Tohoku Mathematical Journal
2020

June
SECOND SERIES VOL. 72, NO. 2

Tohoku Math. J.
72 (2020), 211-233

Title MOLECULAR CHARACTERIZATION OF ANISOTROPIC VARIABLE HARDY-LORENTZ SPACES

Author Xiong Liu, Xiaoli Qiu and Baode Li

(Received July 9, 2018, revised October 17, 2018)
Abstract. Let $A$ be an expansive dilation on $\mathbb{R}^n$, $q\in(0, \infty]$ and $p(\cdot):\mathbb{R}^n\rightarrow(0, \infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. Let $H^{p(\cdot), q}_A({\mathbb{R}}^n)$ be the anisotropic variable Hardy-Lorentz space defined via the radial grand maximal function. In this paper, the authors first establish its molecular characterization via the atomic characterization of $H^{p(\cdot), q}_A(\mathbb{R}^n)$. Then, as applications, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from $H^{p(\cdot), q}_{A}(\mathbb{R}^n)$ to $L^{p(\cdot), q}(\mathbb{R}^n)$ or from $H^{p(\cdot), q}_{A}({\mathbb{R}^n)$ to itself. All these results are still new even in the classical isotropic setting.

Mathematics Subject Classification. Primary 42B30; Secondary 42B20, 46E30.

Key words and phrases. Anisotropy, Hardy-Lorentz space, molecule, Calderón-Zygmund operator.

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