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