number 5 (2018), 1173-1216.

Dates
Received: 5 August 2017
Accepted: 13 November 2017
First available in Project Euclid: 29 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1511924486

Digital Object Identifier
doi:10.11650/tjm/171101

Mathematical Reviews number (MathSciNet)
MR3859372

Zentralblatt MATH identifier
06965415

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 42B30: $H^p$-spaces 42B25: Maximal functions, Littlewood-Paley theory 42B08: Summability 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

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

Citation

Liu, Jun; Weisz, Ferenc; Yang, Dachun; Yuan, Wen. Variable Anisotropic Hardy Spaces and Their Applications. Taiwanese J. Math. 22 (2018), no. 5, 1173--1216. doi:10.11650/tjm/171101. https://projecteuclid.org/euclid.twjm/1511924486


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