LITTLEWOOD-PALEY CHARACTERIZATIONS OF ANISOTROPIC HARDY SPACES OF MUSIELAK-ORLICZ TYPE

Abstract

Let $\varphi : \mathbb{R}^n\times [0,\,\infty)\to[0,\infty)$ be a Musielak-Orlicz function and $A$ an expansive dilation. Let $H^\varphi_A({\mathbb {R}}^n)$ be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. Its atomic characterization and some other maximal function characterizations of $H^\varphi_A({\mathbb {R}^n})$, in terms of the radial, the non-tangential and the tangential maximal functions, are known. In this article, the authors further obtain their characterizations in terms of the Lusin-area function, the $g$-function or the $g^\ast_\lambda$-function via first establishing an anisotropic Peetre's inequality of Musielak-Orlicz type. Moreover, the range of $\lambda$ in the $g^\ast_\lambda$-function characterization of $H^\varphi_A({\mathbb {R}}^n)$ coincides with the known best conclusions in the case when $H^\varphi_A({\mathbb {R}}^n)$ is the classical Hardy space $H^p({\mathbb{R}}^n)$ or the anisotropic Hardy space $H^p_A({\mathbb {R}}^n)$ or their weighted variants, where $p\in(0,\,1]$.