Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Abstract

Let $\vec A: = \left( {{A_1},{A_2}} \right)$ be a pair of expansive dilations and φ: ℝⁿ×ℝ^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space $H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$ via the anisotropic Lusin-area function and establish its atomic characterization, the $\vec g$ -function characterization, the $\vec g_\lambda ^*$-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of $H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$ is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet ($\left( {\varphi ,q,\vec s} \right)$), if T is a sublinear operator and maps all ($\left( {\varphi ,q,\vec s} \right)$)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from $H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$ to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from $H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$ to L ^φ(Rⁿ × R^m) and from $H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$ to itself, whose kernels are adapted to the action of $\vec A$. The results of this article essentially extend the existing results for weighted product Hardy spaces on ℝⁿ × ℝ^m and are new even for classical product Orlicz-Hardy spaces.